A numerical study of fractional rheological models and fractional Newell-Whitehead-Segel equation with non-local and non-singular kernel

Abstract

In the recent years, few type of fractional derivatives which have non-local and non-singular kernel are introduced. In this work, we present fractional rheological models and Newell-Whitehead-Segel equations with non-local and non-singular kernel. For solving these equations, we present a spectral collocation method based on the shifted Legendre polynomials. To do this, we extend the unknown functions and its derivatives using the shifted Legendre basis. These expansions and the properties of the shifted Legendre polynomials along with the spectral collocation method will help us to reduce the main problem to a set of nonlinear algebraic equations. Finally, The accuracy and efficiency of the proposed method are reported by some illustrative examples.

1. Introduction

The study of fractional calculus started at the end of the seventeenth century. It is a branch of mathematical analysis in which integer order derivatives and integrals extend to a real or complex number [1], [2]. In the end of nineteenth century basic theory of fractional calculus was developed with the studies of Liouville, Grünwald, Letnikov, and Riemann. It has been shown that fractional derivative operators are useful in describing dynamical processes with memory or hereditary properties such as creep or relaxation processes in viscoelastoplastic materials [3], [4], impact problem [5], plasma physics [1], diffusion process models [6], [7], [8], [9], chaotic systems [10], control problems [11], [12], dynamics modeling of coronavirus (2019-nCov) [13], etc.

Since in definition of the most important fractional operators such as Riemann-Liouville (RL) and Caputo exists a kernel of type local and sinqular, it is difficult or impossible to describe many non-local dynamics systems. Hence novel definitions for fractional integral and derivative operators have been introduced such as Caputo–Fabrizio (CF) [14] and Atangana–Baleanu (AB) operators [15]. The most important advantage of these operators is the existence of the non-local and non-sinqular kernel which introduced to describe complex physical problems [16], [17], [18], [19], [20], [21], [22], [23].

The AB and CF derivatives show crossover properties for the meansquare displacement, while the RL derivative is scale invariant. Their probability distributions are also a Gaussian to non-Gaussian crossover, with the difference that the CF kernel has a steady state between the transition. Only the AB kernel is a crossover for the waiting time distribution from stretched exponential to power law. The CF derivative is less noisy while the fractional AB derivative provides an excellent description, due to its Mittag-Leffler memory, able to distinguish between dynamical systems taking place at different scales without steady state [24], [25].

Orthogonal basis functions have been generally used to achieve approximate solution for many problems in various fields of science. Approximation of the solution using these functions is known as a useful tool in solving many classes of equations, numerically, e.g., differential equations [26], [27], integro-differential equations [28], [29] and partial differential equations [30] of various orders (fixed, fractional or variable order).

2. Basic concepts

In this section, many definitions of new fractional operators together with their important properties are recall which will be used further.

Definition 2.1

(See Yang [31]) Let 0 < ω  . The RL–integral is defined as

$${{}^{RL}I}_t^{\omega }\epsilon \left(t\right)=\frac{1}{\Gamma \left(\omega \right)}{\int }_0^t{(t-s)}^{\omega -1}\epsilon \left(s\right)ds.$$

The RL–integral of order ω satisfies the following property

$${{}^{RL}I}_t^{\omega }{t}^{\upsilon }=\frac{\Gamma (\upsilon +1)}{\Gamma (\upsilon +1+\omega )}{t}^{\upsilon +\omega },\upsilon \ge 0.$$

Definition 2.2

(See Atangana and Baleanu [15]) Let 0 < ω  ≤ 1, ε ∈ H ¹(0, 1) and AB(ω) be a normalization function suchthat $AB\left(0\right)=AB\left(1\right)=1$ and $AB\left(\omega \right)=1-\omega +\frac{\omega }{\Gamma \left(\omega \right)}$. Then

• i.
  The Caputo AB–derivative is defined as

  $${{}^{ABC}D}_t^{\omega }\epsilon \left(t\right)=\frac{AB\left(\omega \right)}{1-\omega }{\int }_0^t{E}_{\omega }(-\frac{\omega }{1-\omega }{(t-s)}^{\omega })\epsilon {}^{\prime }\left(s\right)ds,$$

  where ${\displaystyle E_{\omega }\left(t\right)=\sum _{i=0}^{\infty }\frac{t^i}{\Gamma (\omega i+1)}}$ is the Mittag-Leffler function.
• ii.
  The AB–integral is given as

  $${{}^{AB}I}_t^{\omega }\epsilon \left(t\right)=\frac{1-\omega }{AB\left(\omega \right)}\epsilon \left(t\right)+\frac{\omega }{AB\left(\omega \right)\Gamma \left(\omega \right)}{\int }_0^t{(t-s)}^{\omega -1}\epsilon \left(s\right)ds.$$ (1)

Let ${\alpha }_{\omega }=\frac{1-\omega }{AB\left(\omega \right)}$ and ${\beta }_{\omega }=\frac{1}{AB\left(\omega \right)\Gamma \left(\omega \right)},$ then we can rewrite (1) as

$${{}^{AB}I}_t^{\omega }\epsilon \left(t\right)={\alpha }_{\omega }\epsilon \left(t\right)+{\beta }_{\omega }\Gamma (\omega +1){{}^{RL}I}_t^{\omega }\epsilon \left(t\right).$$

It is easy to report that the AB–integral satisfies the following properties [32]

$$\begin{array}{c}{}^{\mathit{AB}}{I}_t^{\omega }C=C({\alpha }_{\omega }+{\beta }_{\omega }{t}^{\omega }),C\in \mathbb{R},\hfill \\ {}^{\mathit{AB}}{I}_t^{\omega }{t}^{\upsilon }=t^{\upsilon }({\alpha }_{\omega }+{\beta }_{\omega }(\upsilon +\omega +1)B(\upsilon +1,\omega +1)t^{\omega }),\hfill \\ {}^{\mathit{AB}}{I}_t^{\omega }\left({}^{\mathit{ABC}}{D}_t^{\omega }\epsilon \left(t\right)\right)=\epsilon \left(t\right)-\epsilon \left(0\right),\hfill \end{array}$$

where B( · ,  · ) is the Beta function.

Theorem 2.1

Let C[0, 1] be the space of all continuous functions defined on [0, 1] and f, g ∈ C[0, 1]. Then the following inequality can be established [15]

$$\parallel {}^{\mathit{ABC}}{D}_t^{\omega }f\left(t\right)-{}^{\mathit{ABC}}{D}_t^{\omega }g\left(t\right){\parallel }_{\infty }\le \delta \parallel f\left(t\right)-g\left(t\right){\parallel }_{\infty },$$

where δ is a constant number.

Theorem 2.2

Suppose that f and g satisfy the assumptions of Theorem 2.1 , then we have

$$\parallel {}^{\mathit{AB}}{I}_t^{\omega }f\left(t\right)-{}^{\mathit{AB}}{I}_t^{\omega }g\left(t\right){\parallel }_{\infty }\le \epsilon \parallel f\left(t\right)-g\left(t\right){\parallel }_{\infty },$$

where $\epsilon ={\alpha }_{\omega }+{\beta }_{\omega }$ .

Proof

According to definition of the AB–integral, we have

$$\begin{array}{cc}{\parallel }^{\mathit{AB}}{I}_t^{\omega }f\left(t\right){-}^{\mathit{AB}}{I}_t^{\omega }g\left(t\right){\parallel }_{\infty }& ={\parallel }^{\mathit{AB}}{I}_t^{\omega }(f\left(t\right)-g\left(t\right)){\parallel }_{\infty }\hfill \\ & =\parallel {\alpha }_{\omega }(f\left(t\right)-g\left(t\right))+{\beta }_{\omega }\Gamma {(\omega +1)}^{\mathit{RL}}{I}_t^{\omega }(f\left(t\right)-g\left(t\right)){\parallel }_{\infty }\hfill \\ & \le {\alpha }_{\omega }\parallel f\left(t\right)-g\left(t\right){\parallel }_{\infty }+{\beta }_{\omega }\Gamma (\omega +1){\parallel }^{\mathit{RL}}{I}_t^{\omega }(f\left(t\right)-g\left(t\right)){\parallel }_{\infty }\hfill \\ & \le ({\alpha }_{\omega }+{\beta }_{\omega })\parallel f\left(t\right)-g\left(t\right){\parallel }_{\infty }.\hfill \end{array}$$ (2)

Taking $\epsilon ={\alpha }_{\omega }+{\beta }_{\omega },$ the proof is complete. □

Definition 2.3

(See Caputo and Fabrizio [14]) Let ω ∈ (0, 1], ε ∈ H ¹(0, 1) and CF(ω) be a normalization function suchthat $CF\left(0\right)=CF\left(1\right)=1$ and $CF\left(\omega \right)=\frac{2}{2-\omega }$. Then

• i.
  The CF–derivative is defined as

  $${{}^{CF}D}_t^{\omega }\epsilon \left(t\right)=\frac{(2-\omega )CF\left(\omega \right)}{2(1-\omega )}{\int }_0^t{e}^{-\frac{\omega }{1-\omega }{(t-s)}^{\omega }}\epsilon {}^{\prime }\left(s\right)ds.$$
• ii.
  The CF–integral is given as

  $${}^{CF}{I}_t^{\omega }\epsilon \left(t\right)=\frac{2(1-\omega )}{(2-\omega )CF\left(\omega \right)}\epsilon \left(t\right)+\frac{2\omega }{(2-\omega )CF\left(\omega \right)}{\int }_0^t\epsilon \left(s\right)\mathit{ds}.$$ (3)

Let ${\overline{\alpha }}_{\omega }=\frac{2(1-\omega )}{(2-\omega )CF\left(\omega \right)}$ and ${\overline{\beta }}_{\omega }=\frac{2\omega }{(2-\omega )CF\left(\omega \right)}$, then we can rewrite (3) as

$${{}^{CF}I}_t^{\omega }\epsilon \left(t\right)={\overline{\alpha }}_{\omega }\epsilon \left(t\right)+{\overline{\beta }}_{\omega }{}^{RL}{I}_t^1\epsilon \left(t\right).$$

The CF–integral satisfies the following properties

$$\begin{array}{c}{}^{CF}{I}_t^{\omega }C=C({\overline{\alpha }}_{\omega }+{\overline{\beta }}_{\omega }t),C\in \mathbb{R},\hfill \\ {}^{CF}{I}_t^{\omega }{t}^{\upsilon }=t^{\upsilon }({\overline{\alpha }}_{\omega }+\frac{{\overline{\beta }}_{\omega }}{\upsilon +1}t),\hfill \\ {}^{CF}{I}_t^{\omega }\left({}^{CF}{D}^{\omega }\epsilon \left(t\right)\right)=\epsilon \left(t\right)-\epsilon \left(0\right).\hfill \end{array}$$

Theorem 2.3

Let C[0, 1] be the space of all continuous functions defined on [0, 1] and f, g ∈ C[0, 1]. Then the following inequality can be established

$$\parallel {}^{\mathit{CF}}{I}_t^{\omega }f\left(t\right){-}^{\mathit{CF}}{I}_t^{\omega }g\left(t\right){\parallel }_{\infty }\le \epsilon \parallel f\left(t\right)-g\left(t\right){\parallel }_{\infty },$$

where $\epsilon ={\overline{\alpha }}_{\omega }+{\overline{\beta }}_{\omega }$ .

Proof

According to definition of the CF-integral, we have

$$\begin{array}{cc}{\parallel }^{\mathit{CF}}{I}_t^{\omega }f\left(t\right){-}^{\mathit{CF}}{I}_t^{\omega }g\left(t\right){\parallel }_{\infty }\hfill & ={\parallel }^{\mathit{CF}}{I}_t^{\omega }(f\left(t\right)-g\left(t\right)){\parallel }_{\infty }\hfill \\ & =\parallel {\overline{\alpha }}_{\omega }(f\left(t\right)-g\left(t\right))+{\overline{\beta }}_{\omega }^{\mathit{RL}}{I}_t^1(f\left(t\right)-g\left(t\right)){\parallel }_{\infty }\hfill \\ & \le {\overline{\alpha }}_{\omega }\parallel f\left(t\right)-g\left(t\right){\parallel }_{\infty }+{\overline{\beta }}_{\omega }{\parallel }^{\mathit{RL}}{I}_t^1(f\left(t\right)-g\left(t\right)){\parallel }_{\infty }\hfill \\ & \le ({\overline{\alpha }}_{\omega }+{\overline{\beta }}_{\omega })\parallel f\left(t\right)-g\left(t\right){\parallel }_{\infty }.\hfill \end{array}$$ (4)

Taking $\epsilon ={\overline{\alpha }}_{\omega }+{\overline{\beta }}_{\omega },$ the proof completes. □

3. The shifted Legendre polynomials and their properties

The shifted Legendre polynomials (SLPs) on the interval [0, 1] are defined by

$${\mathcal{L}}_n\left(t\right)=L_n(2t-1),n=0,1,2,\dots ,$$ (5)

where L_n(t) is the well-known Legendre polynomial (LP) of degree n. The recursive formula of LP on $[-1,1]$ is given by

$$\begin{array}{c}{L}_0\left(t\right)=1,\hfill \\ L_1\left(t\right)=t,\hfill \\ L_{n+1}\left(t\right)=\frac{2n+1}{n+1}t{L}_n\left(t\right)-\frac{n}{n+1}{L}_{n-1}\left(t\right),n=1,2,3,\cdots .\hfill \end{array}$$

The given SLPs (${\mathcal{L}}_n\left(t\right)$) in the Eq. (5), could be written the following analytic form

$${\displaystyle {\mathcal{L}}_n\left(t\right)=\sum _{k=0}^n{\varsigma }_{n,k}{t}^k,}$$ (6)

where

$${\varsigma }_{n,k}=\frac{{(-1)}^{n+k}(n+k)!}{(n-k)!{(k!)}^2}.$$ (7)

For two arbitrary functions h, p ∈ L ²[0, 1] the inner product and norm in this space are defined, respectively, by

$$〈h\left(t\right),p\left(t\right)〉={\int }_0^{1}h\left(t\right)p\left(t\right)dt,$$

$${\parallel h\left(t\right)\parallel }_2={〈h\left(t\right),h\left(t\right)〉}^{\frac{1}{2}}={\left({\int }_0^1{\left|h\left(t\right)\right|}^{2}dt\right)}^{\frac{1}{2}}.$$

For the SLPs, the orthogonality condition is as follows

$$〈{\mathcal{L}}_m\left(t\right),{\mathcal{L}}_n\left(t\right)〉=\{\begin{array}{cc}\frac{1}{2m+1},\hfill & m=n,\hfill \\ 0,\hfill & m\ne n.\hfill \end{array}$$

Suppose that ε(t) ∈ L ²[0, 1]. Then, the function ε(t) can be expanded in terms of the SLPs by

$$\epsilon \left(t\right)=\sum _{i=0}^{\infty }{\epsilon }_i{\mathcal{L}}_i\left(t\right),$$ (8)

where

$${\epsilon }_i=\frac{〈\epsilon \left(t\right),{\mathcal{L}}_i\left(t\right)〉}{〈{\mathcal{L}}_i\left(t\right),{\mathcal{L}}_i\left(t\right)〉}=(2i+1){\int }_0^1\epsilon \left(t\right){\mathcal{L}}_i\left(t\right)dt.$$ (9)

By taking only the first $M+1$ terms in (8), ε(t) can be approximated as

$$\epsilon \left(t\right)\simeq {\epsilon }_M\left(t\right)=\sum _{i=0}^M{\epsilon }_i{\mathcal{L}}_i\left(t\right)=C^T\mathfrak{L}\left(t\right),$$ (10)

where $C={[{\epsilon }_0,{\epsilon }_1,\dots ,{\epsilon }_M]}^T$ and

$$\mathfrak{L}\left(t\right)={[{\mathcal{L}}_0\left(t\right),{\mathcal{L}}_1\left(t\right),\dots ,{\mathcal{L}}_M\left(t\right)]}^T.$$ (11)

Theorem 3.1

Suppose that $\epsilon \in C^{M+1}[0,1]$ and $H=span\{{\mathcal{L}}_0\left(t\right),{\mathcal{L}}_1\left(t\right),\dots ,{\mathcal{L}}_M\left(t\right)\}\subset L^2[0,1]$ . Assume ε_M is the best approximation of ε into H, then the error bound is as

$$\parallel \epsilon \left(t\right)-{\epsilon }_M\left(t\right){\parallel }_2\le \frac{\rho }{(M+1)!2^{2M+1}},$$

where $\rho ={\sup }_{\theta \in [0,1]}\left|{\epsilon }^{(M+1)}\left(\theta \right)\right|$ .

Proof

Suppose that P_M is the interpolating polynomials to ε at points t_i, where $t_i,i=0,1,\dots ,M$ are the roots of $(M+1)$-degree shifted Chebysheve polynomials on [0, 1]. Then

$$\epsilon \left(t\right)-P_M\left(t\right)=\frac{{\epsilon }^{(M+1)}\left(\theta \right)}{(M+1)!}\prod _{i=0}^M(t-t_i),\theta \in [0,1].$$

So,

$$|\epsilon \left(t\right)-P_M\left(t\right)|\le \frac{\rho }{(M+1)!2^{2M+1}},$$

where $\rho ={\sup }_{\theta \in [0,1]}\left|{\epsilon }^{(M+1)}\left(\theta \right)\right|$.

Since ε_M is the best approximation of ε in H, we get

$$\begin{array}{cc}\parallel \epsilon \left(t\right)-{\epsilon }_M\left(t\right){\parallel }_2^2& \le \parallel \epsilon \left(t\right)-P_M\left(t\right){\parallel }_2^2={\int }_0^1{|\epsilon \left(t\right)-{\epsilon }_M\left(t\right)|}^2\mathit{dt}\\ & ={\int }_0^1{\left(\frac{\rho }{(M+1)!2^{2M+1}}\right)}^2\mathit{dt}={\left(\frac{\rho }{(M+1)!2^{2M+1}}\right)}^2.\end{array}$$ (12)

By taking the squared root from both sides (12), the proof completes. □

Similarly, any function ε(x, t) in $H^{*}=L^2([0,1]\times [0,1])$ can be approximated in terms of the SLPs as

$$\epsilon (x,t)\simeq {\mathfrak{L}}^T\left(x\right)\mathcal{C}\mathfrak{L}\left(t\right),$$ (13)

where $\mathcal{C}=\left[c_{i,j}\right]$ is an $(M+1)\times (M+1)$ matrix whose elements are given by

$$c_{i,j}=\frac{〈〈\epsilon (t,s),{\mathcal{L}}_i\left(t\right)〉,{\mathcal{L}}_j\left(s\right)〉}{\parallel {\mathcal{L}}_i\left(t\right){\parallel }_2^2\parallel {\mathcal{L}}_j\left(s\right){\parallel }_2^2},i,j=0,1,\cdots ,M.$$

Theorem 3.2

Let $H^{*}=span\{{\mathcal{L}}_0\left(x\right){\mathcal{L}}_0\left(t\right),\dots ,{\mathcal{L}}_0\left(x\right){\mathcal{L}}_M\left(t\right),\dots ,{\mathcal{L}}_M\left(x\right){\mathcal{L}}_0\left(t\right),\dots ,{\mathcal{L}}_M\left(x\right){\mathcal{L}}_M\left(t\right)\}$ and ε ∈ H* is a smooth function defined on $I=[0,1]\times [0,1]$ with bounded derivatives as follows

$${\displaystyle \underset{(x,t)\in I}{\max }\left|\frac{{\partial }^{M+1}\epsilon (x,t)}{\partial x^{M+1}}\right|\le {\theta }_1,\underset{(x,t)\in I}{\max }\left|\frac{{\partial }^{M+1}\epsilon (x,t)}{\partial t^{M+1}}\right|\le {\theta }_2,\underset{(x,t)\in I}{\max }\left|\frac{{\partial }^{2M+2}\epsilon (x,t)}{\partial x^{M+1}\partial t^{M+1}}\right|\le {\theta }_3,}$$

where θ ₁, θ ₂ and θ ₃ are positive constants. If ${\epsilon }_M(x,t)={\mathfrak{L}}^T\left(x\right)\mathcal{C}\mathfrak{L}\left(t\right)$ be the best approximation of ε into H*, then

$$\parallel \epsilon (x,t)-{\epsilon }_M(x,t){\parallel }_2\le \frac{1}{(M+1)!2^{2M+1}}({\theta }_1+{\theta }_2+\frac{{\theta }_3}{(M+1)!2^{2M+1}}).$$

Proof

Let that P_M is the interpolating polynomials to ε at points (x_i, t_j), where $x_i,i=0,1,\dots ,M$ and $t_j,j=0,1,\dots ,M$ are the roots of $(M+1)$-degree shifted Chebysheve polynomials on [0, 1]. Then

$$\begin{array}{c}\epsilon (x,t)-P_M(x,t)=\frac{{\partial }^{M+1}\epsilon (\xi ,t)}{\partial x^{M+1}(M+1)!}\prod _{i=0}^M(x-x_i)+\frac{{\partial }^{M+1}\epsilon (x,\eta )}{\partial t^{M+1}(M+1)!}\prod _{j=0}^M(t-t_j)\hfill \\ -\frac{{\partial }^{2M+2}\epsilon ({\xi }^{\text{’}},{\eta }^{\text{’}})}{\partial x^{M+1}\partial t^{M+1}{((M+1)!)}^2}\prod _{i=0}^M(x-x_i)\prod _{j=0}^M(t-t_j),\hfill \end{array}$$

where ξ, ξ′, η, η′ ∈ [0, 1]. Then we obtain

$$|\epsilon (x,t)-P_M(x,t)|\le {\displaystyle \underset{(x,t)\in I}{\max }\left|\frac{{\partial }^{M+1}\epsilon (\xi ,t)}{\partial x^{M+1}}\right|\frac{{\displaystyle \prod _{i=0}^M|x-x_i|}}{(M+1)!}+\underset{(x,t)\in I}{\max }\left|\frac{{\partial }^{M+1}\epsilon (x,\eta )}{\partial t^{M+1}}\right|\frac{{\displaystyle \prod _{j=0}^M|t-t_j|}}{(M+1)!}}+{\displaystyle \underset{(x,t)\in I}{\max }\left|\frac{{\partial }^{2M+2}\chi ({\xi }^{\prime },{\eta }^{\prime })}{\partial x^{M+1}\partial t^{M+1}}\right|\frac{{\displaystyle \prod _{i=0}^M|x-x_i|\prod _{j=0}^M|t-t_j|}}{{((M+1)!)}^2}.}$$ (14)

Since ε(x, t) is a smooth function on I, then there exist constants θ ₁, θ ₂ and θ ₃, such that

$$\underset{(x,t)\in I}{\max }\left|\frac{{\partial }^{M+1}\epsilon (x,t)}{\partial x^{M+1}}\right|\le {\theta }_1,\underset{(x,t)\in I}{\max }\left|\frac{{\partial }^{M+1}\epsilon (x,t)}{\partial t^{M+1}}\right|\le {\theta }_2,\underset{(x,t)\in I}{\max }\left|\frac{{\partial }^{2M+2}\epsilon (x,t)}{\partial x^{M+1}\partial t^{M+1}}\right|\le {\theta }_3.$$ (15)

By substituting (15) into (14) and employing the estimates for Chebysheve interpolation nodes, we have

$$|\epsilon (x,t)-P_M(x,t)|\le \frac{1}{(M+1)!2^{2M+1}}({\theta }_1+{\theta }_2+\frac{{\theta }_3}{(M+1)!2^{2M+1}}).$$ (16)

Since ε_M is the best approximation of ε in H*, that is

$$\parallel \epsilon (x,t)-{\epsilon }_M(x,t){\parallel }_2\le \parallel \epsilon (x,t)-{\epsilon }^{*}(x,t){\parallel }_2,$$

where ε* is any arbitrary polynomial in H*. Then, using (16) we obtain

$$\begin{array}{cc}{\parallel \epsilon (x,t)-{\epsilon }_M(x,t)\parallel }_2^2\hfill & ={\int }_0^1{\int }_0^1{|\epsilon (x,t)-{\epsilon }_M(x,t)|}^{2}dxdt\hfill \\ & \le {\int }_0^1{\int }_0^1{|\epsilon (x,t)-P_M(x,t)|}^{2}dxdt\hfill \\ & ={\int }_0^1{\int }_0^1{\left(\frac{1}{(M+1)!2^{2M+1}}({\theta }_1+{\theta }_2+\frac{{\theta }_3}{(M+1)!2^{2M+1}})\right)}^{2}dxdt\hfill \\ & ={\left(\frac{1}{(M+1)!2^{2M+1}}({\theta }_1+{\theta }_2+\frac{{\theta }_3}{(M+1)!2^{2M+1}})\right)}^2.\hfill \end{array}$$ (17)

Finally, taking the square root of both sides of (17) completes the proof. □

3.1. Operational matrices of the SLPs

This subsection is devoted to introducing some operational matrices (OMs) of the SLPs basis vector which will be used further.

• (1)
  The OM of the integration of the vector $\mathfrak{L}\left(t\right)$ given by (11) can be approximated as

  $${\int }_0^t\mathfrak{L}\left(s\right)ds\simeq P\mathfrak{L}\left(t\right),$$ (18)

  where P is given as [29]

  $$P=\frac{1}{2}\left[\begin{array}{ccccccc}1& 1& 0& \dots & 0& 0& 0\\ \frac{-1}{3}& 0& \frac{1}{3}& \dots & 0& 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & \frac{-1}{2M-1}& 0& \frac{1}{2M-1}\\ 0& 0& 0& \dots & 0& \frac{-1}{2M+1}& 0\end{array}\right].$$
• (2)
  The OM of AB–integral of order ω of the vector $\mathfrak{L}\left(t\right)$ is obtained as

  $${}^{AB}{I}_t^{\omega }\mathfrak{L}\left(t\right)={\alpha }_{\omega }\mathfrak{L}\left(t\right)+{\beta }_{\omega }\Gamma (\omega +1){}^{RL}{I}_t^{\omega }\mathfrak{L}\left(t\right).$$ (19)

  Now, we must obtain the OM of RL–integral of order ω. To do this, we apply the LR–integral operator, ${}^{RL}{I}_t^{\omega },$ on ${\mathcal{L}}_i\left(t\right),i=0,1,\dots ,M$ as

  $${{}^{RL}I}_t^{\omega }{\mathcal{L}}_i\left(t\right)={{}^{RL}I}_t^{\omega }\left(\sum _{r=0}^i{\varsigma }_{i,r}{t}^r\right)=\sum _{r=0}^i{\varsigma }_{i,r}\left({{}^{RL}I}_t^{\omega }{t}^r\right)=\sum _{r=0}^i\frac{\Gamma (r+1){\varsigma }_{i,r}}{\Gamma (r+\omega +1)}{t}^{r+\omega }.$$

  By approximating the function $t^{r+\omega }$ in terms of the SLPs, we have

  $$t^{r+\omega }\simeq \sum _{k=0}^M{e}_{r,k}{\mathcal{L}}_k\left(t\right).$$ (20)

  In view of (20) and for $i=0,1,\dots ,M,$ we get

  $${{}^{RL}I}_t^{\omega }{\mathcal{L}}_i\left(t\right)\simeq \sum _{r=0}^i\frac{\Gamma (r+1){\varsigma }_{i,r}}{\Gamma (r+\omega +1)}\left(\sum _{k=0}^M{e}_{r,k}{\mathcal{L}}_k\left(t\right)\right)=\sum _{k=0}^M\left(\sum _{r=0}^i\frac{\Gamma (r+1){\varsigma }_{i,r}{e}_{r,k}}{\Gamma (r+\omega +1)}\right){\mathcal{L}}_k\left(t\right)=\sum _{k=0}^M\left(\sum _{r=0}^i{\rho }_{i,k,r}\right){\mathcal{L}}_k\left(t\right).$$

  Therefore, for $i=0,1,\dots ,M,$ we can write

  $${{}^{RL}I}_t^{\omega }\mathfrak{L}\left(t\right)={\mathcal{F}}^{\omega }\mathfrak{L}\left(t\right),$$ (21)

  where

  $${\mathcal{F}}^{\omega }=\left[\begin{array}{cccc}{\rho }_{0,0,0}& {\rho }_{0,1,0}& \dots & {\rho }_{0,M,0}\\ {\displaystyle \sum _{r=0}^1{\rho }_{1,0,r}}& {\displaystyle \sum _{r=0}^1{\rho }_{1,1,r}}& \dots & {\displaystyle \sum _{r=0}^1{\rho }_{1,M,r}}\\ ⋮& ⋮& \dots & ⋮\\ {\displaystyle \sum _{r=0}^M{\rho }_{M,0,r}}& {\displaystyle \sum _{r=0}^M{\rho }_{M,1,r}}& \dots & {\displaystyle \sum _{r=0}^M{\rho }_{M,M,r}}\end{array}\right],$$

  with

  $${\rho }_{i,k,r}=\frac{\Gamma (r+1){\varsigma }_{i,r}{e}_{r,k}}{\Gamma (r+\omega +1)}.$$

  By substituting (21) into (19), the proof completes.
• (3)
  The OM of CF–integral of order ω of the vector $\mathfrak{L}\left(t\right)$ is obtained as

  $${}^{CF}{I}^{\omega }\mathfrak{L}\left(t\right)={\overline{\alpha }}_{\omega }\mathfrak{L}\left(t\right)+{\overline{\beta }}_{\omega }{}^{RL}{I}^1\mathfrak{L}\left(t\right).$$

  In view of (18), we have

  $${}^{CF}{I}^{\omega }\mathfrak{L}\left(t\right)={\overline{\alpha }}_{\omega }\mathfrak{L}\left(t\right)+{\overline{\beta }}_{\omega }P\mathfrak{L}\left(t\right)=({\overline{\alpha }}_{\omega }I+{\overline{\beta }}_{\omega }P)\mathfrak{L}\left(t\right)=I^{\omega }\mathfrak{L}\left(t\right),$$ (22)

  where I is an $(M+1)\times (M+1)$ identity matrix and $I^{\omega }={\overline{\alpha }}_{\omega }I+{\overline{\beta }}_{\omega }P$. The matrix I^ω is called the OM of CF–integral based on the SLPs.

4. Applications

4.1. Rheological models

• i.
  Classic approach The behavior of linear viscoelastic materials can be described by linear differential equations. In general, a constitutive equation for a linear viscoelastic material is given by

  $$\mathcal{A}\sigma =\mathcal{B}\epsilon ,$$

  where $\mathcal{A}:={\sum }_{i=0}^n{a}_i{D}_t^i,\mathcal{B}:={\sum }_{i=0}^m{b}_i{D}_t^i,a_i,b_i\in \mathbb{R},i=0,1,\dots ,n,j=0,1,\dots ,m,$ and n ≥ m. Thus, models including connected common mechanical elements such as springs and dampers can be used to visualize the constitutive equation in a convenient way. These descriptions are known as rheological models constructed by combining linear springs and dampers in series and parallel. Three well-known rheological models called Kelvin-Voigt, Maxwell, and Zener models. More complex rheological models with more realistic responses can be constructed by including additional elements [2].
  A Kelvin-Voigt element is composed of a linear spring and damper connected in parallel, and its constitutive equation is given as

  $$\eta D_t\epsilon \left(t\right)+E\epsilon \left(t\right)=\sigma \left(t\right),$$

  where E is the elasticity modulus and η is the viscosity. Under a creep test with $\sigma \left(t\right)={\sigma }_0$ and $\epsilon \left(0\right)=0,$ the response is obtained as

  $$\epsilon \left(t\right)=(1-exp[-\frac{E}{\eta }t])\frac{{\sigma }_0}{E}.$$

  A Maxwell element is composed of a linear spring and damper connected in series, and its constitutive equation is given as

  $$D_t\epsilon \left(t\right)=\frac{D_t\sigma \left(t\right)}{E}+\frac{\sigma \left(t\right)}{\eta }.$$

  Under a creep test with $\sigma \left(t\right)={\sigma }_0$ and $\epsilon \left(0\right)=0,$ the response is

  $$\epsilon \left(t\right)=(\frac{1}{E}+\frac{1}{\eta }t){\sigma }_0.$$

  A Zener element is composed of a linear spring and a Maxwell element connected in parallel, and its constitutive equation is given as

  $$D_t\epsilon \left(t\right)+\frac{E_1{E}_2}{E_1+E_2}\epsilon \left(t\right)=\frac{1}{E_1+E_2}{D}_t\sigma \left(t\right)+\frac{E_2}{\eta (E_1+E_2)}\sigma \left(t\right).$$

  Under a creep test with $\sigma \left(t\right)={\sigma }_0$ and $\epsilon \left(0\right)=0,$ the response of the model is

  $$\epsilon \left(t\right)=(1-exp[-\frac{E_1{E}_2}{E_1+E_2}t])\frac{{\sigma }_0}{\eta E_1}.$$
• ii. Fractional approach
  In general, consider the following FDE:

  $$D_t^{\omega }\epsilon \left(t\right)+{\lambda }_1\epsilon \left(t\right)={\lambda }_2{D}_t^v\sigma \left(t\right)+{\lambda }_3\sigma \left(t\right),$$ (23)

  where 0 < ω ≤ 1 and ${\lambda }_i\in {\mathbb{R}}^{+},i=1,2,3$. $D_t^{\omega }$ is denoted either the AB (${{}^{ABC}D}_t^{\omega }$) derivative or CF (${{}^{CF}D}_t^{\omega }$) derivative. The constitutive equation of the proposed fractional rheological models can be obtained by adjusting the parameters λ_i in the equation (23).

  • a.
    The constitutive equation of the fractional Kelvin-Voigt model is obtained when

    $${\lambda }_1=\frac{E}{\eta },{\lambda }_2=0,{\lambda }_3=\frac{1}{\eta },\omega =v.$$
  • b.
    The constitutive equation of the fractional Maxwell model is obtained when

    $${\lambda }_1=0,{\lambda }_2=\frac{1}{E},{\lambda }_3=\frac{1}{\eta },\omega =v.$$
  • c.
    The constitutive equation of the fractional Zener model is obtained when

    $${\lambda }_1=\frac{E_1{E}_2}{E_1+E_2},{\lambda }_2=\frac{1}{E_1+E_2},{\lambda }_3=\frac{E_2}{\eta (E_1+E_2)},\omega =v.$$

  All the previous settings for λ_i yield the classic rheological models when $\omega =v=1$.
• iii. The method
  In here, we introduce a numerical method for the solution of the form Eq. (23). Let in the Eq. (23), the derivative is described in the AB (or CF) sense. For solving the Eq. (23), first we approximate ${}^{ABC}{D}_t^{\omega }\epsilon \left(t\right)$ and ${}^{ABC}{D}_t^v\sigma \left(t\right)$ as

  $$\begin{array}{cc}& {}^{ABC}{D}_t^{\omega }\epsilon \left(t\right)\simeq C_1^T\mathfrak{L}\left(t\right),\hfill \\ & {}^{ABC}{D}_t^v\sigma \left(t\right)\simeq C_2^T\mathfrak{L}\left(t\right).\hfill \end{array}$$ (24)

  By taking the AB–integral of (24) and using initial conditions ($\epsilon \left(0\right)=0,\sigma \left(0\right)={\sigma }_0$), we have

  $$\begin{array}{c}\epsilon \left(t\right)\simeq C_1^T{J}^{\omega }\mathfrak{L}\left(t\right),\hfill \\ \sigma \left(t\right)\simeq C_2^T{J}^v\mathfrak{L}\left(t\right)+\sigma \left(0\right).\hfill \end{array}$$ (25)

  By approximating $\sigma \left(0\right)\simeq C_3^T\mathfrak{L}\left(t\right),$ σ(t) can be rewritten as

  $$\sigma \left(t\right)\simeq C_4\mathfrak{L}\left(t\right),$$ (26)

  where $C_4=C_2^T{J}^v+C_3^T$. By putting (24)-(26) in (23), we have

  $$C_1^T+{\lambda }_1{C}_1^T{J}^{\omega }-{\lambda }_2{C}_2^T-{\lambda }_3{C}_4=0.$$ (27)

  By solving the system (27), the unknown parameters are obtained. Finally the approximate solution can be computed by (25).
• iv. Test examples

  The creep behavior of the fractional Kelvin-Voigt model for different values of $E=\{0.3,0.5,0.8,1\}$ and $\omega =\{0.5,0.6,0.7,0.8,0.9,0.99,1\}$ is shown in Figs. 1 and 2 , when $\eta =0.5$ and ${\sigma }_0=1$ for the AB and CF derivatives, respectively. We used Mathematica for computation.

Fig. 1.

The creep behavior of the Kelvin-Voigt model for different values of E, ω and $M=5,$ by considering the AB derivative.

Fig. 2.

The creep behavior of the Kelvin-Voigt model for different values of E, ω and $M=5,$ by considering the CF derivative.

4.2. The Newell-Whitehead-Segel equation

• i. Classic approach
  Consider the Newell-Whitehead-Segel equation

  $$D_t\epsilon (x,t)={\lambda }_1{D}_{xx}\epsilon (x,t)+{\lambda }_2\epsilon (x,t)-{\lambda }_3\epsilon {(x,t)}^{{\lambda }_4},$$ (28)

  with initial condition

  $$\epsilon (x,0)=f_0\left(x\right),$$ (29)

  and boundary conditions

  $$\epsilon (0,t)=f_1\left(t\right),\epsilon (1,t)=f_2\left(t\right),$$ (30)

  where λ ₁, λ ₂ and λ ₃ are real numbers with λ ₁ > 0, and λ ₄ is a positive integer number. When ${\lambda }_1=1,$ ${\lambda }_2=1,$ ${\lambda }_3=1$ and ${\lambda }_4=2,$ the Eq. (28) is called the Fishers equation.
• ii. Fractional approach
  Consider the Newell-Whitehead-Segel equation by

  $$D_t^{\omega }\epsilon (x,t)={\lambda }_1{D}_{xx}\epsilon (x,t)+{\lambda }_2\epsilon (x,t)-{\lambda }_3\epsilon {(x,t)}^{{\lambda }_4},(x,t)\in [0,1]\times [0,1],$$ (31)

  where 0 < ω ≤ 1. In the fractional Newell-Whitehead-Segel Eq. (31), $D_t^{\omega }$ is called the AB or CF derivative operator.
• iii.
  The methodHere, we introduce a numerical method for the solution of the form Eq. (31). Let in the Eq. (31), the derivative is described in the AB (or CF) sense. For solving the Eq. (31), first we approximate D_xx ε(x, t) as

  $$D_{xx}\epsilon (x,t)\simeq {\mathfrak{L}}^T\left(x\right)C_1\mathfrak{L}\left(t\right),$$ (32)

  where C ₁ is an $(M+1)\times (M+1)$ unknown vector. By integrating from (32) respect x twice, we get

  $$\epsilon (x,t)\simeq {\mathfrak{L}}^T\left(x\right){\left(P^2\right)}^T{C}_1\mathfrak{L}\left(t\right)+x{\epsilon }_x(0,t)+\epsilon (0,t).$$ (33)

  By putting $x=1$ into (33), we have

  $${\epsilon }_x(0,t)=\epsilon (1,t)-\epsilon (0,t)-{\mathfrak{L}}^T\left(1\right){\left(P^2\right)}^T{C}_1\mathfrak{L}\left(t\right).$$ (34)

  Let

  $$1\simeq C_2^T\mathfrak{L}\left(t\right)\left(or{C}_2^T\mathfrak{L}\left(x\right)\right),x\simeq C_3^T\mathfrak{L}\left(x\right),\epsilon (0,t)\simeq C_4^T\mathfrak{L}\left(t\right),\epsilon (1,t)\simeq C_5^T\mathfrak{L}\left(t\right),$$ (35)

  where the elements of C ₂, C ₃, C ₄ and C ₅ vectors can be calculused by (9). With helping (34) and (35), ε(x, t) can be rewritten as

  $$\epsilon (x,t)\simeq {\mathfrak{L}}^T\left(x\right)C_6\mathfrak{L}\left(t\right),$$ (36)

  where $C_6={\left(P^2\right)}^T{C}_1+C_3{C}_5^T-C_3{\mathfrak{L}}^T\left(1\right){\left(P^2\right)}^T{C}_1-C_3{C}_4^T+C_2{C}_4^T$.
  Let $F(x,t,\epsilon (x,t))={\lambda }_2\epsilon (x,t)-{\lambda }_3\epsilon {(x,t)}^{\lambda 4}$. We approximate F and ε(x, 0) using the shifted Legender basis by

  $$F(x,t,\epsilon (x,t))={\mathfrak{L}}^T\left(x\right)C_7\mathfrak{L}\left(t\right),$$ (37)

  $$\epsilon (x,0)=C_8^T\phi \left(t\right),$$ (38)

  where C ₇ is an $(M+1)\times (M+1)$ unknown vector and the elements of C ₈ vector can be obtained by (9). By taking the AB-integral of both sides of the Eq. (31) and using (32), (36)–(38), we get

  $$C_6-C_8{C}_2^T-{\lambda }_1{C}_1{J}^{\omega }-C_7{J}^{\omega }=0.$$ (39)

  Now, by putting (36) into (37) and using the collocation points $x_i=\frac{i}{M+2},i=1,2,\dots ,M+1$ and $t_j=\frac{j}{M+2},j=1,2,\dots ,M+1,$ gives

  $$F(x_i,t_j,{\mathfrak{L}}^T\left(x_i\right)C_6\mathfrak{L}\left(t_j\right))-{\mathfrak{L}}^T\left(x_i\right)C_7\mathfrak{L}\left(t_j\right)=0.$$ (40)

  Eqs. (39) and (40) form a system of $2(M+1)(M+1)$ nonlinear equations of the vectors of C ₁ and C ₇.

  By solving this system, the unknown parameters of the vectors of C ₁ and C ₇ are obtained. Finally the approximate solution can be computed by (36).

• iv. Test examples
  Consider the Newell-Whitehead-Segel equation in the following cases

  • Case 1.
    By considrting ${\lambda }_1=1,{\lambda }_2=1,{\lambda }_3=1\text{and}{\lambda }_4=2$, the Newell-Whitehead-Segel equation with the exact solution $\epsilon (x,t)=\frac{1}{{(1+e^{\frac{x}{\sqrt{6}}-\frac{5}{6}t})}^2}$ is as follows

    $$D_t^{\omega }\epsilon (x,t)=D_{xx}\epsilon (x,t)+\epsilon (x,t)-\epsilon {(x,t)}^2.$$
  • Case 2.
    By considering ${\lambda }_1=1,{\lambda }_2=1,{\lambda }_3=1\text{and}{\lambda }_4=3$, the Newell-Whitehead-Segel equation with the exact solution $\epsilon (x,t)=\frac{1}{2}(1+\tanh \left(\frac{\sqrt{2}x+3t}{4}\right))$ is as follows

    $$D_t^{\omega }\epsilon (x,t)=D_{xx}\epsilon (x,t)+\epsilon (x,t)-\epsilon {(x,t)}^3.$$
  • Case 3.
    By considering ${\lambda }_1=1,{\lambda }_2=3,{\lambda }_3=4\text{and}{\lambda }_4=3$, the Newell-Whitehead-Segel equation with the exact solution $\epsilon (x,t)=\sqrt{\frac{3}{4}}\frac{e^{\sqrt{6}x}}{e^{\sqrt{6}x}+e^{\frac{\sqrt{6}}{2}x-\frac{9}{2}t}}$ is as follows

    $$D_t^{\omega }\epsilon (x,t)=D_{xx}\epsilon (x,t)+3\epsilon (x,t)-4\epsilon {(x,t)}^3.$$

  By solving the Newell-Whitehead-Segel equation in above cases using the proposed method, the numerical results for different values of ω are reported in Table 1, Table 2, Table 3 and Figs. 3 , 4 and و .

Table 1.

Comparison of the absolute error at some selected points ($M=5,{\lambda }_1=1,{\lambda }_2=1,{\lambda }_3=1,{\lambda }_4=2$).

(x, t)	$\omega =0.5$		$\omega =0.7$		$\omega =0.99$
	AB	CF	AB	CF	AB	CF
(0.1,0.1)	$7.525e-2$	$6.619e-3$	$6.486e-3$	$5.450e-3$	$4.428e-4$	$2.744e-4$
(0.3,0.3)	$1.494e-3$	$1.062e-2$	$1.161e-2$	$6.956e-3$	$3.825e-4$	$7.925e-6$
(0.5,0.5)	$1.472e-2$	$7.619e-3$	$1.042e-2$	$3.408e-3$	$1.690e-4$	$1.527e-4$
(0.7,0.7)	$9.731e-3$	$2.521e-3$	$6.184e-3$	$3.423e-4$	$7.833e-6$	$1.907e-4$
(0.9,0.9)	$2.974e-3$	$5.007e-4$	$1.574e-3$	$1.288e-3$	$2.798e-5$	$7.392e-5$

Table 2.

Comparison of the absolute error at some selected points ($M=5,{\lambda }_1=1,{\lambda }_2=1,{\lambda }_3=1,{\lambda }_4=3$).

(x, t)	$\omega =0.5$		$\omega =0.7$		$\omega =0.99$
	AB	CF	AB	CF	AB	CF
(0.1,0.1)	$1.264e-2$	$1.112e-2$	$1.091e-2$	$9.147e-3$	$7.272e-4$	$4.29809e-4$
(0.3,0.3)	$1.977e-2$	$1.325e-2$	$1.495e-2$	$7.890e-3$	$2.671e-4$	$2.60225e-4$
(0.5,0.5)	$1.316e-2$	$3.731e-3$	$8.061e-3$	$1.091e-3$	$2.576e-4$	$5.29674e-4$
(0.7,0.7)	$3.730e-3$	$4.513e-3$	$5.362e-4$	$6.466e-3$	$4.170e-4$	$4.66574e-4$
(0.9,0.9)	$9.471e-4$	$4.303e-3$	$1.770e-3$	$4.135e-3$	$1.778e-4$	$1.23433e-4$

Table 3.

Comparison of the absolute error at some selected points ($M=5,{\lambda }_1=1,{\lambda }_2=3,{\lambda }_3=4,{\lambda }_4=3$).

(x, t)	$\omega =0.5$		$\omega =0.7$		$\omega =0.99$
	AB	CF	AB	CF	AB	CF
(0.1,0.1)	$2.488e-2$	$2.203e-2$	$2.165e-2$	$1.823e-2$	$1.253e-3$	$5.482e-4$
(0.3,0.3)	$1.729e-2$	$8.122e-3$	$1.119e-2$	$1.119e-3$	$1.064e-3$	$1.610e-3$
(0.5,0.5)	$2.379e-3$	$1.197e-2$	$6.092e-3$	$1.486e-2$	$1.106e-3$	$8.904e-4$
(0.7,0.7)	$8.982e-3$	$1.514e-2$	$9.800e-3$	$1.391e-3$	$8.813e-4$	$6.349e-4$
(0.9,0.9)	$4.675e-3$	$6.516e-3$	$4.367e-3$	$4.932e-3$	$4.051e-7$	$1.552e-4$

Fig. 3.

The numerical results for different values of ω and $M=5,$ by considering the AB derivative.

Fig. 4.

The numerical results for different values of ω and $M=5,$ by considering the CF derivative.

5. Conclusion

In this work, we have presented a numerical method for solving fractional rheological models and Newell-Whitehead-Segel equations.

The derivative is considered in the Caputo–Fabrizio and Atangana–Baleanu sense. Our numerical method is based on the operational matrices of the shifted Legendre polynomials . By this way, the main problem is reduced to a system of nonlinear algebraic equations which greatly simplifies the problem. An error estimation is proved for the approximate solution. Finally, some examples have been presented to demonstrate the accuracy and efficiency of the proposed method.

Author contributions

All authors discussed the results and contributed to the final manuscript.

Financial disclosure

This research received no external funding.

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.