Analysis of nonlinear Burgers equation with time fractional Atangana-Baleanu-Caputo derivative

Abstract

This paper demonstrates, a numerical method to solve the one and two dimensional Burgers' equation involving time fractional Atangana-Baleanu Caputo $(\mathcal{ABC})$ derivative with a non-singular kernel. The numerical stratagem consists of a quadrature rule for time fractional $(\mathcal{ABC})$ derivative along with Haar wavelet (HW) approximations of one and two dimensional problems. The key feature of the scheme is to reduce fractional problems to the set of linear equations via collocation procedure. Solving the system gives the approximate solution of the given problem. To verify the effectiveness of the developed method five numerical examples are considered. Besides this, the obtained simulations are compared with some published work and identified that proposed technique is better. Moreover, computationally the convergence rate in spatiotemporal directions is presented which shows order two convergence. The stability of the proposed scheme is also described via Lax-Richtmyer criterion. From simulations it is obvious that the scheme is quite useful for the time fractional problems.

1. Introduction

Fractional calculus (FC) remains an active area of research from the past few decades. It generalizes the concept of integrals and derivatives to an arbitrary order (fractional order), which is hard to describe in classical calculus. The recent growing attraction of researchers in this particular area reveals that it has distinctive applications in the disparate sites of science and engineering disciplines. Some recent papers which involve fractional formulations include, anomalous heat conduction and quantum mechanics processes [1], [2], [3], diffusion processes [4], [5], [6], Stokes problem [7], financial models [8], stochastic processes [9], [10], infiltration phenomena [11], biological models [12], and many others for which we refer to. The fundamental results which are devoted to FC relative to classical calculus is the long and short term memories. These results can be distinguished via global and local fractional derivatives.

In this work, we will examine the numerical solutions of the following global time fractional Burgers' equation (TFBE) involving Atangana-Baleanu-Caputo $(\mathcal{ABC})$ operator [13]:

$${}_0{}^{\mathcal{ABC}}\mathcal{D}_t^{\alpha }\mathcal{C}(\mathbb{X},t)+\mathcal{C}(\mathbb{X},t)\mathrm{\nabla }\mathcal{C}(\mathbb{X},t)-\varrho {\mathrm{\nabla }}^2\mathcal{C}(\mathbb{X},t)=q(\mathbb{X},t),\mathbb{X}\in \Omega \subset {\mathcal{R}}^{\mathbf{s}},\mathbf{s}=1,2,t\in [t_0,T],0<\alpha <1,$$ (1.1)

where $\mathbb{X}=\xi$ for $s=1$ and $\mathbb{X}=(\xi ,\zeta )$ for $s=2$, ${}_0{}^{\mathcal{ABC}}\mathcal{D}_t^{\alpha }$ is the time-fractional $\mathcal{ABC}$ operator, ∇ is the gradient operator, ϱ represents the kinematic viscosity, $q(\mathbb{X},t)$, is the smooth source function, and $\mathcal{C}(\mathbb{X},t)$ is the unknown function. The associated conditions are prescribed below:

$$\mathcal{C}(\mathbb{X},t_0)=\epsilon (\mathbb{X}),\mathbb{X}\in \Omega ,\mathcal{C}(\mathbb{X},t)=\mu (t),\mathbb{X}\in \partial \mathrm{\Omega },t\ge 0,$$

where Ω and ∂Ω indicate the solution domain and its boundary, respectively.

The TFBE is like a sub-diffusion convection model which is ubiquitous in different research areas. For instance, it describes the weak shock propagation, turbulence phenomena, unidirectional propagation of acoustic waves, etc. [14], [15], [16]. In previous work, Eq. (1.1) has been investigated by several authors with integer and time fractional derivatives. For instance, Mukundan and Awasthi suggested an implicit technique [17]. Shao et al. [18] proposed local discontinuous Galerkin finite element method. Guo [19] and his co-authors implemented fifth order finite volume technique. Quartic b-spline collocation procedure was addressed in [20]. Besides these, several other methods like, differential quadrature method [21], Lagrange polynomials based method [22], radial basis functions (RBFs) based strategies [23] and some wavelet numerical methods [24], [25], [26], have been studied in literature.

In the recent past, wavelet numerical techniques became fabulous in numerical computations. These methods have pretty good features like regularity, supporting width, symmetry, vanishing moments, and orthogonality. Broadly, wavelet transforms are divided into continuous and discrete forms. Poisson wavelet, Meyer wavelet, Shannon wavelet etc. are the examples of continuous transforms while discrete wavelet transforms include Haar wavelet (HW), the dual-tree complex wavelet etc. Amongst the discrete wavelets, HW attained special attraction, because they are composed of a pair of piecewise constant mathematical functions which are easy to analyze. Few finest aspects of HW include normalization, orthogonality and the existence of the closed form expressions of their integrals. The foundation of HW started from Haar functions which were introduced by Alfréd Haar in the 1990s. Initially, HW based numerical strategies were addressed by Chen and Hsiao [27] in 1997s. Chen and Hsiao proposed HW numerical scheme for the solutions of ordinary differential equations (ODEs), via approximating the highest order derivative with truncated Haar series. Later on, Lepik contributed in this direction and advised several numerical strategies [28]. In further studies, HW technique has been used in astrophysical Lane-Emden equations [29], biharmonic and Poisson equations [30], fluid flow layer problems [31] and telegraph equations [32]. To further explore the role of HW in different numerical computations, the interested readers are referred to see [33], [34], [35], [36].

In general, the analytical techniques are difficult to apply for the fractional models and require rigorous mathematical analysis. In order to cope with these challenges, numerical techniques are preferable. This work aims, to propose a new numerical strategy to solve the specified problem by coupling HW with quadrature rule for the $\mathcal{ABC}$ derivative. Based on our analysis, the proposed idea has not described in existing work for the solutions of the aforementioned models in one and two space dimensions. This would be a fruitful contribution in this particular area of research and will provide an easy and efficient numerical way to tackle such complex problems.

Rest of the manuscript is divided into the following sections. Some fundamental results are reported in Section 2. Proposed method and stability analysis are portrayed in Sections 3 and 4, respectively. Illustrative numerical experiments are discussed in Section 6. Finally, conclusion is given in Section 8.

2. Preliminaries

In this section, we recall some basic definitions which will be used in main results.

2.1. Definition

Let us consider $\mathbb{X}\in \mathrm{\aleph }=[\upsilon ,\omega )$ such that $\upsilon ={\mathbb{X}}_0<{\mathbb{X}}_1<\cdots <{\mathbb{X}}_{2\varpi }$, where ℵ is the space interval and is divided into N=2ϖ equivalent subintervals, where width of each sub interval is $\mathrm{\Delta }\mathbb{X}=\frac{\omega -\upsilon }{2\varpi }$, $\varpi =2^j$ and j stand for the resolution level. Let $i=m+k+1$ represent the wavelet number, where $m=2^{\lambda },\lambda =0,1,...,j$, and $k=0,1,...,2^{\lambda }-1$. The HW functions for $ı\ge 1$ are defined as [33]:

$${\mathit{\Upsilon }}_1(\mathbb{X})=\{\begin{array}{cc}1,\hfill & \mathbb{X}\in [\upsilon ,\omega )\hfill \\ 0,\hfill & \text{otherwise},\hfill \end{array}$$ (2.1)

$${\mathit{\Upsilon }}_{ı}(\mathbb{X})=\{\begin{array}{cc}{(-1)}^{ȷ+1},\hfill & \mathbb{X}\in [{\varsigma }_{ȷ}(ı),{\varsigma }_{ȷ+1}(ı)),ȷ=1,2\hfill \\ 0,\hfill & \text{otherwise},\hfill \end{array}$$ (2.2)

where ${\varsigma }_{\eta +1}(ı)=\upsilon +(2\kappa +\eta )\mu \mathrm{\Delta }\mathbb{X},\eta =0,1,2\text{and}\mu =\varpi /m=2^{j-\lambda }$. To estimate the numerical solution of TFBE by using HW, the following recurrent integrals are required:

$${\mathcal{P}}_{ı,\gamma }(\mathbb{X})=\underset{\mathcal{A}}{\overset{\mathbb{X}}{\int }}\underset{\upsilon }{\overset{\mathbb{X}}{\int }}\dots \underset{\upsilon }{\overset{\mathbb{X}}{\int }}{\Upsilon }_{ı}(z)d{z}^{\gamma }=\frac{1}{(\gamma -1)!}\underset{\upsilon }{\overset{\mathbb{X}}{\int }}{(\mathbb{X}-z)}^{\gamma -1}{\Upsilon }_{ı}(z)dz,$$ (2.3)

where $\gamma =1,2,\dots n,ı=1,2,\dots 2\varpi$. Utilizing Eqs. (2.1)-(2.3) the resultant expressions are given by:

$${\mathcal{R}}_{1,\gamma }(\mathbb{X})=\frac{1}{\gamma !}{(\mathbb{X}-\upsilon )}^{\gamma },$$ (2.4)

$${\mathcal{P}}_{ı,\gamma }(\mathbb{X})=\{\begin{array}{cc}0,\hfill & \text{if }\mathbb{X}<{\varsigma }_1(ı),\hfill \\ \frac{1}{\gamma !}{\{\mathbb{X}-{\varsigma }_1(ı)\}}^{\gamma },\hfill & \text{if }\mathbb{X}\in [{\varsigma }_1(ı),{\varsigma }_2(ı)\text{)},\hfill \\ \frac{1}{\gamma !}[{\{\mathbb{X}-{\varsigma }_1\left(ı\right)\}}^{\gamma }-2{\{\mathbb{X}-{\varsigma }_2(ı)\}}^{\gamma }],\hfill & \text{if }\mathbb{X}\in [{\varsigma }_2(ı),{\varsigma }_3(ı)\text{)},\hfill \\ \frac{1}{\gamma !}[{\{\mathbb{X}-{\varsigma }_1(ı)\}}^{\gamma }-2{\{(\mathbb{X}-{\varsigma }_2(ı)\}}^{\gamma }+{\{\mathbb{X}-{\varsigma }_3(ı)\}}^{\gamma }],\hfill & \text{if }\mathbb{X}\in [{\varsigma }_3(ı),\omega \text{)}.\hfill \end{array}$$ (2.5)

2.2. Definition

The $\mathcal{ABC}$ fractional derivative is denoted by $\frac{{\partial }^{\alpha }\mathcal{C}(\mathbb{X},t)}{\partial t^{\alpha }}$ and is defined as follows: [37]:

$${}_0{}^{\mathcal{ABC}}\mathcal{D}_t^{\alpha }\mathcal{C}(\mathbb{X},t)=\frac{{\partial }^{\alpha }\mathcal{C}(\mathbb{X},t)}{\partial t^{\alpha }}=\{\begin{array}{cc}\frac{R(\alpha )}{1-\alpha }{\int }_0^t\frac{\partial \mathcal{C}(\mathbb{X},\nu )}{\partial \nu }{E}_{\alpha }(-\frac{\alpha }{1-\alpha }{(t-\nu )}^{\alpha })d\nu ,\hfill & 0<\alpha <1,\hfill \\ \hfill \\ \frac{\partial \mathcal{C}}{\partial t},\hfill & \alpha =1,\hfill \end{array}$$ (2.6)

where $R(\alpha )$ is a normalization function which satisfies $R(0)=R(1)=1$ and $E_{\alpha }$ is the Mittag-Leffler function. The $\mathcal{ABC}$ derivative is important because it has non-local and non-singular kernel which was a critical problem in the Riemann-Liouville and Caputo derivative definitions. This is the recent and generalized definition which uses the Mittag-Leffler function in the kernel which remove the singularity problem.

This definition has been used by different authors in the analysis of various problems for which the readers may refer to see ([38], [39], [40]), and the references therein.

2.3. Definition

The Mittag-Leffler function has one or two parameters are defined as follows [37]:

$$E_{\theta }(\phi )=\sum _{k=0}^{\infty }\frac{{\phi }^k}{\mathrm{\Gamma }(\theta k+1)},\theta >0,$$ (2.7)

$$E_{\theta ,\vartheta }(\phi )=\sum _{k=0}^{\infty }\frac{{\phi }^k}{\mathrm{\Gamma }(\theta k+\vartheta )},\theta ,\vartheta >0.$$ (2.8)

Incorporating Mittag-Leffler function (Eqs. (2.7)-(2.8)), the $\mathcal{ABC}$ derivative of algebraic, exponential and trigonometric functions can be expressed as:

$${}_0{}^{\mathcal{ABC}}\mathcal{D}_t^{\alpha }(t^n)=\frac{R(\alpha )}{1-\alpha }n!t^n{E}_{\alpha ,n+1}(-\frac{\alpha }{1-\alpha }{(t)}^{\alpha }),{}_0{}^{\mathcal{ABC}}\mathcal{D}_t^{\alpha }(e^{\lambda t})=\sum _{k=1}^{\infty }\frac{R(\alpha )}{1-\alpha }{(\lambda t)}^k{E}_{\alpha ,k+1}(-\frac{\alpha }{1-\alpha }{(t)}^{\alpha }),{}_0{}^{\mathcal{ABC}}\mathcal{D}_t^{\alpha }\sin (\lambda t)=\sum _{k=1}^{\infty }\frac{R(\alpha )}{1-\alpha }{(-1)}^{k-1}{(\lambda t)}^{2k-1}{E}_{\alpha ,2k}(-\frac{\alpha }{1-\alpha }{(t)}^{\alpha }),{}_0{}^{\mathcal{ABC}}\mathcal{D}_t^{\alpha }\cos (\lambda t)=\sum _{k=1}^{\infty }\frac{R(\alpha )}{1-\alpha }{(-1)}^k{(\lambda t)}^{2k}{E}_{\alpha ,2k+1}(-\frac{\alpha }{1-\alpha }{(t)}^{\alpha }).$$ (2.9)

2.4. Quadrature rule for $\mathcal{ABC}$ derivative

Partitioning the time domain into M equally subintervals as $0=t_0<t_1<\cdots <t_M$ with interval width $\mathrm{\Delta }t=\frac{t}{M}$. The quadrature rule for the $\mathcal{ABC}$ derivative (Eq. (2.6)) is given below [37]:

$${}_0{}^{\mathcal{ABC}}\mathcal{D}_t^{\alpha }\mathcal{C}(\xi ,t_{n+1})=\frac{R(\alpha )}{1-\alpha }\underset{0}{\overset{t_{n+1}}{\int }}\frac{\partial \mathcal{C}(\mathbb{X},\nu )}{\partial \nu }{E}_{\alpha }(-\frac{\alpha }{1-\alpha }{(t_{n+1}-\nu )}^{\alpha })d\nu ,0<\alpha <1,=\frac{R(\alpha )}{1-\alpha }\sum _{ȷ=0}^n\underset{t_{ȷ}}{\overset{t_{ȷ+1}}{\int }}\frac{\partial \mathcal{C}(\mathbb{X},\nu )}{\partial \nu }{E}_{\alpha }(-\frac{\alpha }{1-\alpha }{(t_{n+1}-\nu )}^{\alpha })d\nu .$$

Using $\frac{\partial \mathcal{C}(\mathbb{X},\nu )}{\partial \nu }=\frac{{\mathcal{C}}^{ȷ+1}(\mathbb{X})-{\mathcal{C}}^{ȷ}(\mathbb{X})}{\mathrm{\Delta }t}$, where ${\mathcal{C}}^n(\mathbb{X})=\mathcal{C}(\mathbb{X},t_n)$ in the above integral we have:

$${}_0{}^{\mathcal{ABC}}\mathcal{D}_t^{\alpha }\mathcal{C}(\mathbb{X},t_{n+1})=\frac{R(\alpha )}{1-\alpha }\sum _{ȷ=0}^n\frac{{\mathcal{C}}^{ȷ+1}(\mathbb{X})-{\mathcal{C}}^{ȷ}(\mathbb{X})}{\mathrm{\Delta }t}\underset{t_{ȷ}}{\overset{t_{ȷ+1}}{\int }}{E}_{\alpha }(-\frac{\alpha }{1-\alpha }{(t_{n+1}-\nu )}^{\alpha })d\nu +{\chi }_{\mathrm{\Delta }t}^{n+1},=\frac{R(\alpha )}{1-\alpha }\sum _{ȷ=0}^n[{\mathcal{C}}^{n-ȷ+1}(\mathbb{X})-{\mathcal{C}}^{n-ȷ}(\mathbb{X})]\{(ȷ+1)E_{\alpha ,2}(-\frac{\alpha }{1-\alpha }{((ȷ+1)\mathrm{\Delta }t)}^{\alpha })-ȷE_{\alpha ,2}(-\frac{\alpha }{1-\alpha }{((ȷ)\mathrm{\Delta }t)}^{\alpha })\}+{\chi }_{\mathrm{\Delta }t}^{n+1},=\frac{R(\alpha )}{1-\alpha }\sum _{ȷ=0}^n[{\mathcal{C}}^{n-ȷ+1}(\mathbb{X})-{\mathcal{C}}^{n-ȷ}(\mathbb{X})][(ȷ+1)E_{ȷ+1}-ȷE_{ȷ}]+{\chi }_{\mathrm{\Delta }t}^{n+1},$$

$${}_0{}^{\mathcal{ABC}}\mathcal{D}_t^{\alpha }\mathcal{C}(\mathbb{X},t_{n+1})=\frac{R(\alpha )}{1-\alpha }\sum _{ȷ=0}^n{\gamma }_{ȷ}[{\mathcal{C}}^{n-ȷ+1}(\mathbb{X})-{\mathcal{C}}^{n-ȷ}(\mathbb{X})]+{\chi }_{\mathrm{\Delta }t}^{n+1},$$ (2.10)

where $E_{ȷ}=E_{\alpha ,2}(-\frac{\alpha }{1-\alpha }{(ȷ\mathrm{\Delta }t)}^{\alpha })$ and ${\gamma }_{ȷ}=(ȷ+1)E_{ȷ+1}-ȷE_{ȷ}$.

Following are the properties of ${\gamma }_{ȷ}$:

• •${\gamma }_{ȷ}>0$ and ${\gamma }_0=E_1$, $ȷ=0,1,\dots ,n$,
• •${\gamma }_0>{\gamma }_1>\cdots >{\gamma }_{ȷ},{\gamma }_{ȷ}\to 0$ as $ȷ\to \infty$

3. Proposed methodology

Here, we discuss the proposed methodology for one and two dimensional problems.

3.1. Case 1

First, consider the one space dimension problem for which we take $s=1$ and $\mathbb{X}=\xi$. Using Eq. (2.10) and the following implicit scheme to Eq. (1.1) the resultant is:

$$\frac{R(\alpha )}{1-\alpha }\sum _{ȷ=0}^n{\gamma }_{ȷ}[{\mathcal{C}}^{n-ȷ+1}(\xi )-{\mathcal{C}}^{n-ȷ}(\xi )]+[{\mathcal{C}}^{n+1}(\xi ){\mathcal{C}}_{\xi }^{n+1}(\xi )-\varrho {\mathcal{C}}_{\xi \xi }^{n+1}(\xi )]=q(\xi ,t_{n+1}).$$ (3.1)

In Eq. (3.1) the nonlinear term is linearized as follows [37]:

$${\mathcal{C}}^{n+1}(\xi ){\mathcal{C}}_{\xi }^{n+1}(\xi )\approx {\mathcal{C}}^{n+1}(\xi ){\mathcal{C}}_{\xi }^n(\xi )+{\mathcal{C}}^n(\xi ){\mathcal{C}}_{\xi }^{n+1}(\xi )-{\mathcal{C}}^n(\xi ){\mathcal{C}}_{\xi }^n(\xi ).$$ (3.2)

Incorporating Eq. (3.2) in Eq. (3.1) and some algebraic manipulation, gives the following equation:

$$(E_1+\rho {\mathcal{C}}_{\xi }^n(\xi )){\mathcal{C}}^{n+1}(\xi )+\rho {\mathcal{C}}^n{\mathcal{C}}_{\xi }^{n+1}(\xi )-\rho \varrho {\mathcal{C}}_{\xi \xi }^{n+1}(\xi )=E_1{\mathcal{C}}^n(\xi )+\rho {\mathcal{C}}^n(\xi ){\mathcal{C}}_{\xi }^n(\xi )+\rho q^{n+1}-\sum _{ȷ=1}^n{\gamma }_{ȷ}[{\mathcal{C}}^{n-ȷ+1}(\xi )-{\mathcal{C}}^{n-ȷ}(\xi )],$$ (3.3)

where $\rho =\frac{1-\alpha }{R(\alpha )}$ and $q(\xi ,t_{n+1})=q^{n+1}$.

The method is based on integral approach, therefore, the highest order derivative is estimated by a truncated Haar wavelet series as:

$${\mathcal{C}}_{\xi \xi }^{n+1}(\xi )=\sum _{i=1}^{2\varpi }{\eta }_i^{n+1}{\Upsilon }_i(\xi ),$$ (3.4)

where ${\eta }_i^{n+1}$ represent the unknown coefficients of wavelet and ${\Upsilon }_i$ stand for HW basis. Twice integration of Eq. (3.4) gives:

$${\mathcal{C}}_{\xi }^{n+1}(\xi )=\sum _{i=1}^{2\varpi }{\eta }_i^{n+1}{\mathcal{P}}_{i,1}(\xi )+{\mathcal{C}}_{\xi }^{n+1}(0),$$ (3.5)

$${\mathcal{C}}^{n+1}(\xi )=\sum _{i=1}^{2\varpi }{\eta }_i^{n+1}{\mathcal{P}}_{i,2}(\xi )+\xi {\mathcal{C}}_{\xi }^{n+1}(0)+{\mathcal{C}}^{n+1}(0).$$ (3.6)

Integration of Eq. (3.5) from 0 to 1 gives the following unknown value:

$${\mathcal{C}}_{\xi }^{n+1}(0)={\mathcal{C}}^{n+1}(1)-{\mathcal{C}}^{n+1}(0)-\sum _{i=1}^{2\varpi }{\eta }_i^{n+1}{\mathcal{P}}_{i,2}(1).$$ (3.7)

Substituting Eq. (3.7) into and Eqs. (3.5)-(3.6), the resulting equations can be written as:

$${\mathcal{C}}_{\xi }^{n+1}(\xi )=\sum _{i=1}^{2\varpi }{\eta }_i^{n+1}[{\mathcal{P}}_{i,1}(\xi )-{\mathcal{P}}_{i,2}(1)]+{\mathcal{C}}^{n+1}(1)-{\mathcal{C}}^{n+1}(0).$$ (3.8)

$${\mathcal{C}}^{n+1}(\xi )=\sum _{i=1}^{2\varpi }{\eta }_i^{n+1}[{\mathcal{P}}_{i,2}(\xi )-\xi {\mathcal{P}}_{i,2}(1)]+\xi [{\mathcal{C}}^{n+1}(1)-{\mathcal{C}}^{n+1}(0)]+{\mathcal{C}}^{n+1}(0).$$ (3.9)

Substitution of Eqs. (3.4), (3.8) and (3.9) in Eq. (3.3) and also the evaluation at $\xi \to {\xi }_{\sigma }=\frac{\sigma -0.5}{2\varpi }$, $\sigma =1,2,...,2\varpi$, yields:

$$\sum _{i=1}^{2\varpi }{\eta }_i^{n+1}[(E_1+\rho {\mathcal{C}}_{\xi }^n(\xi )){\mathrm{\Omega }}_{i,2}({\xi }_{\sigma })+{\xi }_{\sigma }{\mathcal{C}}^n(\xi ){\mathrm{\Omega }}_{i,1}({\xi }_{\sigma })-{\xi }_{\sigma }\varrho \Upsilon ({\xi }_{\sigma })]=B({\xi }_{\sigma }),$$ (3.10)

where

$${\mathrm{\Omega }}_{i,1}({\xi }_{\sigma })={\mathcal{P}}_{i,1}({\xi }_{\sigma })-{\mathcal{P}}_{i,1}(1),{\mathrm{\Omega }}_{i,2}({\xi }_{\sigma })={\mathcal{P}}_{i,2}({\xi }_{\sigma })-{\xi }_{\sigma }{\mathcal{P}}_{i,2}(1)B({\xi }_{\sigma })=E_1{\mathcal{C}}^n({\xi }_{\sigma })+\rho {\mathcal{C}}^n({\xi }_{\sigma }){\mathcal{C}}_{\xi }^n({\xi }_{\sigma })+\rho q^{n+1}-\sum _{ȷ=1}^n{\gamma }_{ȷ}[{\mathcal{C}}^{n-ȷ+1}-{\mathcal{C}}^{n-ȷ}]-(E_1+\rho {\mathcal{C}}_{\xi }^n({\xi }_{\sigma }))\times [{\xi }_{\sigma }\{{\mathcal{C}}^{n+1}(1)-{\mathcal{C}}^{n+1}(0)\}+{\mathcal{C}}^{n+1}(0))]-\rho {\mathcal{C}}^n({\xi }_{\sigma })[{\mathcal{C}}^{n+1}(1)-{\mathcal{C}}^{n+1}(0)].$$

Solution of the system (Eq. (2.10)), produces the unknown coefficients and then they can be used in Eq. (3.9) to determine the numerical solution of the given fractional model.

3.2. Case 2

Here, the proposed scheme is described using s=2 for which $\mathbb{X}=(\xi ,\zeta )$. Using Eq. (2.10) and the numerical strategy presented before, Eq. (1.1) transforms to:

$$\frac{R(\alpha )}{1-\alpha }\sum _{ȷ=0}^n{\gamma }_{ȷ}[{\mathcal{C}}^{n-ȷ+1}(\xi ,\zeta )-{\mathcal{C}}^{n-ȷ}(\xi ,\zeta )]+\{{\mathcal{C}}^{n+1}(\xi ,\zeta ){\mathcal{C}}_{\xi }^{n+1}(\xi ,\zeta )+{\mathcal{C}}^{n+1}(\xi ,\zeta ){\mathcal{C}}_{\zeta }^{n+1}(\xi ,\zeta )-\varrho {\mathcal{C}}_{\xi \xi }^{n+1}(\xi ,\zeta )-\varrho {\mathcal{C}}_{\zeta \zeta }^{n+1}(\xi ,\zeta )\}.$$ (3.11)

Further simplification of Eq. (3.11) leads to:

$$(E_1+\rho {\mathcal{C}}_{\xi }^n(\xi ,\zeta )+\rho {\mathcal{C}}_{\zeta }^n(\xi ,\zeta )){\mathcal{C}}^{n+1}(\xi ,\zeta )+\rho {\mathcal{C}}^n{\mathcal{C}}_{\xi }^{n+1}(\xi ,\zeta )+\rho {\mathcal{C}}^n{\mathcal{C}}_{\zeta }^{n+1}(\xi ,\zeta )-\rho \varrho {\mathcal{C}}_{\xi \xi }^{n+1}(\xi ,\zeta )-\rho \varrho {\mathcal{C}}_{\zeta \zeta }^{n+1}(\xi ,\zeta )=E_1{\mathcal{C}}^n(\xi ,\zeta )+\rho {\mathcal{C}}^n(\xi ,\zeta ){\mathcal{C}}_{\xi }^n(\xi ,\zeta )+\rho {\mathcal{C}}^n(\xi ,\zeta ){\mathcal{C}}_{\zeta }^n(\xi ,\zeta )+\rho q^{n+1}-\sum _{ȷ=1}^n{\gamma }_{ȷ}[{\mathcal{C}}^{n-ȷ+1}(\xi ,\zeta )-{\mathcal{C}}^{n-ȷ}(\xi ,\zeta )].$$ (3.12)

Assume the mixed highest order derivative by two dimensional Haar wavelet truncated series as follows:

$${\mathcal{C}}_{\xi \xi \zeta \zeta }^{n+1}(\xi ,\zeta )=\sum _{i=1}^{2\varpi }\sum _{\iota =1}^{2\varpi }{\eta }_{i,\iota }^{n+1}{\Upsilon }_i(\xi ){\Upsilon }_{\iota }(\zeta ),$$ (3.13)

where ${\eta }_{i,\iota }^{n+1}$ represent the coefficient of wavelet to be measured numerically. Taking integration of Eq. (3.13) with respect to y from 0 to ζ one gets:

$${\mathcal{C}}_{\xi \xi \zeta }^{n+1}(\xi ,\zeta )=\sum _{i=1}^{2\varpi }\sum _{\iota =1}^{2\varpi }{\eta }_{i,\iota }^{n+1}{\Upsilon }_i(\xi ){\mathcal{P}}_{\iota ,1}(\zeta )+{\mathcal{C}}_{\xi \xi \zeta }^{n+1}(\xi ,0).$$ (3.14)

Integration of Eq. (3.14) from 0 to 1 with respect to y, produces:

$${\mathcal{C}}_{\xi \xi \zeta }^{n+1}(\xi ,0)={\mathcal{C}}_{\xi \xi }^{n+1}(\xi ,1)-{\mathcal{C}}_{\xi \xi }^{n+1}(\xi ,0)-\sum _{i=1}^{2\varpi }\sum _{\iota =1}^{2\varpi }{\eta }_{i,\iota }^{n+1}{\Upsilon }_i(\xi ){\mathcal{P}}_{\iota ,2}(1).$$ (3.15)

Using Eq. (3.15) in Eq. (3.14) the resultant is:

$${\mathcal{C}}_{\xi \xi \zeta }^{n+1}(\xi ,\zeta )=\sum _{i=1}^{2\varpi }\sum _{\iota =1}^{2\varpi }{\eta }_{i,\iota }^{n+1}{\Upsilon }_i(\xi )[{\mathcal{P}}_{\iota ,1}(\zeta )-{\mathcal{P}}_{\iota ,2}(1)]+{\mathcal{C}}_{\xi \xi }^{n+1}(\xi ,1)-{\mathcal{C}}_{\xi \xi }^{n+1}(\xi ,0).$$ (3.16)

Again integration of Eq. (3.16) with respect to y from 0 to y gives:

$${\mathcal{C}}_{\xi \xi }^{n+1}(\xi ,\zeta )=\sum _{i=1}^{2\varpi }\sum _{\iota =1}^{2\varpi }{\eta }_{i,\iota }^{n+1}{\Upsilon }_i(\xi )[{\mathcal{P}}_{\iota ,2}(\zeta )-\zeta {\mathcal{P}}_{\iota ,2}(1)]+\zeta {\mathcal{C}}_{\xi \xi }^{n+1}(\xi ,1)+(1-\zeta ){\mathcal{C}}_{\xi \xi }^{n+1}(\xi ,0).$$ (3.17)

In light of the above calculations the following results can be extracted:

$${\mathcal{C}}_{\zeta \zeta }^{n+1}(\xi ,\zeta )=\sum _{i=1}^{2\varpi }\sum _{\iota =1}^{2\varpi }{\eta }_{i,\iota }^{n+1}[{\mathcal{P}}_{i,2}(\xi )-\xi {\mathcal{P}}_{i,2}(1)]{\Upsilon }_{\iota }(\zeta )+\xi {\mathcal{C}}_{\zeta \zeta }^{n+1}(1,\zeta )+(1-\xi ){\mathcal{C}}_{\zeta \zeta }^{n+1}(0,\zeta ),$$ (3.18)

$${\mathcal{C}}_{\xi }^{n+1}(\xi ,\zeta )=\sum _{i=1}^{2\varpi }\sum _{\iota =1}^{2\varpi }{\eta }_{i,\iota }^{n+1}[{\mathcal{P}}_{i,2}(\xi )-{\mathcal{P}}_{i,2}(1)][{\mathcal{P}}_{\iota ,2}(\zeta )-\zeta {\mathcal{P}}_{\iota ,2}(1)]+\zeta {\mathcal{C}}_{\xi }^{n+1}(\xi ,1)+(1-\zeta ){\mathcal{C}}_{\xi }^{n+1}(\xi ,0)+{\mathcal{C}}^{n+1}(1,\zeta )-{\mathcal{C}}^{n+1}(0,\zeta )-\zeta {\mathcal{C}}^{n+1}(1,1)+\zeta {\mathcal{C}}^{n+1}(0,1)+(\zeta -1){\mathcal{C}}^{n+1}(1,0)+(1-\zeta ){\mathcal{C}}^{n+1}(0,0),$$ (3.19)

$${\mathcal{C}}_{\zeta }^{n+1}(\xi ,\zeta )=\sum _{i=1}^{2\varpi }\sum _{\iota =1}^{2\varpi }{\eta }_{i,\iota }^{n+1}[{\mathcal{P}}_{i,2}(\xi )-\xi {\mathcal{P}}_{i,2}(1)][{\mathcal{P}}_{\iota ,2}(\zeta )-{\mathcal{P}}_{\iota ,2}(1)]+\xi {\mathcal{C}}_{\zeta }^{n+1}(1,\zeta )+(1-\xi ){\mathcal{C}}_{\zeta }^{n+1}(0,\zeta )+{\mathcal{C}}^{n+1}(\xi ,1)-{\mathcal{C}}^{n+1}(\xi ,0)-\xi {\mathcal{C}}^{n+1}(1,1)+\xi {\mathcal{C}}^{n+1}(1,0)+(\xi -1){\mathcal{C}}^{n+1}(0,1)+(1-\xi ){\mathcal{C}}^{n+1}(0,0),$$ (3.20)

$${\mathcal{C}}^{n+1}(\xi ,\zeta )=\sum _{i=1}^{2\varpi }\sum _{\iota =1}^{2\varpi }{\eta }_{i,\iota }^{n+1}[{\mathcal{P}}_{i,2}(\xi )-\xi {\mathcal{P}}_{i,2}(1)][{\mathcal{P}}_{\iota ,2}(\zeta )-\zeta {\mathcal{P}}_{\iota ,2}(1)]+\zeta \{{\mathcal{C}}^{n+1}(\xi ,1)-{\mathcal{C}}^{n+1}(0,1)\}+(1-\zeta )[{\mathcal{C}}^{n+1}(\xi ,0)-{\mathcal{C}}^{n+1}(0,0)]+\xi \{{\mathcal{C}}^{n+1}(1,\zeta )-{\mathcal{C}}^{n+1}(0,\zeta )\}-\xi \zeta \{{\mathcal{C}}^{n+1}(1,1)-{\mathcal{C}}^{n+1}(0,1)\}+\xi (\zeta -1){\mathcal{C}}^{n+1}(1,0)+\xi (1-\zeta ){\mathcal{C}}^{n+1}(0,0)+{\mathcal{C}}^{n+1}(0,\zeta ).$$ (3.21)

Further, the usage of Eqs. (3.17)-(3.21) in (3.12) and evaluation at $(\xi ,\zeta )\to ({\xi }_{\sigma },{\zeta }_{\delta })=(\frac{\sigma -0.5}{2\varpi },\frac{\delta -0.5}{2\varpi })$ where $\sigma ,\delta =1,2,...,2\varpi$, generates the system:

$$\sum _{i=1}^{2\varpi }\sum _{\iota =1}^{2\varpi }{\eta }_{i,\iota }^{n+1}[(E_1+\rho {\mathcal{C}}_{\xi }^n(\xi ,\zeta )+\rho {\mathcal{C}}_{\zeta }^n(\xi ,\zeta )){\mathrm{\Omega }}_{i,\iota }^1(\sigma ,\delta )+\rho {\mathcal{C}}^n{\mathrm{\Omega }}_{i,\iota }^2(\sigma ,\delta )-\rho \varrho {\mathrm{\Omega }}_{i,\iota }^3(\sigma ,\delta )]=B(\sigma ,\delta ),$$ (3.22)

where

$${\mathrm{\Omega }}_{i,\iota }^1(\sigma ,\delta )=[{\mathcal{P}}_{i,2}({\xi }_{\sigma })-{\xi }_{\sigma }{\mathcal{P}}_{i,2}(1)][{\mathcal{P}}_{\iota ,2}({\zeta }_{\delta })-{\zeta }_{\delta }{\mathcal{P}}_{\iota ,2}(1)],{\mathrm{\Omega }}_{i,\iota }^2(\sigma ,\delta )=[{\mathcal{P}}_{i,2}({\xi }_{\sigma })-{\mathcal{P}}_{i,2}(1)][{\mathcal{P}}_{\iota ,2}({\zeta }_{\delta })-{\zeta }_{\delta }{\mathcal{P}}_{\iota ,2}(1)]+[{\mathcal{P}}_{i,2}({\xi }_{\sigma })-{\xi }_{\sigma }{\mathcal{P}}_{i,2}(1)][{\mathcal{P}}_{\iota ,2}({\zeta }_{\delta })-{\mathcal{P}}_{\iota ,2}(1)],{\mathrm{\Omega }}_{i,\iota }^3(\sigma ,\delta )={\Upsilon }_i({\xi }_{\sigma })[{\mathcal{P}}_{\iota ,2}({\zeta }_{\delta })-{\zeta }_{\delta }{\mathcal{P}}_{\iota ,2}(1)]+[{\mathcal{P}}_{i,2}({\xi }_{\sigma })-{\xi }_{\sigma }{\mathcal{P}}_{i,2}(1)]{\Upsilon }_{\iota }(\zeta ),B(\sigma ,\delta )=E_1{\mathcal{C}}^n({\xi }_{\sigma },{\zeta }_{\delta })+\rho {\mathcal{C}}^n({\xi }_{\sigma },{\zeta }_{\delta }){\mathcal{C}}_{\xi }^n({\xi }_{\sigma },{\zeta }_{\delta })+\rho {\mathcal{C}}^n({\xi }_{\sigma },{\zeta }_{\delta }){\mathcal{C}}_{\zeta }^n({\xi }_{\sigma },{\zeta }_{\delta })+\rho q^{n+1}-\sum _{ȷ=1}^n{\gamma }_{ȷ}\{{\mathcal{C}}^{n-ȷ+1}({\xi }_{\sigma },{\zeta }_{\delta })-{\mathcal{C}}^{n-ȷ}({\xi }_{\sigma },{\zeta }_{\delta })\}-(E_1+\rho {\mathcal{C}}_{\xi }^n({\xi }_{\sigma },{\zeta }_{\delta })+\rho {\mathcal{C}}_{\zeta }^n({\xi }_{\sigma },{\zeta }_{\delta })\{{\zeta }_{\delta }[{\mathcal{C}}^{n+1}({\xi }_{\sigma },1)-{\mathcal{C}}^{n+1}(0,1)]+(1-{\zeta }_{\delta })[{\mathcal{C}}^{n+1}({\xi }_{\sigma },0)-{\mathcal{C}}^{n+1}(0,0)]+{\xi }_{\sigma }[{\mathcal{C}}^{n+1}(1,{\zeta }_{\delta })-{\mathcal{C}}^{n+1}(0,{\zeta }_{\delta })]-{\xi }_{\sigma }{\zeta }_{\delta }[{\mathcal{C}}^{n+1}(1,1)-{\mathcal{C}}^{n+1}(0,1)]+{\xi }_{\sigma }({\zeta }_{\delta }-1){\mathcal{C}}^{n+1}(1,0)+{\xi }_{\sigma }(1-{\zeta }_{\delta }){\mathcal{C}}^{n+1}(0,0)+{\mathcal{C}}^{n+1}(0,{\zeta }_{\delta })\}-\rho {\mathcal{C}}^n({\xi }_{\sigma },{\zeta }_{\delta })\{{\zeta }_{\delta }{\mathcal{C}}_{\xi }^{n+1}({\xi }_{\sigma },1)+(1-{\zeta }_{\delta }){\mathcal{C}}_{\xi }^{n+1}({\xi }_{\sigma },0)+{\mathcal{C}}^{n+1}(1,{\zeta }_{\delta })-{\mathcal{C}}^{n+1}(0,{\zeta }_{\delta })-{\zeta }_{\delta }{\mathcal{C}}^{n+1}(1,1)+{\zeta }_{\delta }{\mathcal{C}}^{n+1}(0,1)+({\zeta }_{\delta }-1){\mathcal{C}}^{n+1}(1,0)+(1-{\zeta }_{\delta }){\mathcal{C}}^{n+1}(0,0)+{\xi }_{\sigma }{\mathcal{C}}_{\zeta }^{n+1}(1,{\zeta }_{\delta })+(1-{\xi }_{\sigma }){\mathcal{C}}_{\zeta }^{n+1}(0,{\zeta }_{\delta })+{\mathcal{C}}^{n+1}({\xi }_{\sigma },1)-{\mathcal{C}}^{n+1}({\xi }_{\sigma },0)-{\xi }_{\sigma }{\mathcal{C}}^{n+1}(1,1)+{\xi }_{\sigma }{\mathcal{C}}^{n+1}(1,0)+({\xi }_{\sigma }-1){\mathcal{C}}^{n+1}(0,1)+(1-{\xi }_{\sigma }){\mathcal{C}}^{n+1}(0,0)\}+\rho \varrho \{{\zeta }_{\delta }{\mathcal{C}}_{\xi \xi }^{n+1}({\xi }_{\sigma },1)+(1-{\zeta }_{\delta }){\mathcal{C}}_{\xi \xi }^{n+1}({\xi }_{\sigma },0)+{\xi }_{\sigma }{\mathcal{C}}_{\zeta \zeta }^{n+1}(1,{\zeta }_{\delta })+(1-\xi ){\mathcal{C}}_{\zeta \zeta }^{n+1}(0,{\zeta }_{\delta })\}.$$

The above system can be solved to obtain the required unknown wavelet coefficients which can be substituted in Eqs. (3.17)-(3.21) to refine the solution and derivative at arbitrary time.

4. Stability analysis

In this part of the paper, the computational stability of the proposed scheme for two space dimension problems is elucidated via sufficient condition.

Theorem

Suppose $\mathcal{C}$ is the approximate solution of problem Eq. (1.1) , then the amplification matrix can be defined as $\Im =\mathcal{Z}{\mathcal{A}}^{-1}\mathcal{B}{\mathcal{Z}}^{-1}$ . The method will be stable if $||\Im ||\le 1$ .

Proof

The matrix forms of Eqs. (3.17)-(3.21) are:

$${\mathcal{C}}_{\zeta \zeta }^{n+1}(\xi ,\zeta )=\mathcal{V}{\eta }^{n+1}+{\tilde{\mathcal{V}}}^{n+1},$$ (4.1)

$${\mathcal{C}}_{\xi \xi }^{n+1}(\xi ,\zeta )=\mathcal{W}{\eta }^{n+1}+{\tilde{\mathcal{W}}}^{n+1},$$ (4.2)

$${\mathcal{C}}_{\zeta }^{n+1}(\xi ,\zeta )=\mathcal{X}{\eta }^{n+1}+{\tilde{\mathcal{X}}}^{n+1},$$ (4.3)

$${\mathcal{C}}_{\xi }^{n+1}(\xi ,\zeta )=\mathcal{Y}{\eta }^{n+1}+{\tilde{\mathcal{Y}}}^{n+1},$$ (4.4)

$${\mathcal{C}}^{n+1}(\xi ,\zeta )=\mathcal{Z}{\eta }^{n+1}+{\tilde{\mathcal{Z}}}^{n+1},$$ (4.5)

where ${\eta }^{n+1}={\eta }_{i,\iota }^{n+1}$, $\mathcal{V},\mathcal{W},\mathcal{X},\mathcal{Y},\mathcal{Z}$ and ${\tilde{\mathcal{V}}}^{n+1},{\tilde{\mathcal{W}}}^{n+1},{\tilde{\mathcal{X}}}^{n+1},{\tilde{\mathcal{Y}}}^{n+1},{\tilde{\mathcal{Z}}}^{n+1}$ are the matrices of ${\mathcal{C}}_{\zeta \zeta }^{n+1},{\mathcal{C}}_{\xi \xi }^{n+1},{\mathcal{C}}_{\zeta }^{n+1},{\mathcal{C}}_{\xi }^{n+1}$ and ${\mathcal{C}}^{n+1}$ at collocation points and boundary terms, respectively. Substituting Eqs. (4.1)-(4.5) in Eq. (3.22), we have:

$$[(E_1+\rho {\mathcal{C}}_{\xi }^n(\xi ,\zeta )+\rho {\mathcal{C}}_{\zeta }^n(\xi ,\zeta ))\mathcal{Z}+\rho \mathcal{Y}+\rho \mathcal{X}-\rho \varrho \mathcal{W}-\rho \varrho \mathcal{V}]{\eta }^{n+1}=[(E_1+\rho {\mathcal{C}}_{\xi }^n(\xi ,\zeta )+\rho {\mathcal{C}}_{\zeta }^n(\xi ,\zeta ))\mathcal{Z}]{\eta }^n+{\Re }^{n+1},$$ (4.6)

where

$${\Re }^{n+1}=-[(E_1+\rho {\mathcal{C}}_{\xi }^n(\xi ,\zeta )+\rho {\mathcal{C}}_{\zeta }^n(\xi ,\zeta )){\tilde{\mathcal{Z}}}^{n+1}+\rho {\tilde{\mathcal{Y}}}^{n+1}+\rho {\tilde{\mathcal{X}}}^{n+1}-\rho \varrho {\tilde{\mathcal{W}}}^{n+1}-\rho \varrho {\tilde{\mathcal{V}}}^{n+1}]+[(E_1+\rho {\mathcal{C}}_{\xi }^n(\xi ,\zeta )+\rho {\mathcal{C}}_{\zeta }^n(\xi ,\zeta )){\tilde{\mathcal{Z}}}^n]+\rho q^{n+1}-\sum _{ȷ=1}^n{\gamma }_{ȷ}[{\mathcal{C}}^{n-ȷ+1}(\xi ,\zeta )-{\mathcal{C}}^{n-ȷ}(\xi ,\zeta )].$$

The above equation (Eq. (4.6)) can also be written as:

$${\eta }^{n+1}={\mathcal{A}}^{-1}\mathcal{B}{\eta }^n+{\mathcal{A}}^{-1}{\Re }^{n+1},$$ (4.7)

where $\mathcal{A}=(E_1+\rho {\mathcal{C}}_{\xi }^n(\xi ,\zeta )+\rho {\mathcal{C}}_{\zeta }^n(\xi ,\zeta ))\mathcal{Z}+\rho \mathcal{Y}+\rho \mathcal{X}-\rho \varrho \mathcal{W}-\rho \varrho \mathcal{V}$ and $\mathcal{B}=(E_1+\rho {\mathcal{C}}_{\xi }^n(\xi ,\zeta )+\rho {\mathcal{C}}_{\zeta }^n(\xi ,\zeta ))\mathcal{Z}$. Inserting Eq. (4.7) in Eq. (4.5) leads to:

$${\mathcal{C}}^{n+1}=\mathcal{Z}{\mathcal{A}}^{-1}\mathcal{B}{\eta }^n+\mathcal{Z}{\mathcal{A}}^{-1}{\Re }^{n+1}+{\tilde{\mathcal{Z}}}^{n+1}.$$ (4.8)

Getting the value of ${\eta }^n$ from Eq. (4.5) then using in Eq. (4.8), we get:

$${\mathcal{C}}^{n+1}=\mathcal{Z}{\mathcal{A}}^{-1}\mathcal{B}{\mathcal{Z}}^{-1}{\mathcal{C}}^n-\mathcal{Z}{\mathcal{A}}^{-1}\mathcal{B}{\mathcal{Z}}^{-1}{\tilde{\mathcal{Z}}}^n+\mathcal{Z}{\mathcal{A}}^{-1}{\Re }^{n+1}+{\tilde{\mathcal{Z}}}^{n+1}.$$ (4.9)

If $\widetilde{{\mathcal{C}}^{n+1}}$ is approximate solution then:

$$\widetilde{{\mathcal{C}}^{n+1}}=\mathcal{Z}{\mathcal{A}}^{-1}\mathcal{B}{\mathcal{Z}}^{-1}\widetilde{{\mathcal{C}}^n}-\mathcal{Z}{\mathcal{A}}^{-1}\mathcal{B}{\mathcal{Z}}^{-1}{\tilde{\mathcal{Z}}}^n+\mathcal{Z}{\mathcal{A}}^{-1}{\Re }^{n+1}+{\tilde{\mathcal{Z}}}^{n+1}.$$ (4.10)

From Eqs. (4.9) - Eq. (4.10) it follows that:

$$e^{n+1}=\Im e^n,$$

where $\Im =\mathcal{Z}{\mathcal{A}}^{-1}\mathcal{B}{\mathcal{Z}}^{-1}$ is the amplification matrix. If $||\Im ||\le 1$ then the scheme will be stable according to Lax-Richtmyer criterion.

5. Algorithm

Input: Specify the space domain [0,1], j, t, ϱ, and Δt.

Output: Numerical solution of ${\mathcal{C}}^n(\xi ,t)$.

Step 1: Define ${\mathbb{X}}_i,i=1,2,3,...,2\varpi$.

Step 2: At ${\mathbb{X}}_i$ define $\Upsilon ,{\mathcal{P}}_{ı,\gamma }$ given in Eqs. (2.1)-(2.5).

Step 3: For n=0 (do the following steps)

Step 4: Generate the matrices ${\mathrm{\Omega }}_{i,1},{\mathrm{\Omega }}_{i,2}$ in Eq. (3.8).

Step 5: Compute the unknown coefficient ${\eta }_i$ in Eq. (3.10).

Step 6: Approximate the solution ${\mathcal{C}}^{n+1}$ in Eq. (3.9).

Step 7: For $n=1:N$.

Step 8: Repeat steps 4-6.

6. Numerical simulations

Here, the proposed scheme is implemented to solve some benchmark problems. To confirm the effectiveness, the following error measures are incorporated:

$${\mathbb{L}}_{\infty }=\underset{1\le i\le 2\varpi }{\max }|{({\mathcal{C}}_i^n)}^{ext}-{({\mathcal{C}}_i^n)}^{app}|,{\mathbb{L}}_2={\left(\sum _{i=1}^{2\varpi }{({({\mathcal{C}}_i^n)}^{ext}-{({\mathcal{C}}_i^n)}^{app})}^2\right)}^{\frac{1}{2}},\text{Relative Error}=\frac{{\max }_{1\le i\le 2\varpi }|{({\mathcal{C}}_i^n)}^{ext}-{({\mathcal{C}}_i^n)}^{app}|}{{\max }_{1\le i\le 2\varpi }\left|{({\mathcal{C}}_i^n)}^{ext}\right|},{\mathbb{L}}_{RMS}={\left(\frac{1}{2\varpi }\sum _{i=1}^{2\varpi }{({({\mathcal{C}}_i^n)}^{ext}-{({\mathcal{C}}_i^n)}^{app})}^2\right)}^{\frac{1}{2}}.$$

Besides this, the computational rate of convergence in time and space directions are calculated via the following estimation formulae:

$$\text{Order}=\frac{log({\mathbb{L}}_{\infty }^{\mathrm{\Delta }\partial })-log({\mathbb{L}}_{\infty }^{\mathrm{\Delta }\partial +1})}{log(2)},$$

where $\mathrm{\Delta }\partial =\mathrm{\Delta }t,\mathrm{\Delta }\mathbb{X}$, in time and spatial directions, respectively.

Problem 6.1

Consider Eq. (1.1) for $\xi \in [0,1],t\in [0,1],\varrho =2$ and $R(\alpha )=1$. The artificial analytical solution of this problem is $\mathcal{C}(\xi ,t)=t^2\sin (\pi \xi )$. The associated source function is derived using Eq. (2.9) which is given below:

$$2\frac{R(\alpha )}{1-\alpha }{t}^2\sin (\pi \xi )E_{\alpha ,3}\left(\frac{-\alpha }{1-\alpha }{t}^{\alpha }\right)+\pi t^4\cos (\pi \xi )\sin (\pi \xi )+\varrho {\pi }^2{t}^2\sin (\pi \xi ).$$

The proposed scheme is implemented and the error norms, ${\mathbb{L}}_{\infty }$, ${\mathbb{L}}_2$, and Relative error (RE), for various values of α, M, N, and time are presented Table 1.

This table indicates that increasing the value of M also increases the accuracy. In Table 2, the outcomes of the current scheme in terms of ${\mathbb{L}}_{\infty }$ are compared to the existing work in literature [37] for $\alpha =0.2$ and 0.5 at different times. From comparison it is clear that for small number of collocation points computed results are better from the cited work. In Table 3, Table 4 the computational order of convergence, in terms of ${\mathbb{L}}_{\infty }$ error norm are matched with those given in [37], [41] in the temporal and spatial directions, respectively. In Table 5, the error norms ${\mathbb{L}}_{\infty }$, ${\mathbb{L}}_2$ and RE, along with the spectral radius (SR) and computational time (CT) for $\alpha =0.5$ and $M=2^7$, are presented which demonstrate that the accuracy of the norms improves with the variation of resolution level. From tables one can see that present scheme converges fast in spatial direction because the computed values are more near to 2. The similar results are true in temporal direction. Additionally, the results are plotted at different times when $\mathrm{\Delta }t=0.01,0.001$, $N=100,500$, and $\alpha =0.2,0.9$, in two dimensional and three dimensional forms, in Figure 1, Figure 2, respectively. The close agreement of the computational and exact solutions is obvious from figures.

Table 1.

Numerical solutions for Problem 6.1 for different N, t, Δt and α.

		Proposed method
		${\mathbb{L}}_{\infty }$		${\mathbb{L}}_2$		RE
t	M	α = 0.2,N = 512	α = 0.5,N = 256	α = 0.2,N = 512	α = 0.5,N = 256	α = 0.2,N = 512	α = 0.5,N = 256
0.2	8	2.13397e-06	4.01746e-06	2.90303e-05	3.83350e-05	5.33494e-05	1.00439e-04
0.4	8	2.95357e-05	3.84600e-05	4.43698e-04	3.53631e-04	1.84599e-04	2.40380e-04
0.6	8	1.45629e-04	1.67544e-04	2.23932e-03	1.63550e-03	4.04528e-04	4.65408e-04
0.8	8	4.58489e-04	4.99372e-04	7.07294e-03	5.06275e-03	7.16393e-04	7.80284e-04
1.0	8	1.12419e-03	1.18962e-03	1.72612e-02	1.22796e-02	1.12419e-03	1.18965e-03
0.2	16	5.39223e-07	9.07127e-07	7.57134e-06	8.38276e-06	1.34806e-05	2.26786e-05
0.4	16	7.77025e-06	9.66260e-06	1.18234e-04	9.04615e-05	4.85643e-05	6.03924e-05
0.6	16	3.86403e-05	4.34396e-05	5.97541e-04	4.31765e-04	1.07335e-04	1.20668e-04
0.8	16	1.22051e-04	1.31172e-04	1.88774e-03	1.34618e-03	1.90705e-04	2.04961e-04
1.0	16	2.99766e-04	3.14555e-04	4.60678e-03	3.27217e-03	2.99768e-04	3.14561e-04
0.2	32	1.29216e-07	1.35615e-07	1.91639e-06	1.36608e-06	3.23040e-06	3.39045e-06
0.4	32	1.91937e-06	1.98373e-06	3.04471e-05	2.14955e-05	1.19961e-05	1.23986e-05
0.6	32	9.71111e-06	1.00962e-05	1.54129e-04	1.08953e-04	2.69754e-05	2.80456e-05
0.8	32	3.10510e-05	3.19263e-05	4.87082e-04	3.44432e-04	4.85174e-05	4.98858e-05
1.0	32	7.67007e-05	7.82759e-05	1.18875e-03	8.40916e-04	7.67010e-05	7.82774e-05
0.2	64	6.75786e-08	1.88705e-07	9.02243e-07	2.06008e-06	1.68947e-06	4.71770e-06
0.4	64	6.16951e-07	1.00340e-06	8.27416e-06	9.32305e-06	3.85596e-06	6.27138e-06
0.6	64	2.71532e-06	3.49836e-06	3.96560e-05	3.19958e-05	7.54259e-06	9.71784e-06
0.8	64	8.08555e-06	9.40997e-06	1.24134e-04	9.14976e-05	1.26337e-05	1.47034e-05
1.0	64	1.90176e-05	2.10456e-05	3.02177e-04	2.16966e-04	1.90176e-05	2.10460e-05
0.2	128	5.83100e-08	2.19688e-07	9.10151e-07	2.48059e-06	1.45776e-06	5.49231e-06
0.4	128	3.00418e-07	9.01781e-07	4.07910e-06	9.91652e-06	1.87762e-06	5.63624e-06
0.6	128	9.93889e-07	2.24716e-06	1.26941e-05	2.30198e-05	2.76082e-06	6.24223e-06
0.8	128	2.58049e-06	4.70474e-06	3.41495e-05	4.45199e-05	4.03203e-06	7.35130e-06
1.0	128	5.64351e-06	8.89077e-06	7.89210e-05	8.03697e-05	5.64353e-06	8.89094e-06

Table 2.

${\mathbb{L}}_{\infty }$ norm for different γ values when Δt = 0.01 of Problem 6.1.

	${\mathbb{L}}_{\infty }$[37]		${\mathbb{L}}_{\infty }$ Proposed method
Δ t	α = 0.2 , N = 500	α = 0.5 , N = 250	α = 0.2 , N = 256	α = 0.5 , N = 128
0.2	3.08595e-07	3.24140e-07	3.3507e-07	5.1961e-07
0.4	1.48688e-06	2.19986e-06	1.2961e-06	1.4056e-06
0.6	3.46559e-06	5.59933e-06	3.0324e-06	3.7515e-06
0.8	6.21777e-06	1.04261e-05	5.4448e-06	7.0975e-06
1	9.70636e-06	1.66560e-05	8.4964e-06	1.1420e-05

Table 3.

Comparison of ${\mathbb{L}}_{\infty }$ norm when M = 2¹² in spatial direction of Problem 6.1.

		[41]		[37]		Proposed method
α	N	${\mathbb{L}}_{\infty }$	Order	${\mathbb{L}}_{\infty }$	Order	${\mathbb{L}}_{\infty }$	Order
0.2	23	1.24013e-02	...	1.20798e-02	...	1.52190e-03	...
	24	3.12783e-03	1.987262	3.03534e-03	1.992661	3.81452e-04	1.996308
	25	7.84221e-04	1.995831	7.61955e-04	1.994081	9.54691e-05	1.998390
	26	1.99548e-04	1.974526	1.90552e-04	1.999520	2.38710e-05	1.999773
	27	5.47402e-05	1.866062	4.76511e-05	1.999604	5.9687e-06	1.999770
0.3	23	1.23742e-02	...	1.20542e-02	...	1.51861e-03	...
	24	3.12128e-03	1.987124	3.02885e-03	1.992688	3.80630e-04	1.996281
	25	7.82583e-04	1.995819	7.60318e-04	1.994097	9.52652e-05	1.998371
	26	1.99139e-04	1.974472	1.90142e-04	1.999523	2.38201e-05	1.999771
	27	5.46367e-05	1.865829	4.75486e-05	1.999604	5.95582e-06	1.999806
0.4	23	1.23351e-02	...	1.20173e-02	...	1.51401e-03	...
	24	3.11185e-03	1.986927	3.019534e-03	1.992726	3.79462e-04	1.996345
	25	7.80229e-04	1.995801	7.57966e-04	1.994120	9.49711e-05	1.998388
	26	1.98550e-04	1.974394	1.89554e-04	1.999526	2.37470e-05	1.9997417
	27	5.44879e-05	1.865497	4.74015e-05	1.999605	5.93727e-06	1.999872
0.5	23	1.22824e-02	...	1.19678e-02	...	1.50764e-03	...
	24	3.09911e-03	1.986662	3.00696e-03	1.992777	3.77847e-04	1.996292
	25	7.77049e-04	1.995778	7.54794e-04	1.994153	9.45743e-05	1.998374
	26	1.97755e-04	1.974291	1.88760e-04	1.999530	2.36475e-05	1.999786
	27	5.42862e-05	1.865055	4.72029-05	1.999605	5.91227e-06	1.999874

Table 4.

${\mathbb{L}}_{\infty }$ norm comparison when N = 2⁹ in temporal direction of Problem 6.1.

		[41]		[37]		Proposed method
α	M	${\mathbb{L}}_{\infty }$	Order	${\mathbb{L}}_{\infty }$	Order	${\mathbb{L}}_{\infty }$	Order
0.2	23	4.73848e-03	...	1.12312e-03	...	1.12121e-03	...
	24	2.45311e-03	0.949812	2.98708e-04	1.910700	2.96782e-04	1.917578
	25	1.24840e-03	0.974528	7.56481e-05	1.981362	7.49981e-05	1.984470
	26	6.30558e-03	0.985385	2.008720-05	1.913028	1.90182e-05	1.979487
	27	3.17781e-03	0.988597	6.77037e-06	1.568970	5.64352e-06	1.752704
0.3	23	4.75003e-03	...	1.13793e-03	...	1.1360e-03	...
	24	2.45506e-03	0.952182	3.02623e-04	1.910816	3.0070e-04	1.917566
	25	1.24836e-03	0.975728	7.66645e-05	1.980889	7.4758e-05	2.008025
	26	6.30264e-04	0.985999	1.97780e-05	1.954666	1.9019e-05	1.974787
	27	3.17562e-04	0.988917	6.68258e-06	1.565417	5.5615e-06	1.773895
0.4	23	4.76689e-03	...	1.15961e-03	...	1.1576e-03	...
	24	2.45781e-03	0.955677	3.08263e-04	1.911400	3.0631e-04	1.918072
	25	1.24824e-03	0.977474	7.81092e-05	1.980597	7.6192e-05	2.007281
	26	6.29823e-04	0.986879	1.93476e-05	2.013340	1.9384e-05	1.974773
	27	3.17240e-04	0.989371	6.56173e-06	1.560005	5.4489e-06	1.830829
0.5	23	4.79195e-03	...	1.19158e-03	...	1.1892e-03	...
	24	2.46197e-03	0.960800	3.16494e-04	1.912633	3.1445e-04	1.919089
	25	1.24815e-03	0.980017	8.01948e-05	1.980595	7.8249e-05	2.006686
	26	6.29224e-04	0.988149	1.88648e-05	2.087812	1.9902e-05	1.975159
	27	3.16796e-04	0.990020	6.39245e-06	1.561256	5.2927e-06	1.910838

Table 5.

Variation of resolution level j when M = 2⁷ and α = 0.5 of Problem 6.1 with CT and SR.

t	j	${\mathbb{L}}_{\infty }$	${\mathbb{L}}_2$	RE	SR	CT
0.2	3.0	5.90272e-05	1.67680e-04	1.48282e-03	0.089505	0.526569
0.2	4.0	1.48390e-05	5.94109e-05	3.71422e-04	0.089603	0.445822
0.2	5.0	3.70541e-06	2.09630e-05	9.26631e-05	0.089628	0.658877
0.2	6.0	9.16714e-07	7.33284e-06	2.29196e-05	0.089634	1.132115
0.2	7.0	2.19688e-07	2.48059e-06	5.49231e-06	0.089636	2.552979
0.4	3.0	2.37864e-04	6.74768e-04	1.49385e-03	0.089476	0.276035
0.4	4.0	5.97487e-05	2.39038e-04	3.73880e-04	0.089581	0.369590
0.4	5.0	1.49052e-05	8.42881e-05	9.31856e-05	0.089607	0.587915
0.4	6.0	3.68614e-06	2.94150e-05	2.30401e-05	0.089614	1.108766
0.4	7.0	9.01781e-07	9.91652e-06	5.63624e-06	0.089615	2.414764
0.6	3.0	5.38356e-04	1.52401e-03	1.50267e-03	0.092675	0.271826
0.6	4.0	1.35060e-04	5.39825e-04	3.75620e-04	0.092810	0.391243
0.6	5.0	3.37041e-05	1.90283e-04	9.36508e-05	0.092844	0.571425
0.6	6.0	8.40344e-06	6.64152e-05	2.33446e-05	0.092852	1.074917
0.6	7.0	2.24716e-06	2.30198e-05	6.24223e-06	0.092854	2.376389
0.8	3.0	9.61689e-04	2.71577e-03	1.50991e-03	0.101547	0.256217
0.8	4.0	2.40826e-04	9.61866e-04	3.76744e-04	0.101748	0.356182
0.8	5.0	6.02922e-05	3.39010e-04	9.42350e-05	0.101798	0.547009
0.8	6.0	1.53671e-05	1.18766e-04	2.40128e-05	0.101811	1.057461
0.8	7.0	4.70474e-06	4.45199e-05	7.35130e-06	0.101814	2.384513
1.0	3.0	1.50757e-03	4.24749e-03	1.51487e-03	0.117683	0.255672
1.0	4.0	3.78245e-04	1.50418e-03	3.78702e-04	0.117983	0.355794
1.0	5.0	9.50983e-05	5.30232e-04	9.51269e-05	0.118059	0.554251
1.0	6.0	2.51631e-05	1.87493e-04	2.51650e-05	0.118078	1.069269
1.0	7.0	8.89077e-06	8.03697e-05	8.89094e-06	0.118082	2.301954

Figure 1.

Comparison of two-dimensional exact and approximate solutions at various time points for Problem 6.1.

Figure 2.

Three dimensional exact and approximate solutions with corresponding error and the two dimensional error at t = 1, when N = 100, Δt = 0.01, α = 0.25 and ξ ∈ [0,1] for Problem 6.1.

Problem 6.2

Consider Eq. (1.1) for $\xi \in [0,1]\subset {\mathcal{R}}^{\mathbf{s}},\mathbf{s}=1,t\in [0,1],\varrho =1$ and $R(\alpha )=1$. The required conditions for this problem are used from the exact solution $\mathcal{C}(\xi ,t)=t^2\cos (\pi \xi )$. The relevant source term is given by:

$$2\frac{R(\alpha )}{1-\alpha }{t}^2\cos (\pi \xi )E_{\alpha ,3}\left(\frac{-\alpha }{1-\alpha }{t}^{\alpha }\right)-\pi t^4\cos (\pi \xi )\sin (\pi \xi )+\varrho {\pi }^2{t}^2\cos (\pi \xi ).$$

Two norms ${\mathbb{L}}_{\infty }$, ${\mathbb{L}}_2$ and RE for this problem are tabulated in Table 6 using different parameters α, M, N and t. In Table 7, the ${\mathbb{L}}_{\infty }$ norm for $\alpha =0.5$ at various Δt are compared with the results in the article [37]. The spatiotemporal order of convergence using ${\mathbb{L}}_{\infty }$ norm are also presented and matched with the same reference work in Table 8 and Table 9, respectively. In Table 10, the error norms ${\mathbb{L}}_{\infty }$, ${\mathbb{L}}_2$, and RE, along with the SR and CT for $\alpha =0.5$ and $M=2^7$, are presented. These results demonstrate that as j is increasing, the accuracy also raises. Approximated and exact measures at different time with $\mathrm{\Delta }t=0.01,0.002$, $N=80,200$ and $\alpha =0.3,0.7$ are displayed in Fig. 3 in the form of two dimension. Also, the three dimensional view of the of computed and closed form solutions with their error, and the error in the two dimensional solutions are plotted in Fig. 4. It is observed from Figure 3, Figure 4 that computational and exact solutions overlap.

Table 6.

Computed results of Problem 6.2 for different values of N, t, Δt and α.

		Proposed Method
		${\mathbb{L}}_{\infty }$		${\mathbb{L}}_2$		RE
t	M	α = 0.2,N = 512	α = 0.5,N = 256	α = 0.2,N = 512	α = 0.5,N = 256	α = 0.2,N = 512	α = 0.5,N = 256
0.2	8	3.15620e-06	2.07610e-06	5.03871e-05	2.30403e-05	7.89053e-05	5.19034e-05
0.4	8	5.36003e-05	4.76953e-05	8.56619e-04	5.37308e-04	3.35004e-04	2.98101e-04
0.6	8	2.75986e-04	2.60030e-04	4.40843e-03	2.93297e-03	7.66633e-04	7.22320e-04
0.8	8	8.82759e-04	8.50398e-04	1.40859e-02	9.58697e-03	1.37932e-03	1.32877e-03
1.0	8	2.18066e-03	2.12449e-03	3.47440e-02	2.39220e-02	2.18067e-03	2.12453e-03
0.2	16	8.50936e-07	6.02097e-07	1.35886e-05	6.71138e-06	2.12735e-05	1.50527e-05
0.4	16	1.43452e-05	1.29385e-05	2.29267e-04	1.45828e-04	8.96581e-05	8.08673e-05
0.6	16	7.37863e-05	6.99264e-05	1.17857e-03	7.88817e-04	2.04963e-04	1.94244e-04
0.8	16	2.35982e-04	2.28090e-04	3.76508e-03	2.57129e-03	3.68724e-04	3.56397e-04
1.0	16	5.83037e-04	5.69267e-04	9.28764e-03	6.40921e-03	5.83040e-04	5.69278e-04
0.2	32	2.28484e-07	1.93255e-07	3.65254e-06	2.17403e-06	5.71214e-06	4.83147e-06
0.4	32	3.73919e-06	3.49692e-06	5.97739e-05	3.94650e-05	2.33700e-05	2.18562e-05
0.6	32	1.91332e-05	1.84211e-05	3.05637e-04	2.07907e-04	5.31480e-05	5.11706e-05
0.8	32	6.10857e-05	5.95821e-05	9.74649e-04	6.71859e-04	9.54468e-05	9.30989e-05
1.0	32	1.50812e-04	1.48152e-04	2.40237e-03	1.66822e-03	1.50812e-04	1.48155e-04
0.2	64	6.73696e-08	8.67971e-08	1.08070e-06	9.88821e-07	1.68425e-06	2.16997e-06
0.4	64	9.87961e-07	1.04302e-06	1.58087e-05	1.18178e-05	6.17478e-06	6.51899e-06
0.6	64	4.94947e-06	5.04306e-06	7.90993e-05	5.70204e-05	1.37486e-05	1.40088e-05
0.8	64	1.56849e-05	1.58193e-05	2.50323e-04	1.78582e-04	2.45077e-05	2.47182e-05
1.0	64	3.85910e-05	3.87848e-05	6.14837e-04	4.37016e-04	3.85912e-05	3.87856e-05
0.2	128	2.65417e-08	6.00764e-08	4.28421e-07	6.88351e-07	6.63545e-07	1.50194e-06
0.4	128	2.88042e-07	4.19176e-07	4.62309e-06	4.78201e-06	1.80027e-06	2.61990e-06
0.6	128	1.33890e-06	1.63745e-06	2.14318e-05	1.86044e-05	3.71919e-06	4.54855e-06
0.8	128	4.12508e-06	4.67725e-06	6.58988e-05	5.29743e-05	6.44546e-06	7.30834e-06
1.0	128	1.00141e-05	1.09324e-05	1.59653e-04	1.23480e-04	1.00141e-05	1.09326e-05

Table 7.

${\mathbb{L}}_{\infty }$ norm at t = 1 of Problem 6.2.

	${\mathbb{L}}_{\infty }$[37]		$L_{\infty }$ [Proposed method]
Δ t	α = 0.2 , N = 500	α = 0.5 , N = 250	α = 0.2 , N = 256	α = 0.5 , N = 128
0.002	2.76831e-05	7.37909e-06	2.56318e-05	6.87741e-06
0.001	2.72299e-05	6.92220e-06	2.51776e-05	6.41905e-06
0.0005	2.71164e-05	6.80815e-06	2.50639e-05	6.30479e-06
0.00025	2.70881e-05	6.77977e-06	2.50354e-05	6.27621e-06

Table 8.

Comparison of ${\mathbb{L}}_{\infty }$ norm when M= 2¹² in spatial direction of Problem 6.2.

		[37]		Proposed method
α	N	${\mathbb{L}}_{\infty }$	Order	${\mathbb{L}}_{\infty }$	Order
0.2	23	2.66277e-03	...	3.93127e-04	...
	24	6.72366e-04	1.985608	1.00340e-04	1.970101
	25	1.69762e-04	1.985730	2.51733e-05	1.994924
	26	4.25054e-05	1.997800	6.31072e-06	1.996020
	27	1.06427e-05	1.997780	1.58526e-06	1.993087
0.3	23	2.65980e-03	...	3.92645e-04	...
	24	6.71585e-04	1.985681	1.00214e-04	1.970137
	25	1.69569e-04	1.985699	2.51420e-05	1.994917
	26	4.24559e-05	1.997832	6.30280e-06	1.996034
	27	1.06304e-05	1.997773	1.58323e-06	1.993121
0.4	23	2.65554e-03	...	3.91950e-04	...
	24	6.70459e-04	1.985785	1.00033e-04	1.970189
	25	1.69289e-04	1.985653	2.50967e-05	1.994906
	26	4.23848e-05	1.997878	6.29137e-06	1.996054
	27	1.06126e-05	1.997762	1.58031e-06	1.993166
0.5	23	2.64977e-03	...	3.91001e-04	...
	24	6.68935e-04	1.985928	9.97861e-05	1.970261
	25	1.68912e-04	1.985592	2.50350e-05	1.994892
	26	4.22884e-05	1.997940	6.27577e-06	1.996082
	27	1.05886e-05	1.997748	1.57632e-06	1.993229

Table 9.

${\mathbb{L}}_{\infty }$ norm comparison when N = 2¹¹ in temporal direction of Problem 6.2.

		[37]		Proposed method
α	M	${\mathbb{L}}_{\infty }$	Order	${\mathbb{L}}_{\infty }$	Order
0.2	23	2.18033e-03	...	2.18032e-03	...
	24	5.82692e-04	1.903742	5.82677e-04	1.903770
	25	1.50464e-04	1.953316	1.50448e-04	1.953435
	26	3.82424e-05	1.976175	3.82259e-05	1.976640
	27	9.66513e-06	1.984311	9.64859e-06	1.986158
0.3	23	2.16774e-03	...	2.16772e-03	...
	24	5.79372e-04	1.903630	5.79356e-04	1.903658
	25	1.49603e-04	1.953348	1.49587e-04	1.953467
	26	3.80217e-05	1.976247	3.80052e-05	1.976714
	27	9.60898e-06	1.984370	9.59245e-06	1.986225
0.4	23	2.14944e-03	...	2.14945e-03	...
	24	5.74606e-04	1.903320	5.745936e-04	1.903354
	25	1.48382e-04	1.953253	1.48367e-04	1.953370
	26	3.77122e-05	1.976219	3.76960e-05	1.976690
	27	9.53096e-06	1.984338	9.51453e-06	1.986206
0.5	23	2.12314e-03	...	2.12338e-03	...
	24	5.67809e-04	1.902718	5.67859e-04	1.902757
	25	1.46655e-04	1.952975	1.46655e-04	1.953101
	26	3.72772e-05	1.976067	3.72648 e-05	1.976544
	27	9.42186e-06	1.984209	9.40643 e-06	1.986094

Table 10.

Variation of resolution level j when M = 2⁷ and α = 0.5 of Problem 6.2 with SR and CT.

t	j	${\mathbb{L}}_{\infty }$	${\mathbb{L}}_2$	RE	SR	CT
0.2	3.0	1.28337e-05	3.66619e-05	3.22395e-04	0.155909	0.087431
0.2	4.0	3.25907e-06	1.32528e-05	8.15750e-05	0.156057	0.043743
0.2	5.0	8.27236e-07	4.74645e-06	2.06871e-05	0.156093	0.055777
0.2	6.0	2.13757e-07	1.73335e-06	5.34433e-06	0.156103	0.108222
0.2	7.0	6.00764e-08	6.88351e-07	1.50194e-06	0.156105	0.313644
0.4	3.0	5.30066e-05	1.51464e-04	3.32894e-04	0.129058	0.022449
0.4	4.0	1.36024e-05	5.52518e-05	8.51174e-05	0.129149	0.031671
0.4	5.0	3.57778e-06	2.05124e-05	2.23679e-05	0.129172	0.051352
0.4	6.0	1.05058e-06	8.51070e-06	6.56662e-06	0.129178	0.105007
0.4	7.0	4.19176e-07	4.78201e-06	2.61990e-06	0.129179	0.278491
0.6	3.0	1.25220e-04	3.58011e-04	3.49516e-04	0.085481	0.022849
0.6	4.0	3.26652e-05	1.32479e-04	9.08460e-05	0.085482	0.031901
0.6	5.0	9.05724e-06	5.18863e-05	2.51666e-05	0.085483	0.052169
0.6	6.0	3.11838e-06	2.52030e-05	8.66281e-06	0.085483	0.128524
0.6	7.0	1.63745e-06	1.86044e-05	4.54855e-06	0.085483	0.346534
0.8	3.0	2.37249e-04	6.78987e-04	3.72495e-04	0.037350	0.021950
0.8	4.0	6.31826e-05	2.55816e-04	9.88419e-05	0.037407	0.031219
0.8	5.0	1.86254e-05	1.06655e-04	2.91110e-05	0.037425	0.050431
0.8	6.0	7.46237e-06	6.01336e-05	1.16608e-05	0.037429	0.111896
0.8	7.0	4.67725e-06	5.29743e-05	7.30834e-06	0.037430	0.291517
1.0	3.0	4.00012e-04	1.14666e-03	4.01947e-04	0.090391	0.024632
1.0	4.0	1.09035e-04	4.40795e-04	1.09167e-04	0.090922	0.035285
1.0	5.0	3.43354e-05	1.95995e-04	3.43457e-05	0.091057	0.053884
1.0	6.0	1.56015e-05	1.25286e-04	1.56027e-05	0.091091	0.110523
1.0	7.0	1.09324e-05	1.23480e-04	1.09326e-05	0.091099	0.289042

Figure 3.

Comparison of two-dimensional exact and approximate solutions at various time points for Problem 6.2.

Figure 4.

Three dimensional exact and approximate solutions with absolute error and absolute error in two dimension at t = 1, N = 100, Δt = 0.001, α = 0.5 and ξ ∈ [0,1] for Problem 6.2.

Problem 6.3

Consider Eq. (1.1) for $\xi \in [0,1]\subset {\mathcal{R}}^{\mathbf{s}},\mathbf{s}=1,t\in [0,1],\varrho =1$ and $R(\alpha )=1$. Here, we use a different exact solution given by:

$$\mathcal{C}(\xi ,t)=t^2\exp (\xi ).$$

The associated source term is given by:

$$2\frac{R(\alpha )}{1-\alpha }{t}^2\exp (\xi )E_{\alpha ,3}\left(\frac{-\alpha }{1-\alpha }{t}^{\alpha }\right)-t^4\exp (2\xi )-\varrho t^2\exp (\xi ).$$

The various values of the parameters α, M, N and t are used for the numerical computations of this problem. The estimated ${\mathbb{L}}_{\infty }$, ${\mathbb{L}}_2$ error norms and RE are recorded in Table 11. The spatiotemporal convergence order and ${\mathbb{L}}_{\infty }$ norms are compared with the work presented in [37], in Table 12, Table 13, respectively. In Table 14, the SR and CT for various values of resolution level is given which shows that accuracy increases as j increases. Numerical estimations and exact solutions at different time for $\mathrm{\Delta }t=0.01,0.002$, $N=150,70$ and $\alpha =0.1,0.6$ are shown in Figure 5, Figure 6. Numerical estimations clearly visualize that proposed method works well.

Table 11.

Numerical solutions of Problem 6.3 for different N, t, Δt and α.

		[Proposed Method]
		${\mathbb{L}}_{\infty }$		${\mathbb{L}}_2$		RE
t	Δt	α = 0.2,N = 512	α = 0.5,N = 256	α = 0.2,N = 512	α = 0.5,N = 256	α = 0.2,N = 512	α = 0.5,N = 256
0.2	8	3.23244e-05	4.08329e-05	5.30825e-04	4.75576e-04	2.97578e-04	3.76274e-04
0.4	8	4.83283e-04	5.17499e-04	7.91986e-03	6.00584e-03	1.11227e-03	1.19218e-03
0.6	8	2.36247e-03	2.42635e-03	3.86021e-02	2.80602e-02	2.41654e-03	2.48430e-03
0.8	8	7.16950e-03	7.25839e-03	1.16589e-01	8.35210e-02	4.12514e-03	4.18037e-03
1.0	8	1.65951e-02	1.67150e-02	2.67882e-01	1.90905e-01	6.11097e-03	6.16111e-03
0.2	16	8.63579e-06	1.07925e-05	1.41814e-04	1.25688e-04	7.95008e-05	9.94522e-05
0.4	16	1.28943e-04	1.37619e-04	2.11292e-03	1.59698e-03	2.96762e-04	3.17038e-04
0.6	16	6.29386e-04	6.45608e-04	1.02822e-02	7.46504e-03	6.43790e-04	6.61029e-04
0.8	16	1.90579e-03	1.92870e-03	3.09823e-02	2.21868e-02	1.09654e-03	1.11081e-03
1.0	16	4.39802e-03	4.43007e-03	7.09657e-02	5.05778e-02	1.61952e-03	1.63292e-03
0.2	32	2.23394e-06	2.77492e-06	3.66858e-05	3.23153e-05	2.05656e-05	2.55708e-05
0.4	32	3.32993e-05	3.54810e-05	5.45653e-04	4.11722e-04	7.66381e-05	8.17390e-05
0.6	32	1.62424e-04	1.66513e-04	2.65340e-03	1.92526e-03	1.66141e-04	1.70490e-04
0.8	32	4.91476e-04	4.97295e-04	7.98936e-03	5.72022e-03	2.82782e-04	2.86410e-04
1.0	32	1.13324e-03	1.14151e-03	1.82843e-02	1.30315e-02	4.17301e-04	4.20757e-04
0.2	64	5.68185e-07	7.01147e-07	9.33093e-06	8.16488e-06	5.23070e-06	6.46105e-06
0.4	64	8.46053e-06	8.99669e-06	1.38637e-04	1.04396e-04	1.94719e-05	2.07261e-05
0.6	64	4.12538e-05	4.22511e-05	6.73927e-04	4.88509e-04	4.21979e-05	4.32602e-05
0.8	64	1.24798e-04	1.26200e-04	2.02867e-03	1.45161e-03	7.18055e-05	7.26830e-05
1.0	64	2.87685e-04	2.89660e-04	4.64163e-03	3.30671e-03	1.05937e-04	1.06768e-04
0.2	128	1.42657e-07	1.73156e-07	2.34267e-06	2.01609e-06	1.31330e-06	1.59563e-06
0.4	128	2.12928e-06	2.25059e-06	3.48907e-05	2.61135e-05	4.90051e-06	5.18478e-06
0.6	128	1.03865e-05	1.06004e-05	1.69674e-04	1.22556e-04	1.06242e-05	1.08536e-05
0.8	128	3.14237e-05	3.16952e-05	5.10809e-04	3.64560e-04	1.80804e-05	1.82544e-05
1.0	128	7.24388e-05	7.27811e-05	1.16875e-03	8.30835e-04	2.66748e-05	2.68270e-05

Table 12.

Error norms comparison in spatial direction with M = 2¹¹ of Problem 6.3.

		[37]		Proposed method
α	N	${\mathbb{L}}_{\infty }$	Order	${\mathbb{L}}_{\infty }$	Order
0.2	23	2.20006e-04	...	7.51978e-05	...
	24	5.43064e-05	2.018348	1.85909e-05	2.016092
	25	1.33344e-05	2.025971	4.43224e-06	2.068489
	26	3.12217e-06	2.094532	8.94453e-07	2.308958
	27	5.69465e-07	2.454869	1.54090e-08	5.859161
0.3	23	2.19273e-04	...	7.48632e-05	...
	24	5.41247e-05	2.018367	1.85063e-05	2.016235
	25	1.32887e-05	2.026090	4.41077e-06	2.068917
	26	3.11030e-06	2.095072	8.88650e-07	2.311344
	27	5.66088e-07	2.457956	1.38810e-08	6.000433
0.4	23	2.18223e-04	...	7.43810e-05	...
	24	5.38646e-05	2.018391	1.83846e-05	2.016435
	25	1.32233e-05	2.026251	4.37997e-06	4.384150
	26	3.09344e-06	2.095803	8.80428e-07	2.314644
	27	5.61387e-07	2.462145	1.18634e-08	6.213608
0.5	23	2.16816e-04	...	7.37203e-05	...
	24	5.35164e-05	2.018423	1.82178e-05	2.016717
	25	1.31358e-05	2.026479	4.33769e-06	2.070344
	26	3.07075e-06	2.096839	8.69089e-07	2.319353
	27	5.54967e-07	2.468117	9.20033e-09	6.561673

Table 13.

Comparison of error norm in temporal direction of Problem 6.3 with N = 2⁴.

		[37]		Proposed method
α	M	${\mathbb{L}}_{\infty }$	Order	${\mathbb{L}}_{\infty }$	Order
0.2	23	1.65532e-02	...	1.65784e-02	...
	24	4.34726e-03	1.928926	4.33871e-03	1.933965
	25	1.08086e-03	2.007927	1.06200e-03	2.030483
	26	2.34688e-04	2.203366	2.13336e-04	2.315582
	27	1.97043e-05	3.574163	4.22874e-06	5.656760
0.3	23	1.65765e-02	...	1.66019e-02	...
	24	4.35399e-03	1.928729	4.34563e-03	1.933707
	25	1.08284e-03	2.007517	1.06415e-03	2.029861
	26	2.35347e-04	2.201958	2.14160e-04	2.312947
	27	1.99857e-05	3.557752	3.82638e-06	8.119509
0.4	23	1.66119e-02	...	1.663732e-02	...
	24	4.36370e-03	1.928594	4.35560e-03	1.933479
	25	1.08558e-03	2.007075	1.06714e-03	2.029121
	26	2.36243e-04	2.200133	2.15290e-04	2.309398
	27	2.03716e-05	3.535642	3.29441e-06	6.030117
0.5	23	1.66737e-02	...	1.66969e-02	...
	24	4.37994e-03	1.928590	4.37174e-03	1.933303
	25	1.09000e-03	2.006583	1.07177e-03	2.028208
	26	2.37608e-04	2.197676	2.16953e-04	2.304541
	27	2.09231e-05	3.505417	2.58682e-06	6.390067

Table 14.

Variation of resolution level j when M = 2⁷ and α = 0.5 of Problem 6.3 with SR and CT.

t	j	${\mathbb{L}}_{\infty }$	${\mathbb{L}}_2$	RE	SR	CT
0.2	3.0	1.07971e-06	3.15736e-06	1.02453e-05	0.169377	0.103236
0.2	4.0	1.39096e-07	5.74600e-07	1.29941e-06	0.169544	0.042454
0.2	5.0	9.95041e-08	5.75263e-07	9.22316e-07	0.169586	0.055443
0.2	6.0	1.58384e-07	1.30306e-06	1.46235e-06	0.169596	0.108798
0.2	7.0	1.73156e-07	2.01609e-06	1.59563e-06	0.169599	0.318984
0.4	3.0	3.75902e-06	1.09961e-05	8.91726e-06	0.182993	0.022072
0.4	4.0	7.78124e-07	3.10708e-06	1.81727e-06	0.183161	0.031000
0.4	5.0	1.89772e-06	1.09882e-05	4.39755e-06	0.183203	0.050477
0.4	6.0	2.17998e-06	1.78797e-05	5.03194e-06	0.183213	0.103772
0.4	7.0	2.25059e-06	2.61135e-05	5.18478e-06	0.183216	0.313853
0.6	3.0	6.41881e-06	1.87820e-05	6.76752e-06	0.206032	0.024106
0.6	4.0	6.41637e-06	2.60569e-05	6.66007e-06	0.206186	0.035494
0.6	5.0	9.60339e-06	5.54555e-05	9.89056e-06	0.206225	0.055316
0.6	6.0	1.04010e-05	8.50131e-05	1.06702e-05	0.206235	0.121972
0.6	7.0	1.06004e-05	1.22556e-04	1.08536e-05	0.206237	0.344757
0.8	3.0	6.63895e-06	1.93678e-05	3.93728e-06	0.236289	0.024286
0.8	4.0	2.23104e-05	9.03413e-05	1.30262e-05	0.236393	0.033284
0.8	5.0	2.94587e-05	1.69293e-04	1.70660e-05	0.236419	0.050288
0.8	6.0	3.12469e-05	2.54109e-04	1.80314e-05	0.236425	0.101618
0.8	7.0	3.16952e-05	3.64560e-04	1.82544e-05	0.236427	0.290042
1.0	3.0	2.57975e-06	6.48945e-06	9.79162e-07	0.270739	0.023646
1.0	4.0	5.46254e-05	2.19868e-04	2.04120e-05	0.270723	0.033891
1.0	5.0	6.84531e-05	3.90540e-04	2.53800e-05	0.270720	0.052610
1.0	6.0	7.19155e-05	5.80453e-04	2.65598e-05	0.270719	0.104322
1.0	7.0	7.27811e-05	8.30835e-04	2.68270e-05	0.270719	0.269335

Figure 5.

This figure illustrates the comparison between two-dimensional Exact versus approximate solutions at different time points for Problem 6.3.

Figure 6.

Exact and approximate solutions with absolute error at t=1 when N = 100, Δt = 0.002, α = 0.5 and ξ ∈ [0,1] for Problem 6.3.

Problem 6.4

Here, we choose Eq. (1.1) in two space dimensions for $(\xi ,\zeta )\in {[0,1]}^2\subset {\mathcal{R}}^{\mathbf{s}},\mathbf{s}=2,t\in [0,1],\varrho =100$ and $R(\alpha )=1$. To artificial closed form solution in two space variables is given by:

$$\mathcal{C}(\xi ,\zeta ,t)=t({\xi }^2-\xi )({\zeta }^2-\zeta ).$$

The following is the function $q(\xi ,\zeta ,t)$:

$$q(\xi ,\zeta ,t)=2\frac{R(\alpha )}{1-\alpha }t({\xi }^2-\xi )({\zeta }^2-\zeta )E_{\alpha ,2}\left(\frac{-\alpha }{1-\alpha }{t}^{\alpha }\right)-2\varrho t({\xi }^2+{\zeta }^2-\xi -\zeta ).$$

The initial and boundary conditions are derived from the exact solutions. The proposed numerical strategy is implemented and the error norms ${\mathbb{L}}_{\infty }$, ${\mathbb{L}}_2$ and RMS are listed for the distinct values of α, M, N and t in Table 15. From table the computed norms are decreasing as N and M are increasing. The order of convergence along with ${\mathbb{L}}_{\infty }$, ${\mathbb{L}}_2$ and ${\mathbb{L}}_{RMS}$ norms are outlined Table 16 which predict that the scheme is nearly second order convergent for two space dimensional problem too. The graphical comparison of the numerical and exact solutions together with an absolute error at $t=1$, $M=128$, $N=16$ and $\alpha =0.25$ are depicted in Fig. 7. From figure the closed coincidence of both solutions are visible.

Table 15.

Numerical results of Problem 6.4 for different N, t, Δt and α.

		Proposed method
		${\mathbb{L}}_{\infty }$		${\mathbb{L}}_2$		${\mathbb{L}}_{RMS}$
t	M	α = 0.2,N = 8	α = 0.5,N = 16	α = 0.2,N = 8	α = 0.5,N = 16	α = 0.2,N = 8	α = 0.5,N = 16
0.2	8	9.91792e-10	9.90983e-10	3.79314e-09	7.69625e-09	7.11061e-08	7.21514e-08
0.4	8	3.96720e-09	3.96406e-09	1.51726e-08	3.07859e-08	1.42213e-07	1.44307e-07
0.6	8	8.92624e-09	8.91932e-09	3.41385e-08	6.92694e-08	2.13320e-07	2.16464e-07
0.8	8	1.58689e-08	1.58568e-08	6.06907e-08	1.23147e-07	2.84427e-07	2.88622e-07
1.0	8	2.47953e-08	2.47766e-08	9.48294e-08	1.92419e-07	3.55534e-07	3.60781e-07
0.2	16	2.47948e-10	2.47746e-10	9.48284e-10	1.92406e-09	1.77765e-08	1.80379e-08
0.4	16	9.91799e-10	9.91016e-10	3.79315e-09	7.69647e-09	3.55532e-08	3.60767e-08
0.6	16	2.23156e-09	2.22983e-09	8.53462e-09	1.73174e-08	5.33300e-08	5.41161e-08
0.8	16	3.96724e-09	3.96421e-09	1.51727e-08	3.07868e-08	7.11068e-08	7.21556e-08
1.0	16	6.19884e-09	6.19416e-09	2.37074e-08	4.81049e-08	8.88836e-08	9.01954e-08
0.2	32	6.19870e-11	6.19366e-11	2.37071e-10	4.81016e-10	4.44413e-09	4.50947e-09
0.4	32	2.47950e-10	2.47754e-10	9.48289e-10	1.92412e-09	8.88831e-09	9.01919e-09
0.6	32	5.57890e-10	5.57459e-10	2.13365e-09	4.32935e-09	1.33325e-08	1.35290e-08
0.8	32	9.91810e-10	9.91053e-10	3.79317e-09	7.69671e-09	1.77767e-08	1.80389e-08
1.0	32	1.54971e-09	1.54854e-09	5.92684e-09	1.20262e-08	2.22209e-08	2.25489e-08
0.2	64	1.54968e-11	1.54841e-11	5.92678e-11	1.20254e-10	1.11103e-09	1.12737e-09
0.4	64	6.19874e-11	6.19386e-11	2.37072e-10	4.81030e-10	2.22208e-09	2.25480e-09
0.6	64	1.39473e-10	1.39365e-10	5.33414e-10	1.08234e-09	3.33312e-09	3.38226e-09
0.8	64	2.47952e-10	2.47763e-10	9.48293e-10	1.92418e-09	4.44417e-09	4.50973e-09
1.0	64	3.87427e-10	3.87135e-10	1.48171e-09	3.00656e-09	5.55523e-09	5.63722e-09
0.2	128	3.87418e-12	3.87105e-12	1.48169e-11	3.00635e-11	2.77758e-10	2.81842e-10
0.4	128	1.54968e-11	1.54846e-11	5.92680e-11	1.20258e-10	5.55519e-10	5.63700e-10
0.6	128	3.48681e-11	3.48411e-11	1.33353e-10	2.70584e-10	8.33281e-10	8.45564e-10
0.8	128	6.19881e-11	6.19408e-11	2.37073e-10	4.81045e-10	1.11104e-09	1.12743e-09
1.0	128	9.68569e-11	9.67839e-11	3.70428e-10	7.51640e-10	1.38881e-09	1.40931e-09

Table 16.

Comparison of ${\mathbb{L}}_{\infty }$ norm when N = 2³ in temporal direction of Problem 6.4.

		Proposed method
α	M	${\mathbb{L}}_{\infty }$	${\mathbb{L}}_2$	${\mathbb{L}}_{RMS}$	Order
0.2	23	2.47759e-08	1.92414e-07	3.60771e-07	...
	24	6.19396e-09	4.81034e-08	9.01927e-08	2.000000
	25	1.54849e-09	1.20259e-08	2.25482e-08	2.000000
	26	3.87123e-10	3.00646e-09	5.63704e-09	1.999998
	27	9.67808e-11	7.51616e-10	1.40926e-09	1.999999
0.3	23	2.47762e-08	1.92416e-07	3.60775e-07	...
	24	6.19404e-09	4.81040e-08	9.01938e-08	2.000000
	25	1.54851e-09	1.20260e-08	2.25484e-08	2.000000
	26	3.87128e-10	3.00650e-09	5.63711e-09	1.999998
	27	9.67820e-11	7.51625e-10	1.40928e-09	2.000000
0.4	23	2.47764e-08	1.92418e-07	3.60778e-07	...
	24	6.19410e-09	4.81044e-08	9.01946e-08	2.000000
	25	1.54853e-09	1.20261e-08	2.25487e-08	1.999995
	26	3.87132e-10	3.00653e-09	5.63717e-09	2.000000
	27	9.67830e-11	7.51633e-10	1.40929e-09	2.000000
0.5	23	2.47766e-08	1.92419e-07	3.60781e-07	...
	24	6.19416e-09	4.81049e-08	9.01954e-08	1.999997
	25	1.54854e-09	1.20262e-08	2.25489e-08	2.000000
	26	3.87135e-10	3.00656e-09	5.63722e-09	2.000000
	27	9.67839e-11	7.51640e-10	1.40931e-09	1.999997

Figure 7.

Comparison of exact and numerical solutions and absolute error at t = 1, N = 16, M = 128, α = 0.25 and (ξ,ζ)∈[0,1]² for Problem 6.4.

Problem 6.5

Finally, we consider the two dimensional problem Eq. (1.1) for $(\xi ,\zeta )\in {[0,1]}^2\subset {\mathcal{R}}^{\mathbf{s}},\mathbf{s}=2,t\in [0,1],\varrho =0.1$ and $R(\alpha )=1$ with analytical solution:

$$\mathcal{C}(\xi ,\zeta ,t)=t^3{(1-{\xi }^2)}^2{(1-{\zeta }^2)}^2.$$

Like Problem 6.4 the associated conditions are extracted from the given solution and its corresponding source term is given as follows:

$$2\frac{R(\alpha )}{1-\alpha }{t}^3{(1-{\xi }^2)}^2{(1-{\zeta }^2)}^2{E}_{\alpha ,4}\left(\frac{-\alpha }{1-\alpha }{t}^{\alpha }\right)-4R{t}^3(-3{\xi }^4{\zeta }^2+{\xi }^4-3{\xi }^2{\zeta }^4+12{\xi }^2{\zeta }^2-5{\xi }^2+{\zeta }^4-5{\zeta }^2+2)-4t^6{({\xi }^2-1)}^3{({\zeta }^2-1)}^3(-{\xi }^2\zeta -\xi {\zeta }^2+\xi +\zeta ).$$

The numerical simulation in terms of ${\mathbb{L}}_{\infty }$, ${\mathbb{L}}_2$ and ${\mathbb{L}}_{RMS}$ norms for the parameters α, M, N t are presented in Table 17. The order of convergence, ${\mathbb{L}}_{\infty }$, ${\mathbb{L}}_2$ and RMS norms are shown in Table 18. Similarly, graphical solutions and error plots for $t=1$, $M=128$, $N=16$ and $\alpha =0.25$ are displayed in Fig. 8. Data analysis discloses, the mutual agreement of exact and numerical solutions.

Table 17.

Numerical results for Problem 6.5 for different values of N, t, Δt and α.

		Proposed method
		${\mathbb{L}}_{\infty }$		${\mathbb{L}}_2$		${\mathbb{L}}_{RMS}$
t	M	α = 0.2,N = 8	α = 0.5,N = 16	α = 0.2,N = 8	α = 0.5,N = 16	α = 0.2,N = 8	α = 0.5,N = 16
0.2	8	1.83187e-06	2.56874e-06	6.39399e-06	1.94700e-05	2.45863e-04	3.74332e-04
0.4	8	1.36496e-04	7.84613e-05	4.94747e-04	5.39846e-04	2.37801e-03	1.29739e-03
0.6	8	1.61574e-03	1.31114e-03	6.01400e-03	9.30321e-03	8.56487e-03	6.62460e-03
0.8	8	9.69571e-03	8.45119e-03	3.66335e-02	6.14560e-02	2.20100e-02	1.84618e-02
1.0	8	4.22522e-02	3.68063e-02	1.54516e-01	2.67115e-01	4.75320e-02	4.10845e-02
0.2	16	1.07798e-06	5.13826e-07	3.76533e-06	3.99167e-06	1.44785e-04	7.67441e-05
0.4	16	4.38459e-05	2.44799e-05	1.58468e-04	1.68912e-04	7.61679e-04	4.05939e-04
0.6	16	4.81627e-04	3.83315e-04	1.79964e-03	2.72518e-03	2.56297e-03	1.94054e-03
0.8	16	2.88764e-03	2.46301e-03	1.08279e-02	1.78736e-02	6.50559e-03	5.36935e-03
1.0	16	1.23510e-02	1.05209e-02	4.53250e-02	7.73073e-02	1.39428e-02	1.18905e-02
0.2	32	8.50204e-07	5.97519e-08	2.97290e-06	3.36917e-07	1.14315e-04	6.47759e-06
0.4	32	1.67297e-05	7.79972e-06	5.99678e-05	5.37779e-05	2.88237e-04	1.29242e-04
0.6	32	1.50138e-04	1.07853e-04	5.59594e-04	7.66228e-04	7.96948e-04	5.45613e-04
0.8	32	8.53281e-04	6.78042e-04	3.18003e-03	4.91387e-03	1.91061e-03	1.47616e-03
1.0	32	3.54060e-03	2.86267e-03	1.29960e-02	2.10595e-02	3.99779e-03	3.23914e-03
0.2	64	7.89860e-07	1.32855e-07	2.76379e-06	8.89880e-07	1.06274e-04	1.71089e-05
0.4	64	9.43080e-06	3.21501e-06	3.35230e-05	2.21193e-05	1.61129e-04	5.31584e-05
0.6	64	6.08307e-05	3.31814e-05	2.25046e-04	2.34680e-04	3.20501e-04	1.67110e-04
0.8	64	3.01887e-04	1.94172e-04	1.11157e-03	1.40128e-03	6.67847e-04	4.20954e-04
1.0	64	1.17070e-03	7.96908e-04	4.26650e-03	5.84971e-03	1.31245e-03	8.99736e-04
0.2	128	7.74796e-07	1.64359e-07	2.71175e-06	1.12415e-06	1.04273e-04	2.16130e-05
0.4	128	7.54267e-06	2.01907e-06	2.67142e-05	1.38889e-05	1.28402e-04	3.33787e-05
0.6	128	3.77117e-05	1.37784e-05	1.38497e-04	9.66376e-05	1.97242e-04	6.88134e-05
0.8	128	1.58683e-04	6.84456e-05	5.75154e-04	4.88765e-04	3.45561e-04	1.46829e-04
1.0	128	5.56567e-04	2.60994e-04	2.00247e-03	1.90063e-03	6.15995e-04	2.92333e-04

Table 18.

Comparison of ${\mathbb{L}}_{\infty }$ norm when N = 2³ in temporal direction of Problem 6.5.

		Proposed method
α	N	${\mathbb{L}}_{\infty }$	${\mathbb{L}}_2$	${\mathbb{L}}_{RMS}$	Order
0.2	8	4.18476e-02	3.08247e-01	4.74110e-02	0.0
0.2	16	1.20496e-02	8.90183e-02	1.36918e-02	1.796153
0.2	32	3.28109e-03	2.42032e-02	3.72265e-03	1.876742
0.2	64	9.11738e-04	6.70833e-03	1.03180e-03	1.847485
0.2	128	2.97005e-04	2.16976e-03	3.33728e-04	1.618133
0.3	8	4.06580e-02	2.98421e-01	4.58997e-02	0.0
0.3	16	1.16677e-02	8.62115e-02	1.32601e-02	1.801022
0.3	32	3.17909e-03	2.34473e-02	3.60640e-03	1.875833
0.3	64	8.83832e-04	6.50124e-03	9.99946e-04	1.846768
0.3	128	2.88211e-04	2.10470e-03	3.23721e-04	1.616646
0.4	8	3.90051e-02	2.84889e-01	4.38184e-02	0.0
0.4	16	1.11654e-02	8.23572e-02	1.26672e-02	1.804634
0.4	32	3.03861e-03	2.24124e-02	3.44722e-03	1.877544
0.4	64	8.45494e-04	6.21845e-03	9.56452e-04	1.845547
0.4	128	2.76143e-04	2.01598e-03	3.10075e-04	1.614380
0.5	8	3.68063e-02	2.67115e-01	4.10845e-02	0.0
0.5	16	1.05209e-02	7.73073e-02	1.18905e-02	1.806690
0.5	32	2.86267e-03	2.10595e-02	3.23914e-03	1.877834
0.5	64	7.96908e-04	5.84971e-03	8.99736e-04	1.844875
0.5	128	2.60994e-04	1.90063e-03	2.92333e-04	1.610398

Figure 8.

Exact and approximate solutions with absolute error t = 1, N = 16, M = 128, α = 0.25 and (ξ,ζ)∈[0,1]² for Problem 6.5.

7. Stability verification

Here, the stability condition discussed in Section 4, is verified computationally and graphically. In Table 19 the SR $||\Im ||$ is elucidated for all problems at various times. From table it is clear that $||\Im ||<1$ for one and two dimensional problems and also from the Figure 9, Figure 10, Figure 11 shows that the scheme is stable. Computational and graphical verification guarantees that the scheme is stable.

Table 19.

${\mathbb{L}}_{\infty }$, ${\mathbb{L}}_2$, RMS norms and spectral radius for different time values when α = 0.5, M = 128 and N = 64 of Problem 6.1, Problem 6.2, Problem 6.5.

t	${\mathbb{L}}_{\infty }$	${\mathbb{L}}_2$	RE	SR
Problem 6.1
0.200000	3.70541e-06	2.09630e-05	9.26631e-05	0.089628
0.400000	1.49052e-05	8.42881e-05	9.31856e-05	0.089607
0.600000	3.37041e-05	1.90283e-04	9.36508e-05	0.092844
0.800000	6.02922e-05	3.39010e-04	9.42350e-05	0.101798
1.000000	9.50983e-05	5.30232e-04	9.51269e-05	0.118059
Problem 6.2
0.200000	8.27236e-07	4.74645e-06	2.06871e-05	0.156093
0.400000	3.57778e-06	2.05124e-05	2.23679e-05	0.129172
0.600000	9.05724e-06	5.18863e-05	2.51666e-05	0.085483
0.800000	1.86254e-05	1.06655e-04	2.91110e-05	0.037425
1.000000	3.43354e-05	1.95995e-04	3.43457e-05	0.091057
Problem 6.5
0.200000	7.31140e-07	2.54361e-06	9.28346e-05	0.430729
0.400000	7.20769e-06	2.53956e-05	1.14397e-04	0.409372
0.600000	3.64262e-05	1.33152e-04	1.71301e-04	0.354840
0.800000	1.54212e-04	5.58423e-04	3.05947e-04	0.250659
1.000000	5.47349e-04	1.96370e-03	5.55986e-04	0.158460

Figure 9.

Two dimensional graph of time versus spectral radius for Problem 6.1.

Figure 10.

Two dimensional graph of time versus spectral radius for Problem 6.2.

Figure 11.

Two dimensional graph of time versus spectral radius for Problem 6.5.

8. Conclusions

In this paper, a hybrid numerical method has been implemented for the time fractional Burgers' equations involving $\mathcal{ABC}$ derivative in one and two space dimension problems. Stability analysis for the numerical has been analyzed theoretically and verified computationally. The attained outcomes have been equated in tabulated and graphical forms with closed form solutions, showing fabulous agreement. Besides this, a comparative analysis has been made with some prior work [37], [41], where the suggested technique exhibited superiority in terms of error norms. The performance of the method has also been judged using distinct error norms with various resolution levels and time. Numerical computations verified that HW hybrid approximations technique is simple to tackle the generalized time fractional problems.

9. Merits, demerits and future plan

As usual each numerical method has merits and demerits. The proposed method uses Haar basis which are mathematically simple as define in Eqs. (2.1)-(2.2). Mostly, the coefficients of the Haar wavelets are near to zero which eases the computations. Moreover, when resolution level increases the accuracy increases. The computational cost of the proposed method increases (specifically for two dimensional problems) when the resolution level exceeds 5. Also when abrupt changes occur in a system then the method loses accuracy. In the future, one can extend this method to three-dimensional single and coupled equations with $\mathcal{ABC}$ derivative. Besides, some numerical experiments can be conducted using irregular domain as well. Moreover, rigorous theorems regarding the convergence and consistency of such scheme can also be explored.

Funding

No funding is available.

CRediT authorship contribution statement

Abdul Ghafoor: Supervision, Formal analysis, Data curation, Conceptualization. Muhammad Fiaz: Writing – original draft, Software, Methodology, Formal analysis. Kamal Shah: Validation, Project administration, Investigation. Thabet Abdeljawad: Writing – review & editing, Visualization, Funding acquisition.

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

The data would be provided on demand.