Rapid variable-step computation of dynamic convolutions and Volterra-type integro-differential equations: RK45 Fehlberg, RK4

Abstract

We introduce a novel, time-efficient adaptive Runge-Kutta computational scheme tailored for systematically solving linear and nonlinear Volterra-type Integro-Differential Equations (VTIDEs). This scheme is particularly effective for equations featuring a specific class of convolution memory, $\mathcal{K}(t,s)=a^{\kappa (t-s)}$ within convolution integrals, where $a,\kappa \in \mathbb{R}$. Such equations frequently arise in fields like viscoelasticity, population dynamics, epidemiology, control systems, financial mathematics, and neuroscience, where current states are influenced by historical data, especially in dynamical systems within physics. Our proposed scheme demonstrates remarkable computational efficiency, achieving a cost-effective computation time of $\mathcal{O}(N_t)$ for $N_t$ iterations, a significant improvement over existing schemes for VTIDE, which typically exhibit complexities of $\mathcal{O}(N_t\log N_t)$ and $\mathcal{O}{(N_t(\log N_t))}^2$. The key feature of our scheme lies in its ability to elegantly and efficiently handle convoluted integrals of varying complexity such as those including nonlinearities and derivatives. Specifically, our approach eliminates the need for backstage integration, allowing the main Runge-Kutta scheme to operate efficiently with only a few data points at each time step. This not only saves time but also prevents potential data storage issues while rendering numerical implementation straightforward and tractable. Moreover, this article introduces flexible discretizations for integration and differentiation, accommodating unequally spaced data points. We implement a fast RK45-Fehlberg algorithm (TE-RK45-Fehlberg), which we use to solve three examples of nonlinear VTIDE presented in this study. We further demonstrate the effectiveness of our approach by using an example, where we employ a novel parabolic step size generator function to efficiently generate step sizes for a computationally fast RK4 scheme (Flex-RK4). For all examples tackled using TE-RK45-Fehlberg and Flex-RK4, we generate plots that compare numerical data with exact solutions, providing insight into the accuracy and reliability of our approach. We also show how the scheme provides straightforward ways of transforming it into implicit forms and obtaining stability curves.

1. Introduction

The physics underlying some dynamical systems are typically represented by equations that combine integration and differentiation (e.g., [1], [2], [3]). These equations often necessitate computational schemes to process the data and resolve complex problems. Computational time in such schemes can be a significant issue, sometimes requiring days of runtime, even on supercomputers. Integro-differential equations (IDE), in particular, present both analytical and numerical challenges, hence finding closed-form solutions to IDEs is generally difficult due to the combined presence of differential and integral terms.

One of the most complex and widely applicable integro-differential equations is the Volterra Integro-Differential Equation (VIDE). VIDEs involve both differentiation and integration, where the integration is typically a convolution. A convolution integral is a mathematical operation that measures the degree of overlap between two functions as one is shifted over the other. It effectively combines these functions to generate a third function that characterizes the modification of one function's shape by the other. This process is fundamental in systems analysis, offering crucial insights into the behavior of various systems. For example, the convolution of two functions $f(t)$ and $g(t)$ is defined by the integral:

$$(f⁎g)(t)=\underset{0}{\overset{t}{\int }}f(\tau )g(t-\tau )d\tau$$ (1)

in practical applications, the limits of integration are often finite.

These equations combine convolution integral and differential equations into a single formulation, making them versatile tools in various scientific and engineering disciplines, thus VIDEs are used to model systems where the future state depends on both current and past states.

For example, VIDEs are employed in population dynamics to model growth rates that depend on the current population size and the cumulative effect of past populations [4]. They are also used in epidemiology to model the spread of infectious diseases, where the rate of new infections at any time depends on the history of the infection rate [5]. In viscoelasticity and hereditary mechanics, VIDEs model materials where stress depends on both the current strain and the history of strain [6], [7], [8].

In control systems with memory or delay, VIDEs are used to describe scenarios where the current output depends on the integral of past outputs [9]. In finance, VIDEs model the pricing of certain financial derivatives where the option price depends on the current price and the integral of past prices [10]. Furthermore, VIDEs describe the activity of neurons where the membrane potential depends on the current input and the cumulative effect of past inputs [11].

In environmental modeling, VIDEs are important for processes like pollutant dispersion, where the current concentration depends on historical emissions [12]. They are also used in economic models to describe how current economic indicators depend on their historical values [13].

Following this, we take advantage of a property in some convolution integrals to construct a fast numerical algorithm using adaptive Runge-Kutta (RK) methods with an elegant implementation that avoids storage constraints. Moreover, using Newton Cote's formulation, the scheme redefines the discretization of numerical differentiation and integration that elegantly accommodates data points of different step intervals. Thus, for an m-stage RK scheme (e.g. $m=2$ for second-order RK), the resultant computational time generated here for $N_t$ evaluations reduces substantially to $\mathcal{O}(N_t)$ in contrast to $\mathcal{O}(N_t\log N_t)$ and $\mathcal{O}{(N_t(\log N_t))}^2$ for other schemes.

Therefore, the objective of the scheme is to efficiently solve linear and non-linear integro-differential equations characterized by a special kernel, as defined in Eq. (2):

$$\mathcal{K}(t,s)=a^{\kappa (t-s)},a,\kappa \in \mathbb{R},$$ (2)

where a and κ are real constants and $t\mathrm{\&}s$ are the independent variable and dummy variable respectively. Utilizing a distinctive property of Eq. (2), the convolution operation is systematically divided, facilitating the development of a rapid numerical scheme (refer to Sec. 4). This scheme optimally exploits pre-calculated convolution integrals at each time step, resulting in a significant reduction in computational time (see Sec. 5.3.1), in contrast to re-computing integrals from scratch at every step (e.g. [14], [15], [7]). The presence of a kernel such as Eq. (2) exemplifies the elegance of the scheme, particularly in cases featuring a trivial unity kernel. Notably, some studies (e.g. [14], [15], [16]) have overlooked the strategy of splitting the convolution, thereby missing the opportunity to circumvent lengthy integrations, even for the special case when $\mathcal{K}(t,s)=1$, leading to less cost-effective schemes.

Furthermore, several other researchers have proposed interesting alternative methods for solving VIDEs with kernels similar to those in Eq. (2). However, some of these investigators have opted for a unity kernel, thereby avoiding the complexities introduced by more intricate kernels [16], [17], [18], [19], [20]. Their choice for unity kernel suggests that the presence of complex kernels may introduce significant challenges in their numerical schemes.

Our scheme's computational efficiency and cost-effectiveness are evident in Fig. 1. While solving for a particular time step, convolution intervals generally consider all previously evaluated dependent variables, reflecting the memory effect, which can be time-consuming if done repeatedly as the RK scheme progresses through each stage, thus cruising on the same time spot. Typically, for input time $t_n$, it is assumed that the data for the dependent variables have already been evaluated ($u(t_0),\dots ,u(t_n)$) at either equal or uneven previous time intervals $\mathrm{\Delta }{t}_i$ up to that point.

Figure 1.

A schematic representation illustrates an m-stage Runge-Kutta (RK) scheme, emphasizing the frontstage operations consisting of m sets of algebraic evaluations and m backstage actions. Each backstage action involves convolution integrals that utilize all previously pre-evaluated values of the dependent variable u(t) up to u(t_n), which can be either evenly spaced or unequally spaced time intervals, particularly in the context of the adaptive scheme.

In this flow diagram, the front stage represents the primary m-stage Runge-Kutta scheme, which consists of m sets of algebraic operations. Each set may involve multiple arithmetic operations to evaluate the source term of the initial value problem (IVP), effectively treating the IVP as if it did not involve convolution integrals. In contrast, the backstage mainly consists of the convolution integration process at a fixed time. Interestingly, this backstage is eventually eliminated by our time-efficient scheme, a key to this article.

Consequently, without our scheme, the integrals would be computed backstage over the intervals $[t_0,t_n]$ for $stage-1$, and for every subsequent stage up to $stage-m$ where the integration would extend from $[t_0,(t_n+\mathrm{\Delta }{t}_m)]$. This would result in an exorbitant amount of computational time. While some researchers have eliminated all the backstage integration except for the first one, $stage-1$ [14], [15], [7] though for unity kernel, we have eliminated all for non-unity kernels in Eq. (2).

The underlying rationale for this approach is elaborated upon in Sec. 4, where $S(u(t_0),..,u(t_n))$ is the set of already evaluated dependent variable data at generally unequally time intervals (adaptative).

While currently insignificant, we will later observe that for the specified step size (ħ), the two outputs ($u^{(m)}(t_n+ħ)$, $u^{(m+1)}(t_n+ħ)$) correspond to the two successive orders of the adaptive scheme, for example Rk4 and RK5.

Recently, Jafarzadeh et al. [21] used fast Fourier Transform on a fast-convolution-based method to approach linear and a certain class of nonlinear peridynamic transient diffusion problems. They achieve a substantial time reduction in evaluating the spatial convolution, but not on time. Here, we streamline flexible numerical differentiation and integration discretizations on ordinary differential equation (ODE) platforms. Building robust solving schemes that efficiently tackle complicated forms of Volterra-type Integro Differential Equations (IDE) found in Physics Problems. The method used here is so robust that it is capable of handling complicated integral convolutions that may include nonlinearities and derivatives of any order.

Thus, we use the Runge Kutta scheme to solve the complex VIDE here, focusing on embedded RK methods [22], [23]. In particular, we elect the Fehlberg-type (RK45F) [24], [25] and use Cash-Karp formulation [25], [26]. Moreover, we appealingly construct a quadratic equation that generates stepsizes following a parabolic trend, eventually using them to systematically feed a flexible RK4 scheme targeting a desired section to take small step sizes. Additionally, our proposed time-efficient convolution implementation makes it easy to construct schemes that allow for calculations of internal and global scheme error [27], [28].

1.1. Referential background

We recall that the derivative and the integral of two functions $f(x)$ and $g(x)$, given by

$$\frac{df}{dx}=\underset{a}{\overset{b}{\int }}g(s)ds,$$ (3)

bear a strong correlation. Accordingly, one is the double inverse of the other. Thus, in the equality, the RHS is interpreted as the average rate of the causality function $g(x)$, weighted on the domain span of the independent variable.

We recall that a relation is termed an Integro-Differential Equation (IDE) when combinations of these two co-exist. A popular one is the Volterra Integro Differential Equation (VIDE) [29], [30] which in its simplest form is

$$\frac{d^{n}f}{d{x}^n}=\mu (x,f(x))+\lambda \underset{0}{\overset{b(x)}{\int }}K(x,s)F(x,s,f(s))ds,$$ (4)

where λ is a constant parameter, $\mu (x,f(x))$ a function of x and $f(x)$, $K(x,s)$ the kernel function, and $F(x,s,f(s))$ a function that depends on $x,s,\mathrm{\&}f(s)$. When the function $F(x,s,f(s))$ contains a combination of derivatives of the dependent variable, we say here that we have a Volterra-type Integro Differential Equation (VTIDE). The Leibniz rule can be applied to an initial value problem, converting it to a VIDE, providing more insight into Eq. (4). Applications and Implications of VIDE can be seen in Biology, Engineering, and the industrial world. These include logistics studies of rates in co-existing biological species, the process of diffusion, nano hydrodynamics, and viscoelasticity, to name a few.

Several variants of Eq. (4) have been studied by several authors [31], [32], [33], [34], [35], [36], [37], [38], [39], [15], [14]. Research has been made on some extended models of the VIDE, for example, modifications of the kernel [9], the fuzzy-type, [40], the Fredholm (fractional) type [41] or cases with degenerate kernels [42]. Consequently, as the integrant is a function of the dependent variable, the variation of the dependent variable appears to rely on its earlier instances and is weighted by a kernel, thus a convolution. While some VIDE are linear [34], [39], [15], [43], [44], [40], hence fewer complexities, others could, however, be nonlinear, generally challenging to tackle [35], [36], [45], [46], [47], presenting stability, convergence, and error-of-convergence issues.

On some numerical solution techniques, a multistep block approach has been used where computations within the RK scheme are evaluated in blocks [19], [48], [49], [18], [50], [51]. Meanwhile, others have formulated time-efficient schemes that work well for IVP and VTIDE having no convolution kernels, thus analogous to $K(x,s)=1$, for instance, the works in references [24], [14], [15], [40]

A precursor parallel study of a time-efficient (TE) scheme for VTIDE is treated by M.N. Azese [52], where convolutions with decay-kernel are accommodated. Therein, the TE-VTIDE is systematically tested and proved not only to be time-saving but also elegant in its implementation. Moreover, the convolution proposed therein is noteworthy for its nonlinear nature and inclusion of derivatives. It is pertinent to observe that this article also investigates the time-efficient scheme developed in the concurrent study [52], expanding upon its examination and implications; adaptative schemes.

1.2. Motivation of the study and organization of this work

The motivation behind this work shares similarities with the concurrent paper [52]. Firstly, it addresses the issue of inconsistency in the order of convergence. Secondly, it aims to tackle the significant computational time requirements observed in previous studies on fluid dynamics involving viscoelastic fluids, in which the author was involved. These studies focused on solving spatio-temporal integro-differential equations related to viscoelastic fluids [7], [8], with integral kernels representing the fading memory of the viscoelastic fluid's kinematics. Furthermore, in a departure from the parallel study [52], we aim to extend the application of the TE scheme to adaptive schemes. Our objective is to develop preliminary discretizations tailored for adaptive schemes and to facilitate the utilization of user-defined generated step sizes.

We address the question of how the scheme is formulated and implemented on changing step size, how it is cost-effective, and how the scheme allows for easier maneuvers in obtaining implicit forms and stability functions.

To achieve these objectives, we outline the structure of our work as follows: In Sec. 2, we introduce three distinct VTIDEs characterized by varying degrees of nonlinearity and convolution complexities. Moving on to Sec. 3, we review and construct the necessary schemes for differentiation, integration, and discretization of ordinary differential equations (ODEs) essential for our study. Subsequently, Sec. 4 is dedicated to the development of the crucial time-efficient scheme, with a specific focus on the integration of convolutions, aimed at reducing the computational burden associated with integration. In Sec. 5, we demonstrate the utilization of these discretizations in the formulation of a Runge-Kutta 45 Fehlberg (RK45F) scheme. We also establish relationships between computation time and numerical errors. In Sec. 6, we integrate our scheme with the Runge-Kutta (RK) platform to compare it with other schemes, while outlining the advantages of our scheme. During this process, we develop stability functions by constructing implicit numerical algorithms. Additionally, in Sec. 7, we validate our RK45F scheme through its application to the three constructed examples. Furthermore, we test the RK4 scheme with varying step sizes, employing a parabolic-varying (user-defined) step size. Finally, in Sec. 8, we provide concluding remarks on the findings and developments presented in this research.

2. Proposed Volterra-type integro-differential equation (VTIDE) examples

In this section, we introduce and elaborate on three instances of non-trivial nonlinear Volterra-type Integro-Differential Equations. These examples feature nonlinearities throughout, with convolution kernels (within integrals) characterized by integrants that are both nonlinear and contain derivatives of the dependent functions. We anticipate that such intricate formulations will facilitate the implementation of a comprehensive discretization package, enabling us to evaluate their compatibility with the underlying RK schemes effectively.

In all that follows, we denote

$$u=u(\overrightarrow{l},t),\text{where}\overrightarrow{l},t\text{represent spatial coordinates and time respectively,}$$ (5)

which can be considered here as velocity. Although our suggested schemes can solve spatio-temporal (Eq. (5)) VTIDE, we shall present only time-dependent cases, thus $u=u(t)$. However, we reference the work by Udugama et al. [7] where spatio-temporal cases have been treated.

2.1. Proposed nonlinear VTIDE: Example 1: linear convolution with derivative

We present here, Example 1, a first-order initial value non-linear VTIDE given by

$$\frac{du}{dt}=u(t)\{\alpha +\beta [\gamma e^{-\mathcal{W}t}+\underset{0}{\overset{t}{\int }}{e}^{-\mathcal{W}(t-s)}(\mathcal{W}u(s)+\frac{du}{ds})ds\left]\right\},$$ (6)

where $\alpha ,\beta ,\gamma ,\text{and}\mathcal{W}$ are constant parameters. The initial condition is

$$u(0)=\gamma .$$ (7)

The integrant is linear but has a first-order derivative and a decaying kernel, $K(t,s)=e^{-\mathcal{W}(t-s)}$. For simplicity, we shall consider $\mathcal{W}=1$, although it doesn't change the intricacies.

2.2. Proposed nonlinear VTIDE: Example 2: non-linear convolution with derivative

Our Example 2 is shown here, which is also first-order nonlinear initial value VTIDE,

$$\frac{du}{dt}=bu(t)-u{(t)}^2+d{e}^{-\xi t}+\underset{0}{\overset{t}{\int }}{e}^{-\mathcal{W}(t-s)}u(s)(u(s)+2\frac{du}{ds})ds,$$ (8)

where $b,d,\xi ,\text{and}\mathcal{W}$ are constant parameters, such that the initial condition is

$$u(0)=0.$$ (9)

Here also, the integrant is nonlinear and includes a first-order derivative, along with a decaying kernel, $K(t,s)=e^{-\mathcal{W}(t-s)}$. It's worth noting that Eq. (8) can also be derived from a Volterra-type Integro-Differential Equation (VTIDE) outlined in [52] by selecting $a=0$, $p=\gamma =0$, $c=-\alpha =-1$, $\mathcal{W}=1$, and consequently $\beta =2$.

2.3. Proposed nonlinear VTIDE: Example 3: linear convolution without a derivative

Our final example here, Example 3, is a first-order VTIDE(or simply VIDE) of the nonlinear type given by

$$\begin{array}{cc}\hfill \frac{du}{dt}& =u(t)(\alpha -\beta \underset{0}{\overset{t}{\int }}{e}^{-\mathcal{W}(t-s)}u(s)ds)\hfill \end{array}$$ (10a)

$$\begin{array}{cc}\hfill u(0)& =u_0\text{and}t\ge 0,\hfill \end{array}$$ (10b)

where, $\alpha ,\beta ,\mathcal{W}$ are constant parameters of choice. Moreover, we consider $u_0$ as the initial velocity at time $t=0$. In Eq. (10a), (10b) the integrant is linear and does not contain a derivative, but also has a decaying kernel, $K(t,s)=e^{-\mathcal{W}(t-s)}$. We note that this example was also addressed in Ali Filiz [14], for a different purpose and rather on a different scheme, but failed to utilize the complete strength of the cost-effective development here.

3. Preliminaries of numerical considerations

This section establishes a robust numerical framework for computationally addressing the VTIDE introduced in Sec. 2. While these equations are typically solved using RK schemes as initial value problems (IVPs), they involve integrations and differentiations, necessitating appropriate discretizations for these operations. The integration and differentiation schemes developed here are designed to be flexible, allowing them to accommodate varying step sizes, making them suitable for adaptive schemes as well as user-defined stepsizes.

The discretization methods for differentiation and integration are extensively covered in numerous textbooks and articles; therefore, we will not delve into the background details here. However, a concise summary of these techniques can be found in the concurrent study [52].

In this context, we employ the Newton-Cotes formula and adapt it to accommodate slightly unequally spaced data points. Polynomial interpolation is a data analytical technique for approximating the governing equation and other data points. Typically, with $n+1$ data points, polynomial interpolation approximates the actual functions using a polynomial of degree n. Among various techniques, we opt for the Lagrange interpolation polynomial. Utilizing the Lagrange interpolating functions, where the number of points determines the degree of accuracy, we can obtain derivatives (and integrals) and higher-order derivatives suitable for unequally spaced data. It's worth noting that here, we interchangeably use Δt and ħ.

Let

$$f:[a,b]\to \mathbb{R},$$

be a well-behaved continuous function in the given interval. Given $y=f(t)$, and for a given $n+1$ unequally spaced data points say from $(t_1,y_1)$ to $(t_{n+1},y_{n+1})$, the Lagrange polynomial, ${\varpi }_n(t)$, of degree n passing through these points is given by

$${\varpi }_n(t)=\sum _{i=1}^{n+1}{l}_i(t)y_i,$$ (11)

where the basis Lagrange polynomials, $l_i$, are calculated from

$$l_i=\prod _{j=1,j\ne i}^{n+1}\frac{(t-t_j)}{(t_i-t_j)}.$$ (12)

If we consider

$$t_{n+1}-t_n=ħ,$$

then for $0\le m\le 1$, we define

$$t_{n+m}=t_n+mħ,\text{such that for}{t}_{n+\frac{3}{8}}\text{say, we have}m=\frac{3}{8}.$$

We also define $t_{ij}$ as the data-difference such that

$$t_{ij}=t_i-t_j,$$ (13)

which will be heavily used henceforward. We note that the matrix $t_{ij}$ is a zero-diagonal and antisymmetric matrix. Moreover, the schemes that would be considered in this article are backward differences such that

$$t_{ij}\ge 0\text{for}i\le j.$$

For example, consider the following backward data points in Table 1. Then, using Table 1 we explore Eq. (13) to generate

$$t_{ij}=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill 0.002\hfill & \hfill 0.006\hfill & \hfill 0.009\hfill \\ \hfill -0.002\hfill & \hfill 0\hfill & \hfill 0.004\hfill & \hfill 0.007\hfill \\ \hfill -0.006\hfill & \hfill -0.004\hfill & \hfill 0\hfill & \hfill 0.003\hfill \\ \hfill -0.009\hfill & \hfill -0.007\hfill & \hfill -0.003\hfill & \hfill 0\hfill \end{array}\right).$$ (14)

Table 1.

Example of points to generate, t_ij.

$t_4$	$t_3$	$t_2$	$t_1$
0.02	0.023	0.027	0.029

Accordingly, we develop discretizations for 2 data points, 3 data points, and 4 data points to be used in this article, although more data points can be used for even more accurate schemes. First, we consider 2 data points,

$$(\stackrel{t_2}{\overbrace{t_n}},\stackrel{t_1}{\overbrace{t_{n+m}}}),$$

having stepsize ħ so that the approximate polynomial is

$${\varpi }_2(t)=l_1(t)y_1+l_2(t)y_2.$$ (15)

Using Eq. (12), the approximation in Eq. (15) takes the form

$$f(t)\approx {\varpi }_2(t)=\frac{1}{t_{12}}[(t-t_2)\varpi (t_1)-(t-t_1)\varpi (t_2)].$$ (16)

Secondly, for available three data points, whether they are equally or unequally spaced, we define

$$(\stackrel{t_3}{\overbrace{t_{n-1}}},\stackrel{t_2}{\overbrace{t_n}},\stackrel{t_1}{\overbrace{t_{n+m}}}),$$

so that we have

$${\varpi }_3(t)=l_1(t)y_1+l_2(t)y_2+l_3(t)y_3,$$ (17)

such that with Eq. (12) we obtain

$$f(t)\approx {\varpi }_3(t)=[\frac{(t-t_2)(t-t_3)}{t_{12}{t}_{13}}\varpi (t_1)+\frac{(t-t_1)(t-t_3)}{t_{21}{t}_{23}}\varpi (t_2)+\frac{(t-t_1)(t-t_2)}{t_{31}{t}_{32}}\varpi (t_3)].$$ (18)

And finally, given four points that are either equally spaced or non-equidistant,

$$(\stackrel{t_4}{\overbrace{t_{n-2}}},\stackrel{t_3}{\overbrace{t_{n-1}}},\stackrel{t_2}{\overbrace{t_n}},\stackrel{t_1}{\overbrace{t_{n+m}}}),$$

we can write

$${\varpi }_4(t)=l_1(t)y_1+l_2(t)y_2+l_3(t)y_3+l_4(t)y_4,$$ (19)

so that using Eq. (12), we eventually get

$$\begin{array}{cc}\hfill f(t)\approx {\varpi }_4(t)\hfill & \hfill =[\frac{(t-t_2)(t-t_3)(t-t_4)}{t_{14}{t}_{13}{t}_{12}}\varpi (t_1)+\frac{(t-t_1)(t-t_3)(t-t_4)}{t_{24}{t}_{23}{t}_{21}}\varpi (t_2)\hfill \\ \hfill \hfill & \hfill +\frac{(t-t_1)(t-t_2)(t-t_4)}{t_{34}{t}_{32}{t}_{31}}\varpi (t_3)+\frac{(t-t_3)(t-t_2)(t-t_1)}{t_{43}{t}_{42}{t}_{31}}\varpi (t_4)]\hfill \end{array}\}.$$ (20)

We formulate numerical derivatives and integration following the aforementioned expressions; Eqs. (16), (18) and (20).

3.1. Adaptable numerical differentiation scheme

We use the polynomial approximations derived in Sec. 3.1 to obtain an approximate first derivative. Accordingly, Eq. (16) yields

$$f^{\prime }(t)\approx \frac{d{\varpi }_2}{dt}=\frac{1}{t_{12}}[\varpi (t_1)-\varpi (t_2)].$$ (21)

Then, using Eq. (18), we obtain

$$\begin{array}{cc}\hfill f^{\prime }(t_1)\approx \frac{d{\varpi }_3}{dt}{|}_{t=t_1}& =[\frac{t_{12}+t_{13}}{t_{12}{t}_{13}}\varpi (t_1)+\frac{t_{13}}{t_{21}{t}_{23}}\varpi (t_2)+\frac{t_{12}}{t_{31}{t}_{32}}\varpi (t_3)]\hfill \end{array}$$ (22a)

$$\begin{array}{cc}\hfill f^{\prime }(t_2)\approx \frac{d{\varpi }_3}{dt}{|}_{t=t_2}& =[\frac{t_{23}}{t_{12}{t}_{13}}\varpi (t_1)+\frac{t_{21}+t_{23}}{t_{21}{t}_{23}}\varpi (t_2)+\frac{t_{21}}{t_{31}{t}_{32}}\varpi (t_3)]\hfill \end{array}$$ (22b)

$$\begin{array}{cc}\hfill f^{\prime }(t_3)\approx \frac{d{\varpi }_3}{dt}{|}_{t=t_3}& =[\frac{t_{32}}{t_{12}{t}_{13}}\varpi (t_1)+\frac{t_{31}}{t_{21}{t}_{23}}\varpi (t_2)+\frac{t_{31}+t_{32}}{t_{31}{t}_{32}}\varpi (t_3)].\hfill \end{array}$$ (22c)

Finally, from Eq. (20), we derive

$$\begin{array}{cc}\hfill f^{\prime }(t_1)\approx \frac{d{\varpi }_4}{dt}{|}_{t=t_1}& =[\frac{(t_{14}{t}_{13})+(t_{14}{t}_{12})+(t_{13}{t}_{12})}{t_{14}{t}_{13}{t}_{12}}\varpi (t_1)+\frac{(t_{14}{t}_{13})}{t_{24}{t}_{23}{t}_{21}}\varpi (t_2)+\frac{(t_{14}{t}_{12})}{t_{34}{t}_{32}{t}_{31}}\varpi (t_3)+\frac{(t_{13}{t}_{12})}{t_{43}{t}_{42}{t}_{41}}\varpi (t_4)]\hfill \end{array}$$ (23a)

$$\begin{array}{cc}\hfill f^{\prime }(t_2)\approx \frac{d{\varpi }_4}{dt}{|}_{t=t_2}& =[\frac{(t_{24}{t}_{23})}{t_{14}{t}_{13}{t}_{12}}\varpi (t_1)+\frac{(t_{24}{t}_{23})+(t_{24}{t}_{21})+(t_{23}{t}_{21})}{t_{24}{t}_{23}{t}_{21}}\varpi (t_2)+\frac{(t_{24}{t}_{21})}{t_{34}{t}_{32}{t}_{31}}\varpi (t_3)+\frac{(t_{23}{t}_{21})}{t_{43}{t}_{42}{t}_{41}}\varpi (t_4)]\hfill \end{array}$$ (23b)

$$\begin{array}{cc}\hfill f^{\prime }(t_3)\approx \frac{d{\varpi }_4}{dt}{|}_{t=t_3}& =[\frac{(t_{34}{t}_{32})}{t_{14}{t}_{13}{t}_{12}}\varpi (t_1)+\frac{(t_{34}{t}_{31})}{t_{24}{t}_{23}{t}_{21}}\varpi (t_2)+\frac{(t_{34}{t}_{32})+(t_{34}{t}_{31})+(t_{32}{t}_{31})}{t_{34}{t}_{32}{t}_{31}}\varpi (t_3)+\frac{(t_{32}{t}_{31})}{t_{43}{t}_{42}{t}_{41}}\varpi (t_4)]\hfill \end{array}$$ (23c)

$$\begin{array}{cc}\hfill f^{\prime }(t_4)\approx \frac{d{\varpi }_4}{dt}{|}_{t=t_4}& =[\frac{(t_{43}{t}_{42})}{t_{14}{t}_{13}{t}_{12}}\varpi (t_1)+\frac{(t_{43}{t}_{41})}{t_{24}{t}_{23}{t}_{21}}\varpi (t_2)+\frac{(t_{42}{t}_{41})}{t_{34}{t}_{32}{t}_{31}}\varpi (t_3)+\frac{(t_{43}{t}_{42})+(t_{43}{t}_{41})+(t_{42}{t}_{41})}{t_{43}{t}_{42}{t}_{41}}\varpi (t_4)].\hfill \end{array}$$ (23d)

3.2. Adaptable numerical integration scheme

We also obtain approximate integration from the approximate polynomials Eqs. (16), (18) and (20) by considering $t_2$ as $t_n$ and $t_1$ as $t_{n+m}$. Firstly, utilizing the two-point approximations from Eq. (16), we derive the following:

$${\mathcal{I}}_2\approx \underset{t_n}{\overset{t_{n+m}}{\int }}{\varpi }_2(t)dx=\underset{t_2}{\overset{t_1}{\int }}{\varpi }_2(t)dx=\frac{t_{12}}{2}(\varpi (t_1)+\varpi (t_2)).$$ (24)

Meanwhile, for three generally unequally spaced data points as shown in Eq. (18), we find the following:

$$\begin{array}{cc}{\mathcal{I}}_3\approx \underset{t_n}{\overset{t_{n+m}}{\int }}{\varpi }_3(t)dx=\underset{t_2}{\overset{t_1}{\int }}{\varpi }_3(t)dx\hfill & =\frac{t_{12}^2(3t_{23}+2t_{12})}{6t_{12}{t}_{13}}{\varpi }_3(t_1)+\frac{t_{12}}{6t_{21}{t}_{23}}[2t_{12}^2+3t_{12}(2t_{21}+t_{13})\hfill \\ \hfill & +6t_{21}\stackrel{t_{23}}{\overbrace{(t_{21}+t_{13})}}]{\varpi }_3(t_2)+\frac{t_{12}^2(3t_{21}+2t_{12})}{6t_{31}{t}_{32}}{\varpi }_3(t_3)\hfill \end{array}\}.$$ (25)

And finally, using the general 4 non-equidistant data points as in Eq. (20), we can approximate that;

$$\begin{array}{cc}{\mathcal{I}}_4\approx \underset{t_n}{\overset{t_{n+m}}{\int }}{\varpi }_4(t)dx\hfill & =\underset{t_2}{\overset{t_1}{\int }}{\varpi }_4(t)dx=\frac{t_{12}^2}{12t_{14}{t}_{13}{t}_{12}}[6t_{24}{t}_{23}+4t_{12}(t_{24}+t_{23})+3t_{12}^2]{\varpi }_4(t_1)\hfill \\ \hfill & +\frac{t_{12}}{12t_{24}{t}_{23}{t}_{21}}\left[4t_{12}^2\right(t_{21}+t_{23}+t_{24})+6t_{12}(t_{24}(t_{23}+t_{21})+t_{23}{t}_{21})+12t_{21}{t}_{23}{t}_{24}+3t_{12}^2]{\varpi }_4(t_2)\hfill \\ \hfill & +\frac{t_{12}^2}{12t_{34}{t}_{32}{t}_{31}}[6t_{24}{t}_{21}+4t_{12}(t_{24}+t_{21})+3t_{12}^2]{\varpi }_4(t_3)\hfill \\ \hfill & +\frac{t_{12}^2}{12t_{43}{t}_{42}{t}_{41}}[6t_{23}{t}_{21}+4t_{12}(t_{23}+t_{21})+3t_{12}^2]{\varpi }_4(t_4)\hfill \end{array}\}.$$ (26)

Implementing the aforementioned flexible step size discretizations, which mitigates data storage concerns, may seem intricate but relatively straightforward. To elucidate the approach, let's consider the integration scheme for four equally or non-equidistant points as illustrated in Eq. (26), denoted as $(\stackrel{t_4}{\overbrace{t_{n-2}}},\stackrel{t_3}{\overbrace{t_{n-1}}},\stackrel{t_2}{\overbrace{t_n}},\stackrel{t_1}{\overbrace{t_{n+m}}})$. Suppose we are at a specific time, $t_n$, within an intermediate step of a multi-stage RK scheme, such as RK4, and intend to compute the integral from $t_n$ to $t_{n+m}$. Initially, it's crucial to note that at this juncture, the value of the integrant (designated as $\varpi (t_{n+m})$) must have been approximated. Consequently, we allocate the 16 $t_{ij}$ values into a 2-dimensional $4\times 4$ array, resembling the structure depicted in Eq. (14). Subsequently, within this allocated array, all $t_{ij}$ values are computed and stored, facilitating their immediate use in evaluating each term in Eq. (26) during that particular stage. Eventually, before exiting the stage, the $4\times 4$ array is deallocated, preparing it for subsequent use in the next stage within the same evaluation, occurring at that precise time instance.

Regarding the error analysis of the schemes above, it's important to recall that for equally spaced data points, the local error in the polynomial approximation is expressed as:

$${\mathcal{E}}_l=f(t)-{\varpi }_n(t)=\frac{f^{n+1}(\xi )}{(n+1)!}\prod _{i=1}^{n+1}(t-t_i),$$ (27)

where ξ is a point lodged between the points of interest. Consequently, one can infer that Eq. (27) can also effectively estimate the upper bound of the local truncation errors in the integrals, such as Eqs. (24) to (26), which involve slightly unequally spaced points. Subsequently, by considering ${ħ}_{\text{min}}$ as the smallest step size and ${ħ}_{\text{max}}$ as the largest, we can intuitively express ${\mathcal{E}}_l(I_2)⪅\mathcal{O}({ħ}_{max}^2)$, ${\mathcal{E}}_l(I_3)⪅\mathcal{O}({ħ}_{max}^4)$, and ${\mathcal{E}}_l(I_4)⪅\mathcal{O}({ħ}_{max}^6)$.

3.3. Synopsis of adaptive Runge-Kutta scheme

In general, the Runge-Kutta method [53] is a numerical technique used in solving first-order initial value ODE such as

$$\frac{dy}{dt}=F(t,y(t)),\text{where}y(0)=y_0,t\ge 0$$ (28)

and where the independent variable, t, is discretized as follows

$$t_n=n\mathrm{\Delta }t=nħ.$$ (29)

The degree of desired accuracy dictates the choice of the RK scheme. For lower-order schemes such as first and second orders, the RK scheme can be re-casted into the theta-methods (θ-method) [54]. We note that S. Gan [45] used the (θ-method) to analyze VIDE with a focus on the dissipativity.

Further, higher-order RK schemes can be represented as below

$$y(t_{n+1})=y(t_n)+ħ\sum _{j=1}^s{b}_j{k}_j,$$ (30)

where $b_j$ are the weights such that $j=1,2,\cdots ,s$ and also so that $k_j$ is defined as

$$k_s=F(t_n+c_sħ,y(t_n)+ħ\sum _j^N{a}_{s,j}{k}_j),$$ (31)

where $c_i$ are considered as the nodes. The RK matrix $a_{i,j}$, is defined such that $1\le j<i\le s$. Additionally, $N=s-1$ for explicit schemes meanwhile $N=s$ for implicit ones. Consistency of the RK scheme requires that

$$\sum _{i=1}^s{b}_i=1$$ (32)

and

$$\sum _{i=1}^{i-1}{a}_{i,j}=c_i\forall i\le s.$$ (33)

To obtain a $p^{th}$-order of the RK-scheme in Eqs. (30) and (31), $s\ge p$. The coefficients $a_{i,j}$, $b_i$, and $c_i$ can be readily obtained using the Butcher table developed by John Butcher [55], [56].

In general, one can re-write Eqs. (6), (8) and (10a), (10b) as follows:

$$\frac{du}{dt}=S(t,u(t),\mathcal{I}(t)),$$ (34)

where $S(t,u(t),\mathcal{I}(t))$ is considered as the source term meanwhile $\mathcal{I}(t)$ is the convolution integral. Until this end, we have described the standard Runge Kutta method with fixed steps.

However, an extension of the standard Runge-Kutta methods having fixed steps for numerical integration is the Adaptive Runge-Kutta method, a numerical technique also used to solve ordinary differential equations or systems of ODEs.

In contrast to the standard Runge-Kutta methods, the adaptive Runge-Kutta method has changing step sizes that are dynamically adjusted during the integration process based on the local error estimation. The method calculates an estimate of the local error at each step and adjusts the step size accordingly to ensure that the error stays within a predefined tolerance.

The advantage of adaptive Runge-Kutta methods is that they can achieve the desired level of accuracy while minimizing computational cost by taking larger steps in regions where the solution varies slowly and smaller steps in areas where it changes rapidly.

Common adaptive Runge-Kutta methods include the Embedded Runge-Kutta (ERK) methods, which use two or more stages of different orders to estimate the error, and the Runge-Kutta-Fehlberg (RKF) method, which combines a fifth-order method with a fourth-order method to calculate the error. An example algorithm describing the process can be seen in Algorithm 1. Nonetheless, following this, the Butcher tableau for RK45-Fehlberg, developed by Fehlberg and re-casted by Cash-Karp can be written as:

$$\begin{array}{cc}\hfill k_1& =ħS(t_n,u(t_n),\mathcal{I}(t_n))\hfill \end{array}$$ (35a)

$$\begin{array}{cc}\hfill k_2& =ħS(t_{n+\frac{1}{5}},u(t_n)+\frac{k_1}{5},{\mathcal{I}}^{(a)}(t_{n+\frac{1}{5}}))\hfill \end{array}$$ (35b)

$$\begin{array}{cc}\hfill k_3& =ħS(t_{n+\frac{3}{10}},u(t_n)+\frac{3}{40}{k}_1+\frac{9}{40}{k}_2,{\mathcal{I}}^{(b)}(t_{n+\frac{3}{10}}))\hfill \end{array}$$ (35c)

$$\begin{array}{cc}\hfill k_4& =ħS(t_{n+\frac{3}{5}},u(t_n)+\frac{3}{10}{k}_1-\frac{9}{10}{k}_2+\frac{6}{5}{k}_3,{\mathcal{I}}^{(c)}(t_{n+\frac{3}{5}}))\hfill \end{array}$$ (35d)

$$\begin{array}{cc}\hfill k_5& =ħS(t_n,u(t_n)-\frac{11}{54}{k}_1+\frac{5}{2}{k}_2-\frac{70}{27}{k}_3+\frac{35}{27}{k}_4,{\mathcal{I}}^{(d)}(t_n))\hfill \end{array}$$ (35e)

$$\begin{array}{cc}\hfill k_6& =ħS(t_{n+\frac{7}{8}},u(t_n)+\frac{1631}{55296}{k}_1+\frac{175}{512}{k}_2+\frac{575}{13824}{k}_3+\frac{44275}{110592}{k}_4+\frac{253}{4096}{k}_5,{\mathcal{I}}^{(e)}(t_{n+\frac{7}{8}}))\hfill \end{array}$$ (35f)

$$\begin{array}{cc}\hfill u(t_{n+1})& =u(t_n)+(\frac{37}{378}{k}_1+\frac{250}{621}{k}_3+\frac{125}{594}{k}_4+\frac{512}{1771}{k}_6).\hfill \end{array}$$ (35g)

$$\begin{array}{cc}\hfill \tilde{u}(t_{n+1})& =u(t_n)+(\frac{2825}{27648}{k}_1+\frac{18575}{48384}{k}_3+\frac{13525}{55296}{k}_4+\frac{277}{14336}{k}_5+\frac{1}{4}{k}_6),\hfill \end{array}$$ (35h)

where $u(t_{n+1})$ is fourth-order, meanwhile $\tilde{u}(t_{n+1})$ is fifth-order corresponding to $u^{(m)}(t_{n+1})$ and $u^{(m+1)}(t_{n+1})$ in Fig. 1.

Algorithm 1.

Adaptative Driver: Stepsize control sub-algorithm.

The integral quantities denoted as ${\mathcal{I}}^{()}(t_{n+m})$ and their successive superscips ($a,b,c,\mathrm{\&}d$), progressively represent improved versions of added m fraction-of-a-time step integrals to $\mathcal{I}(t_n)$, using updated versions of $u(t_n)$ calculated in immediate prior steps.

In Sec. 7.4, we also formulate an RK4 scheme utilizing our time-efficient (TE) approach. It's worth noting that this TE-RK4 scheme is concurrently being developed and implemented in [52] for less complex non-adaptive schemes. Moreover, the RK4 scheme has been effectively employed by this author to solve spatio-temporal initial value problems (IVPs) derived from fluid mechanics applications [8], [57], [3].

4. The crucial time-efficient integration scheme implementation

We outline the essence of our integration scheme, which significantly reduces the computational time for special kernels (Eq. (2)), ultimately resulting in an overall time-effective scheme. While this development is currently being introduced in a parallel investigation [52], we deem it informative to reiterate it here for the sake of completeness, rendering this article standalone.

Our consideration consequently expands to reduce the computational time in the entire RK scheme. Doing so in a fashion that interestingly avoids lengthy integration at each iteration time step. We eventually show later (Sec. 5.3.1) that our hypothesis is successful in substantially reducing computation time, making our algorithm highly time-efficient.

To the best of the author's knowledge, this crucial consideration has yet to be dealt with within the framework of a VIDE or an RK scheme having a special convolution. This section helps elucidate the flow diagram shown in Fig. 1 and reinforces the explanation provided. Hence, reference to Fig. 1 is required for a better understanding of the subsequent discussion.

For simplicity, let's assume we already have a pre-evaluated set of n data points for a dependent function $g(t)$ at $t_0,t_1,\cdots t_n$, and in the current iteration, we are seeking the new data at $t_{n+1}$. Therefore, using precalculated data, we consider having numerically evaluated the integration

$$\mathcal{I}(t_n)=\underset{0}{\overset{t_n}{\int }}\varpi (t_n,s)ds=\underset{0}{\overset{t_n}{\int }}\mathcal{K}(t_n,s)g(s)ds,$$ (36)

where $\mathcal{K}(t_n,s)$ which is defined here as the kernel having a special form

$$\mathcal{K}(t_n,s)=a^{\kappa (t_n-s)},$$ (37)

where a and κ are real numbers. Thus, at each stage, we simply need to evaluate and append the integral from $t_n\to t_{n+m}$ and add it to Eq. (36) in a methodical yet efficient manner. What follows outlines the methodic concatenation process. It can easily be shown that

$$\mathcal{K}(t_{n+1},s)=a^{\kappa (t_n+ħ-s)}=a^{\kappa ħ}{a}^{\kappa (t_n-s)}=a^{\kappa ħ}\mathcal{K}(t_n,s),$$ (38)

where we recall that ħ is the time step here. Also, through Eq. (36) we have the following:

$$\begin{array}{cc}\hfill \mathcal{I}(t_{n+1})& =\underset{0}{\overset{t_{n+1}}{\int }}\varpi (t_{n+1},s)ds=\underset{0}{\overset{t_n}{\int }}\varpi (t_{n+1},s)ds+\underset{t_n}{\overset{t_{n+1}}{\int }}\varpi (t_{n+1},s)ds\hfill \\ \hfill & =\underset{0}{\overset{t_n}{\int }}\mathcal{K}(t_{n+1},s)g(s)ds+\underset{t_n}{\overset{t_{n+1}}{\int }}\mathcal{K}(t_{n+1},s)g(s)ds\hfill \end{array}\}.$$ (39)

Thus, using Eqs. (36) to (39), we can conclude that

$$\begin{array}{cc}\hfill \mathcal{I}(t_{n+1})& =\stackrel{(I)}{\overbrace{\mathcal{K}(ħ,0)\underset{0}{\overset{t_n}{\int }}\varpi (t_n,s)ds}}+\stackrel{(II)}{\overbrace{\underset{t_n}{\overset{t_{n+1}}{\int }}\varpi (t_{n+1},s)ds}}\hfill \\ \hfill & =\stackrel{(I)}{\overbrace{a^{\kappa ħ}\underset{0}{\overset{t_n}{\int }}\mathcal{K}(t_n,s)g(s)ds}}+\stackrel{(II)}{\overbrace{\underset{t_n}{\overset{t_{n+1}}{\int }}\mathcal{K}(t_{n+1},s)g(s)ds}}\hfill \\ \hfill & =\stackrel{(I)}{\overbrace{a^{\kappa ħ}\mathcal{I}(t_n)}}+\stackrel{(II)}{\overbrace{\underset{t_n}{\overset{t_{n+1}}{\int }}\mathcal{K}(t_{n+1},s)g(s)ds}}\hfill \end{array}\}.$$ (40)

Equation (40) outlines the coupling through (I) and (II), where the pre-factor in (I) for te-int-n is recognized.

Up to this point, our focus has been on full-timestep evaluations where we have recognized the two distinctive terms labeled here as (I) and (II). However, for intermediate steps that are not necessarily whole, we can also establish a similar relation as Eq. (40) for any fraction m of a step at a time ($t_n+mħ$):

$$\begin{array}{cc}\hfill \mathcal{I}(t_{n+m})& =\stackrel{(I)}{\overbrace{\mathcal{K}(mħ,0)\underset{0}{\overset{t_n}{\int }}\varpi (t_n,s)ds}}+\stackrel{(II)}{\overbrace{\underset{t_n}{\overset{t_{n+m}}{\int }}\varpi (t_{n+m},s)ds}}\hfill \\ \hfill & =\stackrel{(I)}{\overbrace{a^{m\kappa ħ}\underset{0}{\overset{t_n}{\int }}\mathcal{K}(t_n,s)g(s)ds}}+\stackrel{(II)}{\overbrace{\underset{t_n}{\overset{t_{n+m}}{\int }}\mathcal{K}(t_{n+m},s)g(s)ds}}\hfill \\ \hfill & =\stackrel{(I)}{\overbrace{a^{m\kappa ħ}\mathcal{I}(t_n)}}+\stackrel{(II)}{\overbrace{\underset{t_n}{\overset{t_{n+m}}{\int }}\mathcal{K}(t_{n+m},s)g(s)ds}}\hfill \end{array}\}.$$ (41)

It is crucial to emphasize that through Eqs. (39) and (41), we already have the first terms (I), which have been calculated from the previous time iteration seen in Eq. (36). Furthermore, the second term $(II)$ can be evaluated at the current iteration step by employing the pointwise integration formula Eqs. (24) to (26), or even higher accuracy integration schemes that involve several data points. Therefore, an outline of the steps involved in integrating Eq. (40) at each timestep and iteration is provided below:

$$\left(\begin{array}{cc}\mathcal{I}(t_0)\hfill & =0\hfill \\ \mathcal{I}(t_1)\hfill & =a^{\kappa ħ}\mathcal{I}(t_0)+\stackrel{(II)}{\overbrace{{\int }_{t_0}^{t_1}\varpi (t_1,s)ds}}\hfill \\ \mathcal{I}(t_2)\hfill & =a^{\kappa ħ}\mathcal{I}(t_1)+\stackrel{(II)}{\overbrace{{\int }_{t_1}^{t_2}\varpi (t_2,s)ds}}\hfill \\ \mathcal{I}(t_3)\hfill & =a^{\kappa ħ}\mathcal{I}(t_2)+\stackrel{(II)}{\overbrace{{\int }_{t_2}^{t_3}\varpi (t_3,s)ds}}\hfill \\ \hfill & ⋮\hfill \\ \mathcal{I}(t_n)\hfill & =a^{\kappa ħ}\mathcal{I}(t_{n-1})+\stackrel{(II)}{\overbrace{{\int }_{t_{n-1}}^{t_n}\varpi (t_n,s)ds}}\hfill \end{array}\right)\text{Time Efficient (TE) integration within the iteration.}.$$ (42)

In Eq. (42), the first terms are already pre-evaluated at previous time steps, and the evaluations of $(II)$ can be conducted using:

• •Lower-order two-point integration formulas such as the rectangular ($\mathcal{O}({ħ}^2)$) or Trapezoidal Rule ($\mathcal{O}({ħ}^2)$) methods,
• •Higher-order three-point schemes like Simpson's Rule ($\mathcal{O}({ħ}^4)$), or
• •Other schemes with three or more points, such as those developed in Eqs. (24) to (26),
• •Using Quadrature rules such as Gaussian Quadrature [58], [59].

Interestingly, when the kernel is unity, $\mathcal{K}(t_n,s)=1$, then we obtain

$$\begin{array}{cc}\hfill \mathcal{I}(t_{n+m})& =\stackrel{(I)}{\overbrace{\underset{0}{\overset{t_n}{\int }}\varpi (t_n,s)ds}}+\stackrel{(II)}{\overbrace{\underset{t_n}{\overset{t_{n+m}}{\int }}\varpi (t_{n+m},s)ds}}\hfill \\ \hfill & =\stackrel{(I)}{\overbrace{\underset{0}{\overset{t_n}{\int }}g(s)ds}}+\stackrel{(II)}{\overbrace{\underset{t_n}{\overset{t_{n+m}}{\int }}g(s)ds}}\hfill \\ \hfill & =\stackrel{(I)}{\overbrace{\mathcal{I}(t_n)}}+\stackrel{(II)}{\overbrace{\underset{t_n}{\overset{t_{n+m}}{\int }}g(s)ds}}\hfill \end{array}\}.$$ (43)

The simplified version of the TE scheme, as depicted in Eq. (43), is reminiscent of approaches adopted by certain researchers, particularly when the kernel is assumed to be equal to unity [15], [14]. Despite this simplification, they typically proceeded by conducting one backstage integration, which our hypothesis doesn't advise.

While not applicable within the scope of this paper, it is intriguing to consider that this concept can be extended to scenarios involving double-integral cases, wherein

$$\mathcal{I}(t_n)=\underset{0}{\overset{t_n}{\int }}\underset{0}{\overset{t_n}{\int }}\mathcal{K}(t_n,s_1,s_2)g(s_1,s_2)d{s}_{1}d{s}_2,{\kappa }_i\in \mathbb{R},$$ (44)

where indeed, using $\mathcal{K}(t_n,s_i)=a^{{\kappa }_i(t_n-s_i)}$, which can also be interpreted as double-kernels given that we can express $\mathcal{K}(t_n,s_1,s_2)=a^{{\kappa }_1(t_n-s_1)}{a}^{{\kappa }_2(t_n-s_2)}$, the developments lead us to the following conclusion:

$$\mathcal{I}(t_{n+m})=\stackrel{(I)}{\overbrace{a^{mħ({\kappa }_1+{\kappa }_2)}\mathcal{I}(t_n)}}+\stackrel{(II)}{\overbrace{\underset{t_n}{\overset{t_{n+m}}{\int }}\underset{t_n}{\overset{t_{n+m}}{\int }}\mathcal{K}(t_{n+m},s_1,s_2)g(s_1,s_2)d{s}_{1}d{s}_2.}}$$ (45)

Consequently, Eq. (45) can be generalized for N-degree integrals to give

$$\mathcal{I}(t_{n+m})=\stackrel{(I)}{\overbrace{a^{mħ{\sum }_i^N{\kappa }_i}\mathcal{I}(t_n)}}+\stackrel{(II)}{\overbrace{\underset{\text{N-times}}{\underbrace{\underset{t_n}{\overset{t_{n+m}}{\int }}\underset{t_n}{\overset{t_{n+m}}{\int }}}}\mathcal{K}(t_{n+m},s_1,\cdots ,s_N)g(s_1,s_2,\cdots ,s_N)d{s}_{1}d{s}_2\cdots d{s}_N}}.$$ (46)

Such multi-kernel (or multiple integral convolutions) relations in Eqs. (45) and (46) appear in the literature of viscoelastic fluids [60], [61], [62], where κ typically represents the inverse of the relaxation time.

5. Outline of the time-efficient RK45-Fehlberg

We devise a systematic algorithm that integrates the time-efficient schemes delineated in Sec. 4, augmented by the integration and differentiation discretization techniques elucidated in Sec. 3. It is recognized that a minimum of two data points is necessary to approximate derivatives and integrals, yielding coarse estimations, contingent upon the step size. Nevertheless, for illustrative purposes, we adopt a numerical approximation method that incorporates four data points, whether equidistant or non-equidistant, ensuring that

$$\begin{array}{cc}\hfill \hfill & \hfill \stackrel{\text{earlier iterations}}{\overbrace{{\varpi }_{n-2},{\varpi }_{n-1}{\varpi }_n}}\hfill \\ \hfill \hfill & \hfill \stackrel{\text{projected estimates}}{\overbrace{{\varpi }_{n+m},\text{for}0<m\le 1}}\hfill \end{array}\}.$$ (47)

The examples detailed in Section 2 feature integrands with first derivatives, necessitating a pursuit of reasonable accuracy. To achieve this, we opt for the corresponding four-point integration and differentiation schemes developed in Sections 3.1 and 3.2. Subsequently, we employ these schemes to formulate the RK4 and RK45-Fehlberg schemes.

In the ensuing discussion, the succinct portrayal of the generalized Volterra-type Integro-Differential Equation in Eq. (34) will offer valuable insights.

5.1. Time-efficient flexible stepsize RK4 scheme

It is worth noting that the Fourth-order Runge-Kutta scheme is a widely used RK scheme documented in numerous literature, and its various versions can be derived from Sec. 3.3. Nonetheless, for comprehensive coverage, we provide the formulation of the RK4 scheme. By applying the RK4 scheme to the generalized Volterra-type Integro-Differential Equation (VTIDE) depicted in Eq. (34), which is also concurrently developed in [52], we can integrate the time-efficient algorithms outlined in Sec. 4, resulting in:

$$\begin{array}{cc}\hfill & \text{Given}{u}_{n-1},u_{n-2}.....u_0,n\ge n_p\text{and}\mathcal{I}(t_0)=0,\hfill \end{array}$$ (48a)

$$\begin{array}{cc}\hfill \mathcal{I}(t_n)& =e^{-ħ}\mathcal{I}(t_{n-1})+\mathcal{J}(t_n)\hfill \end{array}$$ (48b)

$$\begin{array}{cc}\hfill k_1& =ħS(t_n,u_n,\mathcal{I}(t_n))\hfill \end{array}$$ (48c)

$$\begin{array}{cc}\hfill u_{n+\frac{1}{2}}^{(a)}& =u_n+\frac{k_1}{2}\text{and}\mathcal{I}(t_{n+\frac{1}{2}})=e^{-\frac{ħ}{2}}\mathcal{I}(t_n)+\stackrel{\text{Eqs.}\mathrm{\text{(23)}}\text{and}\mathrm{\text{(26)}}}{\overbrace{\mathcal{J}(t_{n+\frac{1}{2}}){|}_{u_{n+\frac{1}{2}}=u_{n+\frac{1}{2}}^{(a)}}}}\hfill \end{array}$$ (48d)

$$\begin{array}{cc}\hfill k_2& =ħS(t_{n+\frac{1}{2}},u_{n+\frac{1}{2}},\mathcal{I}(t_{n+\frac{1}{2}}))\hfill \end{array}$$ (48e)

$$\begin{array}{cc}\hfill u_{n+\frac{1}{2}}^{(b)}& =u_n+\frac{1}{2}{k}_2\text{and}\mathcal{I}(t_{n+\frac{1}{2}})=e^{-\frac{1}{2}ħ}\mathcal{I}(t_n)+\stackrel{\text{Eqs.}\mathrm{\text{(23)}}\text{and}\mathrm{\text{(26)}}}{\overbrace{\mathcal{J}(t_{n+\frac{1}{2}}){|}_{u_{n+\frac{1}{2}}=u_{n+\frac{1}{2}}^{(b)}}}}\hfill \end{array}$$ (48f)

$$\begin{array}{cc}\hfill k_3& =ħS(t_{n+\frac{1}{2}},u_{n+\frac{1}{2}},\mathcal{I}(t_{n+\frac{1}{2}}))\hfill \end{array}$$ (48g)

$$\begin{array}{cc}\hfill u_{n+1}& =u_n+k_3\text{and}\mathcal{I}(t_{n+1})=e^{-ħ}\mathcal{I}(t_n)+\stackrel{\text{Eqs.}\mathrm{\text{(23)}}\text{and}\mathrm{\text{(26)}}}{\overbrace{\mathcal{J}(t_{n+1}){|}_{u_{n+1}=u_{n+1}}}}\hfill \end{array}$$ (48h)

$$\begin{array}{cc}\hfill k_4& =ħS(t_{n+1},u_{n+1},\mathcal{I}(t_{n+1}))\hfill \end{array}$$ (48i)

$$\begin{array}{cc}\hfill u(t_{n+1})& =u(t_n)+\frac{1}{6}(k_1+2k_2+2k_3+k_4).\hfill \end{array}$$ (48j)

In the forthcoming section, Sec. 7.4, the parameter $n_p$ represents the initial iteration setup, facilitating the accommodation of array locations situated to the left of the first data ($u(t_0)$). This setup mitigates potential complications arising from cell allocations with indices ≤0. We will utilize this RK4 scheme to investigate computational resolutions featuring changing step sizes, thereby incorporating unequally spaced data.

The differentiation and integration discretization schemes outlined in Sec. 3 are of semi-backward type. For instance, in the case of 4-point discretizations where $n_p\ge 3$, we select two randomly chosen, sufficiently small numbers in descending order, proceeding backward from $n_p$ as the point spaces. This approach ensures the smooth execution of the routine presented in Eq. (13), effectively avoiding divisions by zero, albeit the first three data points may experience slight perturbations.

5.2. Time-efficient adaptable stepsize RK45-Fehlberg

Here, we aim to introduce the Cash-Karp formulation of the Runge-Kutta 45 Fehlberg embedded algorithm. The adaptive Runge-Kutta 45 (RK45) Fehlberg method is a numerical technique utilized for solving ordinary differential equations (ODEs) or systems thereof. It represents an adaptive iteration of the classical Runge-Kutta-Fehlberg approach, which blends a fifth-order technique with a fourth-order one to gauge the local error at each computational step.

Here's a breakdown of its operation:

• —Initially, the method takes a step employing both a fourth-order Runge-Kutta (RK4) and a fifth-order Runge-Kutta (RK5) technique, thus having six stages.
• —The disparity between the solutions derived from RK4 and RK5 serves as the basis for estimating the local truncation error.
• —Depending on this calculated error and a specified tolerance set by the user, the step size for subsequent iterations is adjusted to ensure that the error remains within permissible limits.
• —If the error exceeds the defined threshold, the step size is decreased, and the process is iterated. Conversely, if the error is deemed sufficiently small, the step size may be increased to enhance computational efficiency.

By dynamically adjusting the step size based on the local error estimation, the RK45 Fehlberg method can achieve high accuracy while efficiently allocating computational resources where they are most needed.

This method is widely used in scientific computing and numerical analysis for its balance between accuracy and computational efficiency, particularly when dealing with ODEs where the behavior of the solution may change rapidly over certain intervals.

Worth mentioning that a skeleton of this RK45-Fehlberg scheme was shown in Sec. 5. Once more, it's worth noting that the instances presented in Sec. 2 involve convolution integration with a first derivative. Therefore, considering four discrete points with either equal or unequal spacing, as illustrated in Eq. (47), we opt for Eq. (23a), (23b), (23c), (23d) and Eq. (26). Consequently, enacting the time-efficient formula expanded in Sec. 4, we get

$$\begin{array}{cc}\hfill & \text{Given}{u}_{n-1},u_{n-2}.....u_0,n\ge n_p\text{and}\mathcal{I}(t_0)=0,\hfill \end{array}$$ (49a)

$$\begin{array}{cc}\hfill \mathcal{I}(t_n)& =e^{-ħ}\mathcal{I}(t_{n-1})+\mathcal{J}(t_n)\hfill \end{array}$$ (49b)

$$\begin{array}{cc}\hfill k_1& =ħS(t_n,u_n,\mathcal{I}(t_n))\hfill \end{array}$$ (49c)

$$\begin{array}{cc}\hfill u_{n+\frac{1}{5}}& =u_n+\frac{k_1}{5}\text{and}\mathcal{I}(t_{n+\frac{1}{5}})=e^{-\frac{ħ}{5}}\mathcal{I}(t_n)+\stackrel{\text{Eqs.}\mathrm{\text{(23)}}\text{and}\mathrm{\text{(26)}}}{\overbrace{\mathcal{J}(t_{n+\frac{1}{5}}){|}_{u_{n+\frac{1}{5}}}}}\hfill \end{array}$$ (49d)

$$\begin{array}{cc}\hfill k_2& =ħS(t_{n+\frac{1}{5}},u_{n+\frac{1}{5}},\mathcal{I}(t_{n+\frac{1}{5}}))\hfill \end{array}$$ (49e)

$$\begin{array}{cc}\hfill u_{n+\frac{3}{10}}& =u_n+\frac{3}{40}{k}_1+\frac{9}{40}{k}_2\text{and}\mathcal{I}(t_{n+\frac{1}{5}})=e^{-\frac{3}{10}ħ}\mathcal{I}(t_n)+\stackrel{\text{Eqs.}\mathrm{\text{(23)}}\text{and}\mathrm{\text{(26)}}}{\overbrace{\mathcal{J}(t_{n+\frac{3}{10}}){|}_{u_{n+\frac{3}{10}}}}}\hfill \end{array}$$ (49f)

$$\begin{array}{cc}\hfill k_3& =ħS(t_{n+\frac{3}{10}},u_{n+\frac{3}{10}},\mathcal{I}(t_{n+\frac{3}{10}}))\hfill \end{array}$$ (49g)

$$\begin{array}{cc}\hfill u_{n+\frac{3}{5}}& =u_n+\frac{3}{10}{k}_1-\frac{9}{10}{k}_2+\frac{6}{5}{k}_3\text{and}\mathcal{I}(t_{n+\frac{3}{5}})=e^{-\frac{3}{5}ħ}\mathcal{I}(t_n)+\stackrel{\text{Eqs.}\mathrm{\text{(23)}}\text{and}\mathrm{\text{(26)}}}{\overbrace{\mathcal{J}(t_{n+\frac{3}{5}}){|}_{u_{n+\frac{3}{5}}}}}\hfill \end{array}$$ (49h)

$$\begin{array}{cc}\hfill k_4& =ħS(t_{n+\frac{3}{5}},u_{n+\frac{3}{5}},\mathcal{I}(t_{n+\frac{3}{5}}))\hfill \end{array}$$ (49i)

$$\begin{array}{cc}\hfill u_{n+1}^a& =u_n-\frac{11}{54}{k}_1+\frac{5}{2}{k}_2-\frac{70}{27}{k}_3+\frac{35}{27}{k}_4\text{and}\mathcal{I}(t_{n+1})=e^{-ħ}\mathcal{I}(t_n)+\stackrel{\text{Eqs.}\mathrm{\text{(23)}}\text{and}\mathrm{\text{(26)}}}{\overbrace{\mathcal{J}(t_{n+1}){|}_{u_{n+1}^a}}}\hfill \end{array}$$ (49j)

$$\begin{array}{cc}\hfill k_5& =ħS(t_{n+1},u_{n+1},\mathcal{I}(t_{n+1}))\hfill \end{array}$$ (49k)

$$\begin{array}{cc}\hfill u_{n+\frac{7}{8}}& =u_n+\frac{1631}{55296}{k}_1+\frac{175}{512}{k}_2+\frac{575}{13824}{k}_3+\frac{44275}{110592}{k}_4+\frac{253}{4096}{k}_5\text{and}\begin{array}{cc}\mathcal{I}(t_{n+\frac{7}{8}})\hfill & =e^{-\frac{7}{8}ħ}\mathcal{I}(t_n)+\hfill \\ \hfill & \stackrel{\text{Eqs.}\mathrm{\text{(23)}}\text{and}\mathrm{\text{(26)}}}{\overbrace{\mathcal{J}(t_{n+\frac{7}{8}}){|}_{u_{n+\frac{7}{8}}}}}\hfill \end{array}\}\hfill \end{array}$$ (49l)

$$\begin{array}{cc}\hfill k_6& =ħS(t_{n+\frac{7}{8}},u_{n+\frac{7}{8}},\mathcal{I}(t_{n+\frac{7}{8}}))\hfill \end{array}$$ (49m)

$$\begin{array}{cc}\hfill u(t_{n+1})& =u(t_n)+(\frac{37}{378}{k}_1+\frac{250}{621}{k}_3+\frac{125}{594}{k}_4+\frac{512}{1771}{k}_6).\hfill \end{array}$$ (49n)

$$\begin{array}{cc}\hfill \tilde{u}(t_{n+1})& =u(t_n)+(\frac{2825}{27648}{k}_1+\frac{18575}{48384}{k}_3+\frac{13525}{55296}{k}_4+\frac{277}{14336}{k}_5+\frac{1}{4}{k}_6).\hfill \end{array}$$ (49o)

Here, $\bar{u}(t_{n+1})$ is RK4 ($u^4(t_{n+1})$), meanwhile $\tilde{u}(t_{n+1})$ is RK5 ($u^5(t_{n+1})$). Because of the backward far-reach of Eq. (47) and subsequently Eqs. (23a), (23b), (23c), (23d) and (26), $n_p\ge 3$.

The core of this scheme lies in the stepsize selection process. To elucidate this procedure, at iteration time $t_n$ using embedded integration formulas, in general we derive an m-th order estimation denoted by $u^{(m)}(t_{n+1})$, as well as an $(m+1)$-th order estimation denoted by $u^{(m+1)}(t_{n+1})$. Utilizing these two numerical approximations, the difference between them provides the instantaneous error, defined as:

$$\mathcal{E}=|u^{(m+1)}(t_{n+1})-u^{(m)}(t_{n+1})|,$$ (50)

similar to using Eqs. (49n) and (49o) in the RK45 scheme.

Defining ${\mathcal{E}}_{tol}$ as the error tolerance and considering that the scheme in Eq. (49a), (49b), (49c), (49d), (49e), (49f), (49g), (49h), (49i), (49j), (49k), (49l), (49m), (49n), (49o) encompasses both the 4th order ($p=4$) and the 5th order, we can establish that by defining ${\beta }_f$ as the safety factor with ${\beta }_f\approx 1$, the following relationship holds:

$${ħ}_{\text{new }}=\{\begin{array}{cc}{\beta }_fħ{\left(\frac{{\mathcal{E}}_{tol}}{\mathcal{E}}\right)}^{0.2},\hfill & {\mathcal{E}}_{tol}\ge \mathcal{E}\hfill \\ {\beta }_fħ{\left(\frac{{\mathcal{E}}_{tol}}{\mathcal{E}}\right)}^{0.25},\hfill & {\mathcal{E}}_{tol}<\mathcal{E}\hfill \end{array}$$ (51)

where ${ħ}_{\text{new }}$ is the recommended stepsize.

5.3. Subsidiaries to the TE-RK45-Fehlberg

Assessing the quality of the scheme here in terms of computational time and convergence order is not straightforward due to the potentially changing stepsize. However, we establish a general approach for evaluating these two aspects based on uniform steps.

5.3.1. Computational time for RK scheme

To illustrate the cost-effectiveness of our development, we analyze the iteration flow diagram depicted in Fig. 1 in Sec. 1 and conduct a basic analysis to quantify the computational costs involved. Since the time steps vary, a logical measure of the time performance here should consider a fixed number of iterations for separate runs. We illustrate this by examining the scenario where the step size remains constant throughout the iterations, which is significantly less intricate than analyzing variable step sizes.

Many studies [63], [64], [65] have reported the computational cost as $\mathcal{O}(N_t^2)$. However, for Volterra Integral and Integro-Differential equations, they have achieved a reduction in computational cost to $\mathcal{O}(N_t{(\log N_t)}^2)$ and, in some cases, even as low as $\mathcal{O}(N_t(\log N_t))$ using other cost-effective schemes for Volterra Integral equations.

It is important to note that the computational performance of each computer system may vary. Therefore, our focus is on capturing the time differences between different schemes without delving into specific computer specifications.

At a given time step $t_n$, we assume there are p algebraic evaluations per stage, each taking $\delta t_{rk}$ amount of time. Additionally, at this frozen time step, the convolution integral must perform $N_t$ integration blocks over the $N_t$ stencils. Hence, if each integration stencil entails q algebraic evaluations, each requiring $\delta t_{int}$ time, then the total elapsed time from $t_n$ to $t_{n+1}$ is given by:

$$T_{RK}^{(n)}\approx mp\delta t_{rk}+N_{t}mq\delta t_{int}.$$

Consequently, the total computational time up to $t_n$ is approximated as:

$$T_{RK}\approx mp\delta t_{rk}{N}_t+\frac{mq\delta t_{int}}{2}{N}_t^2.$$ (52)

In the absence of backstage integrations, as assumed by our hypothesis, the computational time for the Runge-Kutta (RK) scheme simplifies to:

$$T_{TE}\approx mp\delta t_{rk}{N}_t.$$ (53)

From these expressions, we calculate the increase in computation time ratio, denoted as the time effectiveness ratio $T{E}_r$:

$$T{E}_r=\frac{T_{RK}}{T_{TE}}\approx 1+\frac{q\delta t_{int}}{2p\delta t_{rk}}{N}_t.$$ (54)

Thus, if smaller time steps are required, particularly with high values of the maximum time, the backstage integration would incur significantly higher computational costs. Even worse is when the algorithm must contemplate suitable choices of step sizes to consider, especially in the case of variable step sizes, this would further escalate computational costs.

To test our naive relations that defend the cost-effectiveness here, we consider the following example of a VTIDE which could also be found in [52]:

$$\frac{du}{dt}=\frac{bu-u^2+d{e}^{-\xi t}}{1+2u}+\frac{1}{(1+2u)}\underset{0}{\overset{t}{\int }}{e}^{-(t-s)}[u^2+4u\frac{du}{ds}+2({\left(\frac{du}{ds}\right)}^2+u\frac{d^{2}u}{d{s}^2}\left)\right]ds,u(0)=0,$$ (55)

where $b,d,\mathrm{\&},\xi$ are real constants. To mitigate the additional cost associated with adaptive step size search, without loss of generality, we adopt a constant step size and apply a second-order implicit Runge-Kutta scheme ($R{K}_2$) thus with 2 stages to solve Eq. (55) numerically. We assess the time for three scenarios: 1) $R{K}_2^2$ with 2 backstage integrations, 2) $R{K}_2^1$ with 1 backstage integration, and 3) $R{K}_2^0=R{K}_2^{TE}$ with no backstage integrations, hence fully time-efficient. Furthermore, we employ Eq. (52) to fit the data for validation. This is depicted in Fig. 2, where it becomes evident that employing the time-efficient algorithm results in a computational cost of $\mathcal{O}(N_t)$, which outperforms other methods as discussed previously.

Figure 2.

Plots of computational time cost versus the number of iterations N_t for $R{K}_2^i$, illustrating the significant reduction in computational cost achieved with our TE-scheme. The corresponding point types represent the fitted lines described by $T_{RK}^i=a_i{N}_t+b_i{N}_t^2$. For the proposed time-efficient scheme, the blue lines and fitted blue circles validate that T_RK ≈ a₀N_t ≈ T_TE, as shown in Eq. (53).

We conclude here that during time iterations, it is customary to impose a maximum set time, $t_{max}$, dictated by the underlying physics. However, enforcing $t_{max}$ on an adaptive scheme, such as Eq. (49a), (49b), (49c), (49d), (49e), (49f), (49g), (49h), (49i), (49j), (49k), (49l), (49m), (49n), (49o), can lead to difficulties in predicting the value of N. Unfortunately, if the constraint in the algorithm, Eq. (51), suggests smaller step sizes, $N_t$ may become unmanageably large. Conversely, fixing $N_t$ on an adaptive scheme offers no assurance of reaching $t_{max}$ accurately.

5.3.2. Numerical error

The onset of calculating the order of convergence in a numerical scheme is the global truncation error. We let $u_{ex}(t)$ be the exact value of $u(t)$ and $u_{num}^{ħ}(t)$ be the numerical approximation for a given time step ħ. Then, for two different uniform time steps, ${ħ}_1,{ħ}_2$, we can write:

$$\begin{array}{c}\hfill u_{ex}(t)=u_{num}^{{ħ}_1}(t)+M{ħ}_1^p\end{array}$$ (56a)

$$\begin{array}{c}\hfill u_{ex}(t)=u_{num}^{{ħ}_2}(t)+M{ħ}_2^p,\end{array}$$ (56b)

where, the factor M is a number bound. Therefore, the error becomes

$$\begin{array}{c}\hfill {\mathcal{E}}_g({ħ}_1)=|u_{ex}(t)-u_{num}^{{ħ}_1}(t)|=M{ħ}_1^p\end{array}$$ (57a)

$$\begin{array}{c}\hfill {\mathcal{E}}_g({ħ}_2)=|u_{ex}(t)-u_{num}^{{ħ}_2}(t)|=M{ħ}_2^p.\end{array}$$ (57b)

The error calculated in Eq. (57a), (57b) includes error from the numerical scheme used, as well as the truncation error, which is the sum of the errors successively inherited from previous approximated data. Hence, we can refer to the error as a global truncation error. That is, at time $t_n$, we use a scheme to calculate $u(t_n)$ using, say $u(t_{n-1})$ having an error due to the scheme that it also inherited from $u(t_{n-2})$ and so forth.

By logarithmically transforming Eq. (57a) and Eq. (57b), and subsequently dividing them, we determine the order as follows:

$$p=\frac{\log \left(\right|u_{ex}(t)-u_{num}^{{ħ}_1}(t)\left|\right)-\log \left(\right|u_{ex}(t)-u_{num}^{{ħ}_2}(t)\left|\right)}{\log \left(\frac{{ħ}_2}{{ħ}_1}\right)}.$$ (58)

Typically, ${ħ}_1,{ħ}_2$ are chosen to have a factor of two, but in general, they can be of different ratios. However, the “log” must be consistent, having the same base of choice, although “base 10” or “base e” are common.

Unfortunately, order-of-convergence calculation from Eq. (58) is not suitable for an adaptive scheme, since the steps ħ are subject to changes. Evaluations of error behavior in schemes can also be done using the standard deviation values ${\mathcal{E}}_{rms}$, and this is also suitable for adaptive schemes. Thus, we define

$${\mathcal{E}}_{rms}=\sqrt{\sum _{i=1}^N\frac{{\left(\right|u_{ex}(t_i)-u_{num}(t_i)\left|\right)}^2}{N}}.$$ (59)

Generally, to construct the choice of the timestep driver we consider that

$$u(t_{n+1})=u_n+\stackrel{\mathcal{O}(p+1)}{\overbrace{C{ħ}^{p+1}+\mathcal{O}({ħ}^{p+2})}},$$ (60)

where C is a constant considered independent of the problem's data and discretization although this independence is not well understood for nonlinear problems. The third term is considered as the numerical difference ($\mathcal{E}$) between the p order and the $p+1$ order approximations schemes. It follows that

$$C{ħ}^{p+1}\approx \mathcal{E}.$$ (61)

So, if we aim for another global truncation error ${\mathcal{E}}_{tol}$ using a new stepsize ${ħ}_{new}$ such that $C{ħ}_{new}^{p+1}\approx {\mathcal{E}}_{tol}$, then a simple ratio analysis discards C and allows the new adaptive value of step size to be determined

$${ħ}_{new}=ħ{\left(\frac{{\mathcal{E}}_{tol}}{\mathcal{E}}\right)}^{\frac{1}{p+1}}.$$ (62)

Equation (62) was used earlier as the driver for the adaptability of the RK45F scheme Eq. (49a), (49b), (49c), (49d), (49e), (49f), (49g), (49h), (49i), (49j), (49k), (49l), (49m), (49n), (49o) presented in Eq. (51).

6. Validation, stability, comparison, and convergence analysis of numerical methods

Numerical techniques for solving VIDE often involve a combination of discretization methods for the differential part and quadrature methods for the integral part. The choice of method depends on the specific form and requirements of the VIDE, including considerations of stability, accuracy, and computational efficiency. The numerical frame of choice for our scheme is the Runge Kutta method. The time-efficient scheme outlined in Sec. 4 renders the standard RK scheme more time efficient, thus a cost advantage. Moreover, our scheme inherits many benefits from the RK scheme over other numerical techniques such as the Adomian Decomposition Method (ADM) and the spectral collocation methods (SCM) for Volterra integro-differential equations. Consequently, the theory developed here closely parallels the RK, though with better computational cost, relative ease of implementation, and interestingly inheriting similar stability analysis and error analysis. Hence, we omit these details as many references carry such analysis [53], [66], [67].

While our scheme enhances the regular RK scheme, the Runge-Kutta method generally offers distinct advantages, for example over SCM, particularly in the following enumerated scenarios:

• —Runge-Kutta methods handle non-smooth solutions and discontinuities better than spectral methods, which suffer from oscillations near discontinuities (Gibbs phenomena), highlighting its Adaptability to Non-Smooth Solutions.
• —Runge-Kutta methods can more easily accommodate irregular or complex domains, whereas spectral methods require structured grids or specific orthogonal basis functions, limiting their flexibility, thus Flexibility in Problem Domains
• —Runge-Kutta methods are generally simpler to implement and understand, making them more accessible for those new to numerical methods, as they do not require complex setups like basis functions or spectral coefficients (Ease of Implementation)
• —Runge-Kutta methods are more effective for stiff differential equations, managing widely varying time scales better with adaptive step-size control (Robustness in Stiff Problems).
• —For small-scale problems, the simplicity and lower overhead of Runge-Kutta methods can make them more computationally efficient compared to the setup costs of spectral collocation matrices (Computational Efficiency for Small Problems).

The choice between Runge-Kutta and spectral collocation methods depends on the problem's specific characteristics, desired accuracy, available computational resources, and user expertise. Benchmarking both methods is common to determine the best fit for the problem.

The choices of the examples VTIDE are such that they can be transformed into ODEs having no convolution integral, resulting in an analytical solution for comparison. However, in their original forms, they appear highly nonlinear. Hence, stability issues, convergence problems, and difficulties in choosing appropriate step sizes are expected.

Several authors have addressed these challenges. For example, Cartwright [53] and his colleagues discussed the connections between the dynamics of a differential equation and its stability. Spijker [68] explored stability and convergence in linear and nonlinear discretizations. Additionally, Sanzserna et al. [69] highlighted the reduction in the order of convergence in Runge-Kutta (RK) schemes.

To have an idea on this issue, we initially focused on obtaining informed estimates for step sizes through a post-step-size test. Using Example 2 from Eq. (83) as an illustration, which is rigorously explored in the upcoming section Sec. 7.2, we demonstrate here how we facilitated this examination by considering its reduced and linearized version given in Eq. (84). This analysis guided us in selecting appropriate step sizes. However, before diving into the analysis, we consider the model equation:

$$\frac{du}{dt}=\lambda u,u(t_0\ne 0),$$ (63)

an autonomous equation having RK coefficients A and b. In general, the stability function is given by

$$\varphi (z)=\frac{\det (\mathbf{I}-z\mathbf{A}+z\mathbf{e}{\mathbf{b}}^{\mathsf{T}})}{\det (\mathbf{I}-z\mathbf{A})},z=\lambda ħ.$$ (64)

Utilizing the RK2 algorithm extracted from Eq. (30) and denoting the dependent variable with a $″{\tilde{\phantom{a}}}^{″}$ to differentiate it from the previous iteration, we derive:

$$\tilde{u}(t_{n+1})=\tilde{u}(t_n)+\frac{ħ}{2}(k_1+k_2),$$ (65)

where

$$\begin{array}{cc}{k}_1=\hfill & b\tilde{u}(t_n)+d{e}^{-\xi t_n}\hfill \\ k_2=\hfill & b(\tilde{u}(t_n)+ħk_1)+d{e}^{-\xi (t_n+ħ)}\hfill \end{array}\}.$$ (66)

Taking ϕ as the stability function, and with the help of Taylor series expansion, we obtain:

$$\varphi (z)=\frac{d{\tilde{u}}_{n+1}}{d{\tilde{u}}_n}=1+bħ+\frac{{(bħ)}^2}{2}.$$ (67)

Consequently, one can pretend that to ensure numerical stability of the ODE in Eq. (83) we need to have $\varphi (z)\le 1$, though analyzed on Eq. (84), considering that we are working in real space. Consequently, for $b\in \mathbb{R}$, the choice of the stepsize ħ for Eq. (84) should satisfy;

$$ħ\le -\frac{2}{b}.$$ (68)

This restriction is informative in choosing stepsizes for Eq. (83).

6.1. On implicit schemes and stability of linear VTIDE

We aim to demonstrate how implicit schemes can be readily constructed, serving as another form of validation, thanks to the TE scheme presented here. This development should apply specifically to a general class of linear VTIDE of the form:

$$\frac{du}{dt}=F(t,u,z)=f(t,u)+\stackrel{z(t)}{\overbrace{\underset{0}{\overset{t}{\int }}G(t,s,u,\frac{du}{ds})ds}},G(t,s,u,\frac{du}{ds})=\mathcal{K}(t,s)g(t,s,u,\frac{du}{ds}).$$ (69)

Without loss of generality, we consider that when calculating $u_{n+1}$, discretization of both the convolution and the derivative utilize at most the $k+1$ enlarged neighboring stencils, $u_{n+2-j}$, where $j=1,\dots ,k+1$. We add that this choice should ensure that $u_{n+1}$ is part of the cluster. To demonstrate the procedure, let's examine the linear Volterra-type integro-differential equation (VTIDE) given by:

$$\frac{du}{dt}=\lambda u(t)+\beta e^{-t}+\stackrel{z(t)}{\overbrace{\underset{0}{\overset{t}{\int }}\underset{\varpi (t,s)}{\underbrace{e^{-(t-s)}(u(s)+\frac{du}{ds})}}ds}},u(0)=\gamma ,$$ (70)

where the constants λ, β, and γ are real numbers. The exact solution to this non-autonomous stiff problem, as described in Eq. (70), is:

$$u(t)=\frac{1}{\lambda +2}\left\{\right(\beta -\gamma \left)\right[1-e^{-(\lambda +2)t}]+\gamma (\lambda +2)\}{e}^{(\lambda +2)t}.$$ (71)

Thus, following our enlarged stencil, defining $z(t_n)$ as the convolution integral value at $t_n$, we obtain

$$z(t_{n+1})={\mathcal{I}}_{n+1}=e^{-ħ}z(t_n)+\stackrel{\mathcal{J}(t_{n+1})}{\overbrace{\sum _{j=1}^{k+1}{\kappa }_j{u}_{n+2-j}}},$$

where ${\kappa }_j$ are coefficients originating from integration and differentiation discretizations, whose choice of nodes and discretization orders dictate $\mathcal{J}(t_{n+1})$. However, because both $u_{n+1}$ and $u_n$ are required, we again develop ${\mathcal{I}}_n$ further to obtain:

$$z(t_{n+1})={\mathcal{I}}_{n+1}=e^{-2ħ}z(t_{n-1})+\sum _{j=1}^{k+1}{\alpha }_j{u}_{n+2-j}$$ (72)

where ${\alpha }_j$ are the new coefficients.

For instance, considering the data points $(u_n,u_{n+1})$, a first-order forward and backward derivative (Eq. (21)), and the second-order two-point trapezoidal integration shown in Eq. (24), can be systematically implemented in the time-efficient convolution scheme described in Sec. 4 to involve only $(u_{n-1},u_n,u_{n+1})$:

$$\frac{d{u}_{n+1}}{dt}=\frac{u_{n+1}-u_n}{ħ},\frac{d{u}_n}{dt}=\frac{u_{n+1}-u_n}{ħ},{\mathcal{I}}_2=\underset{t_n}{\overset{t_{n+1}}{\int }}u(t)ds=ħ\frac{(u_n+u_{n+1})}{2},$$ (73)

which are lower-order discretizations. To better illustrate the subsequent steps, we implement the discretizations outlined in Eq. (73), as follows:

$$z(t_{n+1})={\mathcal{I}}_{n+1}=e^{-2ħ}z(t_{n-1})+{\alpha }_3{u}_{n-1}+{\alpha }_2{u}_n+{\alpha }_1{u}_{n+1},$$ (74)

where

$${\alpha }_3=\frac{e^{-ħ}(ħ-1)-1}{2},{\alpha }_2=\frac{e^{-ħ}(2ħ+e^{-ħ})-1}{2},\text{and}{\alpha }_1=\frac{ħ+1+e^{-ħ}}{2}.$$ (75)

Or, we use

$$z(t_{n+1})=e^{-ħ}{\mathcal{I}}_n+\stackrel{{\kappa }_2}{\overbrace{\left[\frac{e^{-ħ}(ħ-1)-1}{2}\right]}}{u}_n+\stackrel{{\kappa }_1}{\overbrace{\left[\frac{ħ+e^{-ħ}+1}{2}\right]}}{u}_{n+1}.$$ (76)

Then, we utilize the Trapezoidal implicit method described by:

$$u_{n+1}=u_n+\frac{ħ}{2}[F(t_n,u_n,z(t_n))]+\frac{ħ}{2}[F(t_{n+1},u_{n+1},z(t_{n+1}))],$$ (77)

a lower-order scheme as well. Worth noting that this approach can be readily extended to RK schemes with multiple stages and employing higher-order integration and differentiation formulas as compared to those in Eq. (73). Consequently, we obtain:

$$u_{n+1}[1-\frac{ħ}{2}(\lambda +{\alpha }_1\left)\right]=u_{n-1}\left[\frac{ħ}{2}{\alpha }_3\right(1+e^{ħ})]+u_n[1+\frac{ħ}{2}(\lambda +{\alpha }_1+{\alpha }_2\left)\right]+\stackrel{\text{known at previous times}}{\overbrace{\frac{ħ}{2}(1+e^{-ħ})(e^{-ħ}{\mathcal{I}}_{n-1}+\beta e^{-t_n})}},$$ (78)

which is an implicit scheme. Given the value of γ, and noting that ${\mathcal{I}}_1=0$, $u_{n+1}$ is straightforwardly computed from Eq. (78). Subsequently, ${\mathcal{I}}_{n+1}$ is calculated using Eq. (74). It is interesting to note that as a result of Eq. (78), we derive the stability function as follows:

$$\varphi (ħ,\lambda )=\frac{[1+\frac{ħ}{2}(\lambda +{\alpha }_1+{\alpha }_2\left)\right]}{[1-\frac{ħ}{2}(\lambda +{\alpha }_1\left)\right]}<1\lambda ,{\alpha }_1,{\alpha }_2,\mathrm{\&}{\alpha }_3\in \mathbb{R}.$$ (79)

In Fig. 3, the stability region of the VTIDE in Eq. (70) based on the choice of discretization that led to Eq. (78) is depicted. This plot showcases the domain of λ and the corresponding domain of ħ that ensures stability, as dictated by the stability functions Eq. (67) and Eq. (79). The plots suggest, for instance, that for $\lambda \lesssim -4$, the TE scheme will remain stable for step sizes within the range $ħ\lesssim 8.0$, although bigger ħ values would course points and wouldn't resolve the proper trend and physics of the problem. However, for the model problem, stability is ensured for all values of $ħ\in {\mathbb{R}}^{+}$ provided $\lambda \in {\mathbb{R}}^{-}$.

Figure 3.

On the left, the stability region (shaded) from Eq. (67) for the model autonomous problem described in Eq. (63) is shown. On the right, the stability region using time-efficient discretizations based on the implicit method in Eq. (70), is illustrated as given by Eq. (79).

It is worth noting that a similar approach was effectively employed by Mahdi et al. [70], [71] in their investigation of numerical stability for general linear methods (GLMs) applied to diagonally implicit multistage integration methods (DIMSIMs) using a Gregory quadrature rule. Additionally, we observe that quadrature rules can also be applied here, instead of Lagrange polynomials, to approximate the summation in Eq. (72). In Fig. 4, the implicit data from Eqs. (74) and (78) are presented. Even with coarse step sizes ($ħ=0.01$ and 0.02), and despite using low-order schemes, the results in Fig. 4 align well with the exact solution, as evidenced by the error plots on the right. Additionally, the data analysis revealed that for $ħ\ge 0.1$ and $\lambda \gtrsim -1.0$, the scheme produced unstable data, deviating from the exact solution, which is consistent with the instability region depicted in Fig. 3.

Figure 4.

On the left, we have a graph of u versus t for the linear VTIDE described in Eq. (70), using the second-order discretized implicit trapezoidal method as detailed in Eqs. (74) and (78). On the right, the corresponding error plots are generated using Eq. (57a), (57b). The classical convergence order obtained is p = 2.

Despite reports indicating a reduced order of convergence for stiff problems of the general linear type, even those not involving a convolution (linear IVP), it is notable that in our case, despite utilizing first-order derivative schemes, the calculated order of convergence from the data in Fig. 4 averaged $p=2$. This suggests that our approach, possibly through specific algorithmic enhancements or the nature of the problem, achieves a higher convergence rate than typically expected under these conditions.

Consequently, the success of our scheme on RK scheme as compared to other schemes that have solved VIDE with convolution integrals, achieved through its cost-effectiveness, can be summarized by highlighting the following advantages:

• 1.Substantial Reduction in Computational Time: The localized integration scheme (Sec. 4) leads in a substantial reduction in computational time Sec. 5.3.1.
• 2.Straightforward Implementation: The scheme can be readily implemented in adaptive schemes using integration and differentiation discretizations for non-equidistant data points (Sec. 3).
• 3.Increased Accuracy: At each time step, our scheme focuses solely on neighboring well-correlated data points. This targeted approach enhances the likelihood of improving accuracy compared to full integration schemes that span from the initial time (0) to the final time (t), where disparate points with varying orders of magnitude may be encountered (Sec. 4).
• 4.Mitigation of Data Storage Issues: By sidestepping the necessity for a complete memory integral (backstage convolution integration), the scheme effectively addresses data storage challenges caused by the small array sizes required for the convolution integrals. This alleviates the burden of loading all pre-calculated data of the dependent variable into memory.
• 5.Accommodation of User-defined stepsizes: The scheme demonstrates its versatility by accommodating scenarios where varying step sizes are applied to it, all while maintaining its cost-effectiveness, as demonstrated in Sec. 7.4.

In the quest for additional validation of this scheme, the author also successfully tested the time-efficient scheme on a viscoelastic fluid mechanics problem, as found in a previous research article he participated in [7]. Although the results of that test are not presented here, mindful that other tests have been included serving that purpose, incorporating the former would be cumbersome.

Nonetheless, the test involved a spatio-temporal fluid mechanics problem dealing with a viscoelastic fluid driven by surface tension, whose governing equation is presented therein in Eqs. (2.23) and (2.24), a first-order space and time-dependent VTIDE.

Accordingly, the time-efficient convolution integral scheme developed here was used to analyze their convolution in Eq. (4.1), crucial in numerically solving the governing problems therein. Not only did the results match perfectly, but the reduction in computation time was also incredibly substantial, especially considering that the problem in [7] is a spatio-temporal problem that eventually involves spectral decomposition of the spatial variables, as seen in their Eqs. (2.24) and (3.1).

The significant computational cost of the initial code that generated the plots in [7] motivated the development of the time-efficient scheme here, and the author is successfully using this time-efficient scheme to develop time-efficient extensions of that work therein.

7. Corroboration of adaptive TE-VTIDE using examples

In this section, we aim to validate the time-efficient scheme developed using the Runge-Kutta algorithm by numerically solving four examples of VTIDEs presented in Sec. 2. These examples utilize non-uniform, adaptive time step sizes. To achieve this, we will provide an initial time step and assign ${\mathcal{E}}_{tol}$ and ${\beta }_f$, as shown in Sections 7.1 to 7.3. Additionally, we will define a trend in the step sizes and systematically provide them to the RK4 routine, which must intelligently accommodate changing steps. This is the focus of Sec. 7.4.

The version of the adaptive routine adopted from Eq. (51) and used here is summarized and outlined in Algorithm 1.

According to the detailed description in Algorithm 1, the inputs to the algorithm are the tolerance ${\mathcal{E}}_{tol}$, the required error estimate $\mathcal{E}$, the proposed new step size ${ħ}_n$, a tiny number ϵ, the maximum time $t_{max}$, and two pre-calculated values of different orders, $u_5$ and $u_6$. Therefore, the parameters ϵ, ${\mathcal{E}}_{tol}$, ${ħ}_0$, and $t_{max}$ are supplied to the algorithm.

If $\mathcal{R}$ is less than or equal to the tolerance ${\mathcal{E}}_{tol}$, the step is accepted, the value of $u_5$ is retained, and the routine calculates the new time step ${ħ}_{new}$ as indicated in line 7. Conversely, if $\mathcal{R}$ exceeds ${\mathcal{E}}_{tol}$, the step is rejected, and the routine must determine an appropriate step size by recalculating it in line 11. This version of adaptive stepsize control is widely used, and various authors have implemented different versions of this algorithm [72], [73], [74], [75], [76].

7.1. Validation of the TE-RK45F using Example 1

Here, we employ Example 1 (Eq. (6)) to assess the TE-RK45F-Cash Karp adaptive scheme outlined in Eq. (49a), (49b), (49c), (49d), (49e), (49f), (49g), (49h), (49i), (49j), (49k), (49l), (49m), (49n), (49o). To efficiently corroborate our TE-RK45F scheme, we also seek its exact solution. To better utilize the schemes outlined in Eq. (49a), (49b), (49c), (49d), (49e), (49f), (49g), (49h), (49i), (49j), (49k), (49l), (49m), (49n), (49o) and Sec. 4 and identify corresponding expressions, we rewrite the example presented in Eq. (6) as follows:

$$\frac{du}{dt}=\stackrel{{\mathcal{S}}_1(t,u(t),\mathcal{I}(u,\frac{du}{dt}))}{\overbrace{u(t)\{\alpha +\beta [\gamma e^{-\mathcal{W}t}+\underset{0}{\overset{t}{\int }}{e}^{-\mathcal{W}(t-s)}\underset{{\mathcal{F}}_1(u,\frac{du}{ds})}{\underbrace{(\mathcal{W}u(s)+\frac{du}{ds})}}ds\left]\right\}}},u(0)=\gamma$$ (80)

where again $\alpha ,\beta ,\gamma ,\mathrm{\&}\mathcal{W}$ are constants. Consequently, ${\mathcal{F}}_1(u,\frac{du}{ds})$ corresponds to $g(s)$ in Sec. 4 meanwhile ${\mathcal{S}}_1$ is the corresponding source term here.

However, although the numerically solved form of this equation is initially complex (Eq. (6)), a notable simplification emerges upon closer examination. Specifically, when $\mathcal{W}$ is set to 1 and the Leibniz integral rule is applied judiciously, the Eq. (80) simplifies significantly. Consequently, it reduces to:

$$\frac{du}{dt}=\alpha u(t)(1+\frac{\beta }{\alpha }u(t)),u(0)=u_0=\gamma .$$ (81)

We assert that this particular formulation is mathematically and numerically less intricate compared to Eq. (80) as solved numerically in this context. As referenced in Eq. (81), the simplified logistic growth equation is widely recognized as the logistic growth Ordinary Differential Equation. This model is fundamental in population ecology, describing the development of a population within a constrained environment. The equation incorporates the concept of the environment's carrying capacity, the maximum population size that can be sustained indefinitely.

The logistic growth equation has diverse applications across various fields. In ecology, it is used to model the growth of populations within ecosystems. In sociology, it predicts the adoption rates of new technologies. In economics, it describes the diffusion of products in markets. In medicine, it models the growth of bacteria or tumor cells under resource limitations. Researchers, such as Suryaningrat et al. [77], have effectively addressed this simplified variant numerically, and their contributions are acknowledged herein.

Utilizing the partial fractions technique, Eq. (81) is integrated to yield:

$$u(t)=\frac{\alpha \gamma e^{\alpha t}}{[\alpha -\beta \gamma (e^{\alpha t}-1\left)\right]},u(0)=\gamma .$$ (82)

We set $\alpha =-0.5$, $\beta =0.02$, and $\gamma =10$ for our numerical solution. Remarkably, these values align our expression in Eq. (81) exactly with that of Suryaningrat et al. [77], enabling us to also compare our results with theirs directly. For two different time steps, $ħ=0.01$ and 0.002, we respectively set ${\mathcal{E}}_{tol}=1E^{-5}$ and $5E^{-6}$, along with ${\beta }_f=0.95$ and 0.80. Detailed computation parameters, including adaptive parameters, are listed in Table 2. Consequently, we generate TE-RK45F-Cash-Karp data, presented in Fig. 5.

Table 2.

Numerical data using RK45F on Problem 1.

ħ	ħmin	ħmax	ħavg	βf	${\mathcal{E}}_{tol}$	${\mathcal{E}}_{min}$	${\mathcal{E}}_{max}$	${\mathcal{E}}_{rms}$
0.010	2.0059E-38	1.0982E+00	9.8071E-02	0.95	1.0E-05	0.0000E+00	1.3750E-01	3.1812E-02
0.002	4.4118E-26	5.8946E-01	1.9795E-02	0.80	5.0E-06	0.0000E+00	2.4177E-02	3.6743E-03

Figure 5.

Left figure: Plots using the TE-RK45F algorithm on the non-linear VTIDE Eq. (6) or Eq. (80) and its analytical solution Eq. (82). The time-efficient TE-RK45F algorithm described in Eq. (49a), (49b), (49c), (49d), (49e), (49f), (49g), (49h), (49i), (49j), (49k), (49l), (49m), (49n), (49o) is employed, supported by Eqs. (41) and (42). Additionally, the flexible differentiation and integration schemes in Eqs. (23a), (23b), (23c), (23d) and (26) are implemented. Parameters used are α = −0.5, β = 0.02, and γ = 10. The plots are generated with initial time step sizes of Δt₁ = 0.01 for the red star points and Δt₂ = 0.002 for the blue square points, alongside the exact solutions shown as black dash-dots. Further details are available in Table 2.

In Fig. 5(a), it is evident that the TE-RK45F scheme closely matches the exact solution, a conclusion further substantiated by the error plots in Fig. 5(b). Furthermore, Fig. 5(c) illustrates the various time steps generated by the adaptive algorithm as described in Algorithm 1. Finally, Fig. 5(d) presents the corresponding errors, as defined in Eq. (57a), (57b), across the range of utilized step sizes.

7.2. Validation of the TE-RK45F using Example 2

Here, we apply our TE-RK45F-Cash Karp adaptive scheme, as outlined in Eq. (49a), (49b), (49c), (49d), (49e), (49f), (49g), (49h), (49i), (49j), (49k), (49l), (49m), (49n), (49o), to compute Example 2 described in Eq. (8). For similar reasons provided for Eq. (82), example 2 is redefined as follows:

$$\frac{du}{dt}=\stackrel{{\mathcal{S}}_2(t,u(t),\mathcal{I}(u,\frac{du}{dt}))}{\overbrace{bu(t)-u{(t)}^2+d{e}^{-\xi t}+\underset{0}{\overset{t}{\int }}{e}^{-(t-s)}\underset{{\mathcal{F}}_2(u,\frac{du}{dt})}{\underbrace{u(s)(u(s)+2\frac{du}{ds})}}ds}},u(0)=0,$$ (83)

where ${\mathcal{F}}_2(u,\frac{du}{ds})$ corresponds to $g(s)$ in Sec. 4 and ${\mathcal{S}}_2$ the source term.

In addition to featuring a derivative, Eq. (83) is nonlinear both within and outside of the convolution. This complexity could potentially make its computation quite involved, as compared to example 1. Nevertheless, Eq. (83) is remarkably reduced to an ODE given by

$$\frac{du}{dt}=bu+d{e}^{-\xi t},u(0)=0.$$ (84)

Although it is straightforward to verify that Eq. (84) belongs to the class of equations described in Eq. (83), proving this assertion follows a similar approach to the investigation undertaken by Azese [52], where the coefficients of Eq. (83) satisfy the conditions outlined therein. Using the integration factor technique, the solution to the linear first-order ODE in Eq. (84) is straightforward, yielding:

$$u(t)=\frac{d}{b+\xi }[1-e^{-(\xi +b)t}]e^{(bt)}.$$ (85)

This method leverages the properties of the integration factor to simplify the equation, leading to an efficient solution process. Equation (85) which can be conceptualized as a skewed Gaussian distribution, furnishes the exact solution and is subsequently utilized to evaluate the performance of our computational scheme on the complex form given by example 2.

Thus, we utilize the TE-RK45-Fehlberg scheme developed here to compute the solution for Eq. (83). This involves incorporating Eq. (85), Eq. (49a), (49b), (49c), (49d), (49e), (49f), (49g), (49h), (49i), (49j), (49k), (49l), (49m), (49n), (49o), Eq. (23a), (23b), (23c), (23d), Eq. (26), Eq. (41), and Eq. (42), taking advantage of the inter-point distances defined in Eq. (13).

The parameters employed in the TE-RK45-Fehlberg scheme are outlined in Table 3. With initial stepsizes of $ħ=0.05$ and 0.005, we generate two sets of data which are visualized in Fig. 6 alongside the exact solution.

Table 3.

Numerical data using RK45F on Problem 2.

ħ	ħmin	ħmax	ħavg	βf	${\mathcal{E}}_{tol}$	${\mathcal{E}}_{min}$	${\mathcal{E}}_{max}$	${\mathcal{E}}_{rms}$
0.050	1.2021E-02	7.7885E+00	6.8038E-01	0.85	1.0E-04	5.3965E-06	9.6607E-04	4.0173E-04
0.005	9.7801E-04	4.1454E+00	8.8802E-02	0.80	1.0E-05	5.3849E-08	9.4506E-05	2.1743E-05

Figure 6.

Similar to Fig. 5, where the non-linear VTIDE Eqs. (8) and (83) are represented, this exploration includes the following expressions: Eq. (85), Eq. (49a), (49b), (49c), (49d), (49e), (49f), (49g), (49h), (49i), (49j), (49k), (49l), (49m), (49n), (49o), Eqs. (23a), (23b), (23c), (23d) and (26), and Eqs. (41) and (42). Initial time step-sizes are set to Δt₁ = 0.05 for the red star points and Δt₂ = 0.005 for the blue square points, alongside the exact solutions depicted by black dash-dots. Additional data can be found in Table 3.

The plots presented in Fig. 6 demonstrate the effectiveness of our TE-RK45-Fehlberg approach, coupled with the 4-point integration and differentiation discretization. Interestingly, they exhibit a remarkable agreement with the exact graph, with only minor errors, as illustrated in the subfigure on the right, thus validating our time-efficient hypothesis.

7.3. Validation of the TE-RK45F using Example 3

In computing Eq. (10a), (10b), we employ the TE-RK45-Fehlberg algorithm, which addresses a first-order non-linear Volterra-type integro-differential equation previously discussed by Ali Filiz [14]. This allows for a comparison between our scheme and the results obtained by Filiz [14]. It is noteworthy that, although the example is nonlinear, it lacks derivatives within the convolution term, making it relatively less complex than the previous examples. Nonetheless, for the sake of clarity and consistency, and similar reasons provided for Eq. (82), we rewrite Example 3 as follows:

$$\frac{du}{dt}=\stackrel{{\mathcal{S}}_3(t,u(t),\mathcal{I}(u,\frac{du}{dt}))}{\overbrace{u(t)(\alpha -\beta \underset{0}{\overset{t}{\int }}{e}^{-\mathcal{W}(t-s)}\underset{{\mathcal{F}}_3(u,\frac{du}{dt})}{\underbrace{u(s)}}ds)}},u(0)=u_0,$$ (86)

where, here also, $u(t)$ is the temporal dependent variable, and $\alpha ,\beta ,\mathrm{\&}\mathcal{W}$ are constant parameters of choice. Moreover, we consider $u_0$ as the initial velocity at time $t=0$. In the equation, ${\mathcal{F}}_2(u,\frac{du}{ds})$ corresponds to $g(s)$ in Sec. 4, while ${\mathcal{S}}_2$ represents the source term.

As the first step, our objective is to derive an analytical solution. For the trivial case where $\mathcal{W}=0$, indicating a unity kernel, we can readily obtain an exact solution. To facilitate this, we introduce a new variable given by

$$w(t)=\underset{0}{\overset{t}{\int }}u(s)ds,$$

where we can infer that

$$\frac{dw}{dt}=u(t).$$

With these considerations, a modified logistics equation [78], [79] is obtained as follows:

$$\frac{dw}{dt}+\frac{\alpha }{2}w{(t)}^2-\beta w(t)=u_0,$$ (87)

which is a first-order non-linear ODE. With the aid of partial fractions and solutions to quadratic equations, we readily derive the solution in the following form:

$$w(t)=\frac{ab(e^{\psi t}-1)}{a+b{e}^{\psi t}}$$ (88)

where a, b, and ψ are constants calculated from the known values of α, β, and $u_0$ in the original problem described in Eq. (86). Specifically, a and b are determined as follows:

$$a-b=\frac{2\alpha }{\beta },\text{and}ab=\frac{2u_0}{\beta },$$ (89)

where ψ can be subsequently calculated from:

$$\psi =\frac{\beta }{2}(a+b).$$ (90)

Eventually, we obtain

$$u(t)=\frac{u_0{(a+b)}^2{e}^{\psi t}}{{(a+b{e}^{\psi t})}^2}.$$ (91)

To test our scheme here, we recall that $\mathcal{W}=0$ and elect $\alpha =2$ and $\beta =2$, and from Eq. (89) we obtain

$$a=1-\sqrt{1.1},b=-1-\sqrt{1.1},$$

which is eventually used in evaluating ψ.

With the assistance of Eq. (49a), (49b), (49c), (49d), (49e), (49f), (49g), (49h), (49i), (49j), (49k), (49l), (49m), (49n), (49o), Eq. (23a), (23b), (23c), (23d), Eq. (26), Eq. (41), Eq. (42), and the inter-point distance defined in Eq. (13), we utilize the TE-RK45-Fehlberg scheme to numerically solve Eq. (86) and generate exact data from Eq. (91) in the process. The resulting output is presented in Fig. 7.

Figure 7.

Similar to Fig. 5, where the non-linear VTIDE Eqs. (10a), (10b) and (86) are explored, we utilized the following expressions: Eq. (91); Eq. (49a), (49b), (49c), (49d), (49e), (49f), (49g), (49h), (49i), (49j), (49k), (49l), (49m), (49n), (49o); Eqs. (23a), (23b), (23c), (23d) and (26); and Eqs. (41) and (42). Initial time step-sizes were set to Δt₁ = 0.05 for the red star points and Δt₂ = 0.001 for the blue square points, compared with the exact solutions shown as black dash-dots. Additional data can be found in Table 4.

The left subfigure in Fig. 7 confirms that the plots from the exact solution, including two from our numerical scheme generated for two initial stepsizes, are similar. The slight difference in the numerical data from our schemes is depicted by their respective deviations from the exact data presented on the right subfigure of Fig. 7 (also see Table 4 for additional data). Accordingly, one can confidently confirm that our TE-RK45-Fehlberg and its discretization constituents work to agreeable precisions. Consequently, we can conclude here that our time-efficient supposition is corroborated.

Table 4.

Numerical data using RK45F on Problem 3.

ħ	ħmin	ħmax	ħavg	βf	${\mathcal{E}}_{tol}$	${\mathcal{E}}_{min}$	${\mathcal{E}}_{max}$	${\mathcal{E}}_{rms}$
0.050	1.2211E-56	2.6552E-01	6.9674E-02	0.85	1.0E-07	0.0000E+00	2.3974E-05	8.5616E-06
0.001	1.5725E-26	1.6634E-01	5.2935E-02	0.95	1.0E-08	0.0000E+00	6.2666E-06	2.5080E-06

7.4. Parabolic-varying stepsize on TE-RK4 for Example 2

This subsection also aims to validate our hypothesis presented in Sec. 4. Unlike previous tests, this unique assessment examines the capability of our scheme under the time-efficient hypothesis. Specifically, it evaluates how well the underlying Runge-Kutta numerical scheme performs when using controlled step sizes determined by the user, thereby completely avoiding the adaptive step size scenario outlined in Eq. (51). We apply this approach to the most complex of all the examples, VTIDE, specifically Eq. (8), which is revisited in Eq. (83). During this process, we will also compare the accuracy with the exact solution provided in Eq. (85).

The user-defined step-size functions can take any form and can be designed to target specific time intervals during runtime by providing stepsizes of a given order of magnitude. In this particular example, we choose parabolically varying stepsize functions with the maximum step size either at the center or at the ends.

Following this parabolic example, let ${ħ}_{min}$ and ${ħ}_{max}$ represent the minimum and maximum step sizes, respectively, and $t_{min}$ and $t_{max}$ denote the corresponding extrema of the independent time variable. First, we define the stretching factor, s, such that

$$s=4\frac{{ħ}_{max}-{ħ}_{min}}{{(t_{max}-t_{min})}^2}.$$ (92)

And then, the parabolic-stepsizes profile is given by

$$ħ(t)={ħ}_{max}+s(t-t_{min})(t-t_{max}),$$ (93)

which yields an average stepsize of

$${ħ}_{avg}={ħ}_{max}-\frac{2}{3}({ħ}_{max}-{ħ}_{min}).$$ (94)

As previously mentioned, the parabolic-stepsizes can be characterized by the following two trends:

• (1)Trend 1: maximum step size, (${ħ}_{start}={ħ}_{max}$), with its minimum at the center, $t=0.5(t_{max}+t_{min})$, or
• (2)Trend 2: a minimum step size, (${ħ}_{start}={ħ}_{min}$), having a maximum at $t=0.5(t_{max}+t_{min})$.

To emphasize the significance of user-defined trends, we also generate constant step-size data for each trend, using their respective step-size averages ($\mathrm{\Delta }{t}_{avg}$) as the constant step sizes, which are defined as follows:

$${ħ}_{avg}=\frac{1}{t_{max}-t_{min}}\underset{t_{min}}{\overset{t_{max}}{\int }}ħ(s)ds.$$ (95)

Using ${ħ}_{\text{min}}=0.001$, ${ħ}_{\text{max}}=0.1$, $t_{\text{min}}=0$, and $t_{\text{max}}=15$, we generate data for Eq. (8) utilizing the Flex-RK4 algorithm described in Eq. (48a), (48b), (48c), (48d), (48e), (48f), (48g), (48h), (48i), (48j). This involves implementing Eq. (23a), (23b), (23c), (23d), Eq. (26), Eq. (41), and Eq. (42) with a step-size generator function as detailed in Eq. (93) and assisted by Eq. (13). The results are presented in Fig. 8.

Figure 8.

The plots for the non-linear VTIDE Eq. (83) and Eq. (85), where the flex-RK4 routine Eq. (48a), (48b), (48c), (48d), (48e), (48f), (48g), (48h), (48i), (48j) is used, implementing the following components: Eq. (23a), (23b), (23c), (23d), Eq. (26), Eq. (41), and Eq. (42), and utilizing the step-size generator function Eq. (93) with the aid of Eq. (13). The additional data for the root mean square error are as follows: ${\mathcal{E}}_{rmse}=1.89486E^{-02},1.48743E^{-02},2.14202E^{-02},4.39527E^{-02}$ for the respective plots in Fig. 8(c).

In Fig. 8(a), the profiles of the parabolic step sizes for the two trends are displayed. Subsequently, these distinct trends and their respective averages are compared with the exact solution in Fig. 8(b). Meanwhile, Fig. 8(c) illustrates their respective errors.

Notably, in Fig. 8, the changing step size defined following the function in Eq. (93) approximates the exact solution more accurately than their respective constant step sizes, ${ħ}_{\text{avg}}$, calculated from Eq. (94), as revealed in Fig. 8(b). Furthermore, this assertion is confirmed in Fig. 8(c), showcasing the respective deviations from the exact solution—highlighting the effectiveness of our customized parabolic step sizes changing function.

Consequently, we can confidently conclude that not only does our Flex-RK4 scheme perform effectively, but it also has the capability of accommodating user-defined step sizes and adapting accordingly. Thus, a user-defined step size function can be formulated to target specific time frames during runtime, supplying predefined smaller or larger step sizes accordingly.

8. Concluding remarks

In this study, we developed and employed a rapid integration-type adaptive scheme for solving linear and nonlinear Volterra-type Integro-Differential Equations, which incorporate convoluted integrals with derivatives and feature a specific class of exponential kernels (Eq. (2)). We utilized the adaptive Runge-Kutta (RK) numerical algorithm, specifically the RK45-Fehlberg scheme. To demonstrate the effectiveness of our approach, we proposed three VTIDE examples of varying complexity in Sec. 2.

In Sec. 3, we devised numerical discretizations that employ flexible differentiation and integration techniques, capable of handling unequally spaced data points. Using the method outlined in Eq. (13), we systematically provided the routine (integration, differentiation, RK) with these data points, achieving highly reliable solutions.

Sec. 4 elucidates the core of our rapid integration scheme. We reformulated the convolution integral to shift the costly algebraic evaluations typically performed at each time iteration to a background process within the main RK scheme. This “backstage” process, denoted as Convolution Integral: BACKSTAGE in Fig. 1, was brought to the forefront as Main $R{K}_m$-routine: FRONTSTAGE (Fig. 1), substantially reducing the RK program's runtime.

In Sec. 5, we demonstrated how unequally spaced data points could be integrated into a four-point integration and differentiation scheme. This scheme was then used to construct an adaptive Runge-Kutta 45 Fehlberg routine, utilizing the Cash-Karp algorithm. We also highlighted the step selection driver in Eq. (51) and formulated an expression to quantify the computational time-cost for a general uniform step size routine (Eqs. (52) and (53)). Our computational cost analysis was corroborated on a non-linear VTIDE (Eq. (55)), where we also introduced error analysis.

Sec. 6 provided a comprehensive discussion on the validation, stability, and convergence analysis of our time-efficient integration maneuver within a Runge-Kutta platform. We compared the advantages of our time-efficient scheme against other numerical solving techniques for VIDEs, such as ADM and SCM. Using a linear VTIDE as an example (Eq. (70)), we demonstrated the construction of a simple yet efficient implicit scheme (Eq. (78)), tested and plotted in Fig. 4, which corroborated our analysis. We also derived stability functions (Eq. (79)) that highlight stability regions and inform on step sizes used to generate plots for implicit schemes and the exact solution (Fig. 3).

Finally, in Sec. 7, we assessed the RK45-Fehlberg scheme developed in Sec. 5 on the three example problems outlined in Sec. 2. Sec. 7.4 introduced a user-defined parabolic step size generator, which was used to feed the RK4 scheme (Eq. (48a), (48b), (48c), (48d), (48e), (48f), (48g), (48h), (48i), (48j)), providing a flexible step-size scheme. For all examples, we generated plots demonstrating the proximity of our numerical schemes to the exact values (Figs.(5) to (8)), accompanied by their corresponding error evolution graphs.

These results validate our convolution integration approach and its implementation in RK schemes, particularly the RK adaptive scheme and flexible step-size scheme. The VTIDEs proposed in Sections 2.1 to 2.3 exhibit uniqueness, allowing transformation into Ordinary Differential Equations with analytical solutions—a rare characteristic for nonlinear VIDEs. These VTIDEs provide a platform for studying numerical instabilities and step-size behavior.

Future work could extend our scheme to adaptive RK23 schemes and implement our fast-RK adaptive scheme on spatio-temporal VTIDEs, considering spatial discretization. Additionally, using quadrature discretizations could further reduce errors.

CRediT authorship contribution statement

Martin Ndi Azese: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability statement

The data that support the findings of this study are available within the article.