Spectrally accurate approximate solutions and convergence analysis of fractional Burgers’ equation

Abstract

In this paper, a new numerical technique implements on the time-space pseudospectral method to approximate the numerical solutions of nonlinear time- and space-fractional coupled Burgers’ equation. This technique is based on orthogonal Chebyshev polynomial function and discretizes using Chebyshev–Gauss–Lobbato (CGL) points. Caputo–Riemann–Liouville fractional derivative formula is used to illustrate the fractional derivatives matrix at CGL points. Using the derivatives matrices, the given problem is reduced to a system of nonlinear algebraic equations. These equations can be solved using Newton–Raphson method. Two model examples of time- and space-fractional coupled Burgers’ equation are tested for a set of fractional space and time derivative order. The figures and tables show the significant features, effectiveness, and good accuracy of the proposed method.

Introduction

The time- and space-fractional Burgers’ equations arise in areas of many phenomena such as material science, hereditary effects on nonlinear acoustic waves, chemical reaction fluid mechanics, plasma physics, optical fibers, and finance [5, 17, 26, 32].

In this paper, let us consider time-fractional-coupled Burgers’ equation:

$$\begin{aligned} \dfrac{\partial ^{\alpha _{1}}}{\partial t^{\alpha _{1}}} Z_{1}= & {} \dfrac{\partial ^{2}}{\partial x^{2}}Z_{1}+ 2Z_{1} \dfrac{\partial ^{}}{\partial x^{}}Z_{1}-\dfrac{\partial ^{}}{\partial x^{}}(Z_{1}Z_{2}),\qquad (0<\alpha _{1}\le 1),\end{aligned}$$
(1.1)
$$\begin{aligned} \dfrac{\partial ^{\beta _{1}}}{\partial t^{\beta _{1}}} Z_{2}= & {} \dfrac{\partial ^{2}}{\partial x^{2}}Z_{2}+2Z_{2} \dfrac{\partial ^{}}{\partial x^{}}Z_{2}-\dfrac{\partial ^{}}{\partial x^{}}(Z_{1}Z_{2}),\qquad (0<\beta _{1}\le 1), \end{aligned}$$
(1.2)

and space-fractional-coupled Burgers’ equation:

$$\begin{aligned} \dfrac{\partial ^{}}{\partial t^{}} Z_{1}= & {} \dfrac{\partial ^{2}}{\partial x^{2}}Z_{1}+ 2Z_{1} \dfrac{\partial ^{\alpha _{2}}}{\partial x^{\alpha _{2}}}Z_{1}-\dfrac{\partial ^{}}{\partial x^{}}(Z_{1}Z_{2}),\qquad (0<\alpha _{2}\le 1),\end{aligned}$$
(1.3)
$$\begin{aligned} \dfrac{\partial ^{}}{\partial t^{}} Z_{2}= & {} \dfrac{\partial ^{2}}{\partial x^{2}}Z_{2}+2Z_{2} \dfrac{\partial ^{\beta _{2}}}{\partial x^{\beta _{2}}}Z_{2}-\dfrac{\partial ^{}}{\partial x^{}}(Z_{1}Z_{2}),\qquad (0<\beta _{2}\le 1). \end{aligned}$$
(1.4)

with initial condition:

$$\begin{aligned} Z_{1}(x,0) = f_{1}(x),~Z_{2}(x,0) = f_{2}(x),\qquad (x) \in \Omega , \end{aligned}$$

and boundary conditions:

$$\begin{aligned} Z_{1}(x_{L}t)= & {} g_{1}(t), ~~Z_{1}(x_{R},t) = g_{2}(t),~~ \quad t\in [0,1], \quad (x)\in \partial \Omega , \\ Z_{2}(x_{L},t)= & {} g_{3}(t), ~~Z_{2}(x_{R}, t) = g_{4}(t). \\ \end{aligned}$$

where \(\Omega ^{}= \left[ x_{L},x_{R}\right] ^{} \), \( \alpha _{1} \) and \( \beta _{1} \) are order of time-fractional derivative, and \( \alpha _{2} \) and \( \beta _{2} \) are order of space-fractional derivative.

In past years, several analytical and numerical methods have been developed for the time- and space-fractional coupled Burger’s equation. Kurt et al. [16] have proposed exp- function method and perturbation–iteration method for the analytical solutions of nonlinear time-fractional coupled Burgers’ equations. Senol et al. [27] have introduced residual power series method for the numerical solutions of time-fractional burgers’ type equation with conformable fractional derivative. Cenesiz et al. [7] have obtained new type exact solutions for time-fractional Burgers’ equation, modified Burgers’ equation, and Burgers’–Korteweg–de-Vries equation using first integral method. Islam et al. [15] have obtained extract exact solutions for space-time-fractional Burgers’ equation using rational fractional expansion method. Furthermore, authors have also employed exp-function method and the extended tanh method to construct the closed form solutions. Prakash et al. [25] have proposed fractional variational iteration method for the numerical solutions of time- and space-fractional coupled Burgers’ equations. Chen and Li An [8] have introduced the Adomian decomposition method for numerical solutions of coupled Burgers’ equations with time- and space-fractional derivatives. Yildirim and Kelleci [30] have proposed homotopy perturbation method for the numerical solutions of coupled Burgers’ equations with time- and space-fractional derivatives. Asgari and Hosseini [3] have proposed two semi-implicit Fourier spectral schemes for the numerical solution of generalized time-fractional Burger’s equation. In this method, the authors have shown unconditional stability and improve the computational cost. Momani [24] has obtained the non-perturbative analytical solutions of space- and time-fractional Burgers’ equations using Adomian decomposition method and describe the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. Mohammadizadeh et al. [23] have obtained unique solutions under some special conditions for a special class of fractional Burgers’ equation. They have also found at least one optimal solution for this problem. Moreover, this equation was solved by many different numerical methods such as finite-difference method [18, 19], finite-element method [31, 33], variational iteration method [14], spectral method [4, 20,21,22], Adomian decomposition method [1], b-spline method [9, 11, 12], Galerkin method [6], homotopy algorithm [29], generalized exp-function method [10], finite volume method [13], residual power series method [34], and homotopy perturbation method [2], etc.

This paper is organized as follows. In Sect. 2, we describe some basic definitions and notations. Discretizing and description of the methods are presented in Sects. 3 and 4. In Sect. 5, we present the convergence analysis of time- and space-fractional coupled Burgers’ equation. In Sect. 5, we present numerical solutions and errors by the proposed scheme. Finally, the conclusion of our work is given in the last section.

Preliminary

In this section, the definition of the Caputo–Riemann–Liouville fractional derivative is introduced systematically.

Definition 2.1: The partial fractional derivatives of order \( n-1<\nu <n \) of a function \( \L _{M}(t) \), with respect to variable t, in the Caputo–Riemann–Liouville fractional derivative formula are defined by:

$$\begin{aligned} \L _{M}^{'}(t)= & {} \partial ^{\nu }_{t} \L _{M}(t)\nonumber \\= & {} \left\{ \begin{array}{ll} {\frac{1}{\Gamma (-\nu )}\int _{0}^{t}(t-\xi )^{-\nu -1}\left( \L _{M}(\xi )-\L _{M}(0)\right) \mathrm{d}\xi \, }, ~&{} \nu<0,\\ \frac{1}{\Gamma (1-\nu )} \frac{\partial }{\partial t}\int _{0}^{t}(t-\xi )^{-\nu }\left( \L _{M}(\xi )-\L _{M}(0)\right) \mathrm{d}\xi \,, ~&{} 0<\nu<1, \quad \forall M,\\ \frac{1}{\Gamma (n-\nu )} \frac{\partial ^{n}}{\partial t^{n}}\int _{0}^{t}(t-\xi )^{n-\nu -1}\left( \L _{M}(\xi )-\L _{M}(0)\right) \mathrm{d}\xi , &{} n-1\le \nu <n, \end{array}\right. \end{aligned}$$
(2.1)

where \( \nu \ge 0 \) is the order of derivative, \( \Gamma \) is the gamma function, and \( n=\lceil \nu \rceil +1 \) with \( \lceil \nu \rceil \) denoting the integral part of \( \nu \). Caputo–Riemann–Liouville fractional derivatives have some basic properties which are needed in this paper as follows:

$$\begin{aligned} \partial ^{\nu }_{t} C= & {} 0,~~~~ C~~ \text {is constant}, \\ \partial ^{\nu }_{t} t^{\upsilon }= & {} \left\{ \begin{array}{ll} {0}, ~~~~~~&{} \mathrm{for}~~ \upsilon \in N~~ \mathrm{and} ~~ \upsilon <\lceil \nu \rceil ,\\ \frac{\Gamma (\upsilon +1)}{\Gamma (\upsilon +1-\nu )} t^{\upsilon -\nu }, ~~~~~&{} \mathrm{for} ~~\upsilon \in N ~~\mathrm{and} ~~ \upsilon \ge \lceil \nu \rceil . \end{array}\right. \end{aligned}$$

Moreover, construction of the first-order Chebyshev fractional differential matrix is given as:

$$\begin{aligned} Q^{(\nu )}= \left[ \begin{array}{cccc} \L _{0}^{'}(t_{0}) &{} \dots &{} \dots &{} \L _{0}^{'}(t_{M})\\ \ddots &{} \ddots &{} \ddots &{} \ddots \\ \ddots &{} \ddots &{} \ddots &{} \ddots \\ \L _{M}^{'}(t_{0}) &{} \dots &{} \dots &{} \L _{M}^{'}(t_{M})\\ \end{array}\right] . \end{aligned}$$

Pseudospectral methodology

The method allows the representation of functions and their derivatives at given set of grid points and approximate solutions of nonlinear partial differential equations as a sum of basis functions in both space and time directions.

We seek a pseudospectral approximation F(xt), as a finite linear combination of a chosen set of orthogonal basis functions for the one-dimensional form:

$$\begin{aligned} F(x,t) = \sum _{i=0}^{M_{}} \sum _{j=0}^{M_{}}{\tau _i(x) \tau _j(t)}~ { F(x_{i},t_{j})}, \end{aligned}$$
(3.1)

where the trial basis functions defined by:

$$\begin{aligned} \tau _k(z) = {\frac{P(z)}{Q(z_{k})~(z - z_{k})}} , ~~~~k = 0,\ldots ,M, \end{aligned}$$

and

$$\begin{aligned} P(z) = \prod _{k = 0}^{M} (z - z_{k}),~~~~~~~~~ Q(z) =\sum _{l=0}^{M} \prod _{k = 0,k\ne l}^{M} (z_{} - z_{k}). \end{aligned}$$

Here, z is a dummy variable and represents x in spatial and t in time direction, \(\tau _i(x)\) and \(\tau _j(t)\) are \(M^{th}\) degree Lagrange polynomials in spatial and time variables, and \( F(x_{i},t_{j}) \) are unknowns spectral coefficients. Let us define Chebyshev–Gauss–Lobatto points (CGL points):

$$\begin{aligned} z_{k} = -\cos (k\pi /M) , \, \, \forall ~~ k=0,\ldots ,M. \end{aligned}$$
(3.2)

These points are projection of equispaced points on upper half of unit circle on \( [ -1, 1] \) and make cluster at boundaries.

Next, define the Chebyshev differentiation matrix at CGL points \(\left\{ z_{0}, z_{1},\ldots ,z_{M} \right\} \), which is used to approximate the derivatives of F(xt) in the spatial and time directions. The first derivative matrix \(D^{(1)}\) is defined by:

$$\begin{aligned} D^{(1)}_{ij}=\left\{ \begin{array}{ll} {\frac{P^{'}(z_{i})}{Q^{}(z_{j})~(z_{i} - z_{j})}}, &{} i \ne j, \\ \sum _{l=0,l \ne i}^{M} (z_{i} - z_{l})^{-1} , &{} i = j. \end{array}\right. \end{aligned}$$

Then, Eq. (3.1) can be expressed in the form of direct product as:

$$\begin{aligned} F(x,t)= & {} \tau _{0}(x)\left[ \tau _{0}(t)F_{00}+\cdots +\tau _{M}(t)F_{0M}\right] +\cdots + \tau _{M}(x)\left[ \tau _{0}(t)F_{M0}+\cdots +\tau _{M}(t)F_{MM}\right] , \nonumber \\= & {} \left[ (\tau _{0}(x), \ldots ,\tau _{M}(x)) \otimes (\tau _{0}(t), \ldots ,\tau _{M}(t)) \right] \mathbf{F }^{}, \nonumber \\= & {} \left( \Psi _{[0:M]}(x)\otimes \Psi _{[0:M]}(t)\right) ^{T} \mathbf{F }^{}, \end{aligned}$$
(3.3)

where

$$\begin{aligned} \mathbf{F }^{}= & {} \left[ F^{}_{00},\ldots ,F^{}_{0M}\mid \cdots \mid F^{}_{M0}, \ldots ,F^{}_{MM}\right] ^{T},\\ \Psi _{[0:M]}(z)= & {} \left[ \tau _{0}(z), \ldots ,\tau _{M}(z) \right] ^{T}. \end{aligned}$$

In this manner, first spatial derivative of the function can be defined as follows:

$$\begin{aligned} \frac{\partial F^{}}{\partial x}= & {} \left( \frac{\partial }{\partial x}\Psi _{[0:M]}^{T}(x)\otimes \Psi _{[0:M]}^{T}(t) \right) \mathbf{F }^{} = \left( \Psi _{[0:M]}^{T}(x)D^{(1)}_{\left[ 0:M,0:M\right] }\otimes \Psi _{[0:M]}^{T}(t) \right) \mathbf{F }^{}, \nonumber \\= & {} \left( \Psi _{[0:M]}(x) \otimes \Psi _{[0:M]}(t)\right) ^{T}\left( D^{(1)}_{[0:M,0:M]} \otimes I_{M+1}\right) \mathbf{F }^{}. \end{aligned}$$
(3.4)

In similar fashion, first spatial fractional derivative of the function can be defined as follows:

$$\begin{aligned} \frac{\partial ^{\nu } F^{}}{\partial x^{\nu }}= & {} \left( \Psi _{[0:M]}(x) \otimes \Psi _{[0:M]}(t)\right) ^{T}\left( Q^{(\nu )}_{[0:M,0:M]} \otimes I_{M+1}\right) \mathbf{F }^{}. \end{aligned}$$
(3.5)

Next, time derivative and time-fractional derivative of the function can be defined by:

$$\begin{aligned} \frac{\partial F^{}}{\partial t}= & {} \left( \Psi _{[0:M]}(x) \otimes \Psi _{[0:M]}(t)\right) ^{T}\left( I_{M+1} \otimes D^{(1)}_{[0:M,0:M]}\right) \mathbf{F }^{}. \nonumber \\ \frac{\partial ^{\nu } F^{}}{\partial t^{\nu }}= & {} \left( \Psi _{[0:M]}(x) \otimes \Psi _{[0:M]}(t)\right) ^{T}\left( I_{M+1} \otimes Q^{(\nu )}_{[0:M,0:M]}\right) \mathbf{F }^{}. \end{aligned}$$
(3.6)

Similarly, the second spatial derivative of the function is given by:

$$\begin{aligned} \frac{\partial ^{2} F^{}}{\partial x^{2}}= & {} \left( \frac{\partial ^{2}}{\partial x^{2}}\Psi _{[0:M]}^{T}(x)\otimes \Psi _{[0:M]}^{T}(t) \right) \mathbf{F }^{} = \left( \frac{\partial }{\partial x} \left( \frac{\partial }{\partial x}\Psi _{[0:M]}^{T}(x)\right) \otimes \Psi _{[0:M]}^{T}(t) \right) \mathbf{F }^{}, \nonumber \\= & {} \left( \left( \frac{\partial }{\partial x} \Psi _{[0:M]}^{T}(x)D^{(1)}_{\left[ 0:M,0:M\right] }\right) \otimes \Psi _{[0:M]}^{T}(t) \right) \mathbf{F }^{}, \nonumber \\= & {} \left( \left( \Psi _{[0:M]}^{T}(x)D^{(1)}_{\left[ 0:M,0:M\right] } \times D^{(1)}_{\left[ 0:M,0:M\right] }\right) \otimes \Psi _{[0:M]}^{T}(t) \right) \mathbf{F }^{}, \nonumber \\= & {} \left( \Psi _{[0:M]}(x) \otimes \Psi _{[0:M]}(t)\right) ^{T}\left( D^{(2)}_{[0:M,0:M]} \otimes I_{M+1}\right) \mathbf{F }^{}. \end{aligned}$$
(3.7)

where \( D^{(2)}= D^{(1)}\times D^{(1)}\).

Pseudospectral method-based discretization

Furthermore, let us consider following transformations which are used to transform the one-dimensional space \( [x_{L},x_{R}] \), \( [y_{L},y_{R}] \) and time [0, T] in to \( [-1,1] \). We obtain the time fractional coupled Burgers’ equation in new space and time interval:

$$\begin{aligned} \dfrac{\partial ^{\alpha _{1}}}{\partial t^{\alpha _{1}}} Z_{1}= & {} \frac{2^{(2-\alpha _{1})}T^{\alpha _{1}} }{(x_{R} -x_{L})^{2}}\dfrac{\partial ^{2}}{\partial x^{2}}Z_{1}+ \frac{2^{(1-\alpha _{1})}T^{\alpha _{1}} }{(x_{R} -x_{L})^{}}2Z_{1} \dfrac{\partial ^{}}{\partial x^{}}Z_{1} -Z_{1}\frac{2^{(1-\alpha _{1})}T^{\alpha _{1}} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Z_{2})-Z_{2}\frac{2^{(1-\alpha _{1})}T^{\alpha _{1}} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Z_{1}),\\ \dfrac{\partial ^{\beta _{1}}}{\partial t^{\beta _{1}}} Z_{2}= & {} \frac{2^{(2-\beta _{1})}T^{\beta _{1}} }{(x_{R} -x_{L})^{2}}\dfrac{\partial ^{2}}{\partial x^{2}}Z_{2}+ \frac{2^{(1-\beta _{1})}T^{\beta _{1}} }{(x_{R} -x_{L})^{}}2Z_{2} \dfrac{\partial ^{}}{\partial x^{}}Z_{2} -Z_{1}\frac{2^{(1-\beta _{1})}T^{\beta _{1}} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Z_{2}) -Z_{2}\frac{2^{(1-\alpha _{1})}T^{\alpha _{1}}}{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Z_{1}). \end{aligned}$$

and space-fractional coupled Burgers’ equation:

$$\begin{aligned} \dfrac{\partial ^{}}{\partial t^{}} Z_{1}= & {} \frac{2^{}T^{} }{(x_{R} -x_{L})^{2}}\dfrac{\partial ^{2}}{\partial x^{2}}Z_{1}+ \frac{2^{(\alpha _{2}-1)}T^{} }{(x_{R} -x_{L})^{\alpha _{2}}}2Z_{1} \dfrac{\partial ^{\alpha _{2}}}{\partial x^{\alpha _{2}}}Z_{1} -Z_{1}\frac{T^{} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Z_{2})- Z_{2}\frac{T^{} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Z_{1}),\\ \dfrac{\partial ^{}}{\partial t^{}} Z_{2}= & {} \frac{2^{}T^{} }{(x_{R} -x_{L})^{2}}\dfrac{\partial ^{2}}{\partial x^{2}}Z_{2}+ \frac{2^{(\beta _{2}-1)}T^{} }{(x_{R} -x_{L})^{\beta _{2}}}2Z_{2} \dfrac{\partial ^{\beta _{2}}}{\partial x^{\beta _{2}}}Z_{2} -Z_{1}\frac{T^{} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Z_{2})- Z_{2}\frac{T^{} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Z_{1}). \end{aligned}$$

with initial conditions:

$$\begin{aligned} Z_{1}(x,- 1) = p_{1}(x),~ Z_{2}(x,- 1) = p_{2}(x),\qquad x\in [- 1,1], \end{aligned}$$
(4.1)

and boundary conditions:

$$\begin{aligned} Z_{1}(- 1,t)= & {} q_{11}(t), ~~Z_{1}(1,t) = q_{12}(t),~~ \qquad t\in [- 1,1], \end{aligned}$$
(4.2)
$$\begin{aligned} Z_{2}(- 1,t)= & {} q_{21}(t), ~~Z_{2}(1,t) = q_{22}(t),~~ \qquad t\in [- 1,1]. \end{aligned}$$
(4.3)

Furthermore, we consider a mapping for converting the non-homogeneous initial and boundary values to homogeneous initial and boundary values:

$$\begin{aligned} \chi _{k}(x,t) = \frac{1-t}{2} p_{k}(x) + \frac{1-x}{2} q_{1k}(t) +\frac{1+x}{2} q_{2k}(t)- \frac{(1-t)(1-x)}{4} q_{1k}(- 1)- \frac{(1-t)(1+x)}{4}q_{2k}(-1). \nonumber \\ \end{aligned}$$
(4.4)

Define new variables \(Y_{k}(x,t),~\forall k=\left\{ 1,2\right\} \):

$$\begin{aligned} Z_{1}(x,t) = Y_{1}(x,t) +\chi _{1}(x,t),~~Z_{2}(x,t) = Y_{2}(x,t) +\chi _{2}(x,t), \end{aligned}$$
(4.5)

the above time-fractional coupled Burgers’ equation can be modified with new variables and obtained the residuals:

$$\begin{aligned} \dfrac{\partial ^{\alpha _{1}}}{\partial t^{\alpha _{1}}} (Y_{1} +\chi _{1})= & {} \frac{2^{(2-\alpha _{1})}T^{\alpha _{1}} }{(x_{R} -x_{L})^{2}}\dfrac{\partial ^{2}}{\partial x^{2}}(Y_{1} +\chi _{1})+ \frac{2^{(1-\alpha _{1})}T^{\alpha _{1}} }{(x_{R} -x_{L})^{}}2(Y_{1} +\chi _{1}) \dfrac{\partial ^{}}{\partial x^{}}(Y_{1} +\chi _{1})\\&-(Y_{1} +\chi _{1})\frac{2^{(1-\alpha _{1})}T^{\alpha _{1}} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Y_{2} +\chi _{2})-(Y_{2} +\chi _{2})\frac{2^{(1-\alpha _{1})}T^{\alpha _{1}} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Y_{1} +\chi _{1}),\\ \dfrac{\partial ^{\beta _{1}}}{\partial t^{\beta _{1}}} (Y_{2} +\chi _{2})= & {} \frac{2^{(2-\beta _{1})}T^{\beta _{1}} }{(x_{R} -x_{L})^{2}}\dfrac{\partial ^{2}}{\partial x^{2}}(Y_{2} +\chi _{2})+ \frac{2^{(1-\beta _{1})}T^{\beta _{1}} }{(x_{R} -x_{L})^{}}2(Y_{2} +\chi _{2}) \dfrac{\partial ^{}}{\partial x^{}}(Y_{2} +\chi _{2}) \\&-\,\,(Y_{1} +\chi _{1})\frac{2^{(1-\beta _{1})}T^{\beta _{1}} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Y_{2} +\chi _{2}) -(Y_{2} +\chi _{2})\frac{2^{(1-\alpha _{1})}T^{\alpha _{1}}}{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Y_{1} +\chi _{1}), \end{aligned}$$

and obtain the residuals of space-fractional coupled Burgers’ equation:

$$\begin{aligned} \dfrac{\partial ^{}}{\partial t^{}} (Y_{1} +\chi _{1})= & {} \frac{2^{}T^{} }{(x_{R} -x_{L})^{2}}\dfrac{\partial ^{2}}{\partial x^{2}}(Y_{1} +\chi _{1})+ \frac{2^{(\alpha _{2}-1)}T^{} }{(x_{R} -x_{L})^{\alpha _{2}}}2(Y_{1} +\chi _{1}) \dfrac{\partial ^{\alpha _{2}}}{\partial x^{\alpha _{2}}}(Y_{1} +\chi _{1})\\&-\,\,(Y_{1} +\chi _{1})\frac{T^{} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Y_{2} +\chi _{2})- (Y_{2} +\chi _{2})\frac{T^{} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Y_{1} +\chi _{1}),\\ \dfrac{\partial ^{}}{\partial t^{}} (Y_{2} +\chi _{2})= & {} \frac{2^{}T^{} }{(x_{R} -x_{L})^{2}}\dfrac{\partial ^{2}}{\partial x^{2}}(Y_{2} +\chi _{2})+ \frac{2^{(\beta _{2}-1)}T^{} }{(x_{R} -x_{L})^{\beta _{2}}}2(Y_{2} +\chi _{2}) \dfrac{\partial ^{\beta _{2}}}{\partial x^{\beta _{2}}}(Y_{2} +\chi _{2}) \\&-\,\,(Y_{1} +\chi _{1})\frac{T^{} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Y_{2} +\chi _{2})- (Y_{2} +\chi _{2})\frac{T^{} }{(x_{R} -x_{L})^{}}\dfrac{\partial ^{}}{\partial x^{}}(Y_{1} +\chi _{1}). \end{aligned}$$

Now, we apply CGL points and pseudospectral method in coupled equations and obtained the system of nonlinear algebraic equation:

$$\begin{aligned} G_{1}(Y_{1}|Y_{2}) =0 \qquad G_{2}(Y_{1}|Y_{2}) =0. \end{aligned}$$
(4.6)

The system of nonlinear equation (4.6) can be solved using Newton–Raphson method.

Convergence analysis

In this section, author study for the error analysis of the suggested expansion of Chebyshev polynomials is investigated.

Theorem 51

If the series \( \sum _{i=0}^{\infty } \sum _{k=0}^{\infty }{\tau _{i}(x) \tau _{k}(t) \varTheta _{ik}} \) converges uniformly to \( Z_{1}(x,t) \) on the interval \( [-1,1]^{2} \), then we have:

$$\begin{aligned} \varTheta _{ik}= & {} \dfrac{1}{ {\hat{h}}^{}_{i} {\hat{h}}^{}_{k}} \sum _{l=0}^{M} \sum _{n=0}^{M}\tau _{i}(x_{l}) \tau _{k}(t_{n}) w^{}_{l}~w^{}_{m}~w^{}_{n} {Z_{1}^{}(x_{i},t_{k})^{}}. \end{aligned}$$
(5.1)

Proof

We know that:

$$\begin{aligned} Z_{1}(x,t) = \sum _{i=0}^{\infty } \sum _{k=0}^{\infty }{\tau _{i}(x) \tau _{k}(t) \varTheta _{ik}}. \end{aligned}$$
(5.2)

Taking \( L_{w^{}}^{2}[- 1,1] \) norm both side, we get:

$$\begin{aligned} \Arrowvert Z_{1}(x,t)\Arrowvert ^{2}_{L^{2}_{w^{}}[-1,1]}= & {} \sum _{i=0}^{\infty } \sum _{k=0}^{\infty } \Arrowvert \tau _{i}(x) \tau _{k}(t) \varTheta _{ik}\Arrowvert ^{2}_{L^{2}_{w^{}}[-1,1]} =\sum _{i=0}^{\infty } \sum _{k=0}^{\infty } \Arrowvert \tau _{i}(x) \tau _{k}(t) \Arrowvert ^{2}_{L^{2}_{w^{}}[-1,1]} \arrowvert \varTheta _{ik}\arrowvert ^{2}, \\\le & {} \sum _{i=0}^{\infty } \sum _{k=0}^{\infty } \int _{-1}^1\int _{-1}^1 \tau ^{2}_{i}(x) \tau ^{2}_{k}(t) w^{}(x)w^{}(y) w^{}(t)\, dx\ dt\ \arrowvert \varTheta _{ik}\arrowvert ^{2}, \\\le & {} \sum _{i=0}^{\infty } \sum _{k=0}^{\infty } \tau ^{2}_{i}(x) w^{}(x) \, dx\ \int _{-1}^1 dy\ \int _{-1}^1 \tau ^{2}_{k}(t) w^{}(t)\, dt\ \arrowvert \varTheta _{ik}\arrowvert ^{2}, \\\le & {} \sum _{i=0}^{\infty } \sum _{k=0}^{\infty } \left[ \tau _{i}(x),\tau _{i}(x)\right] _{w^{}} \left[ \tau _{k}(t), \tau _{k}(t)\right] _{w^{}} \arrowvert \varTheta _{ik}\arrowvert ^{2}. \end{aligned}$$

Finally, put the value of discrete inner product, and we get:

$$\begin{aligned} \Arrowvert Z_{1}\Arrowvert ^{2}_{L^{2}_{w^{}}[-1,1]} \le \sum _{i=0}^{\infty } \sum _{k=0}^{\infty } {\hat{h}}^{}_{i} {\hat{h}}^{}_{k} \arrowvert \varTheta _{ik}\arrowvert ^{2}. \end{aligned}$$
(5.3)

Hence, function \( Z_{1}(x,t) \) is bounded. In a similar fashion, we can obtain function \( Z_{2}(x,t) \) is bounded. \(\square \)

Numerical results and discussion

In this section, to demonstrate the performance, the proposed method is implemented on test problems of time- and space-fractional Burgers’ equation. Error at different grid points will be expressed in terms of \( L_{2} \)- norm which is defined by:

$$\begin{aligned} L_{2} = \Arrowvert Z_{i}^\mathrm{exa}-Z_{i(\alpha ,\beta )}^\mathrm{num} \Arrowvert _{2}, \end{aligned}$$

where \(Z_{i}^\mathrm{exa}\) and \(Z_{i(\alpha ,\beta )}^\mathrm{num},~~~ \forall i\in {1,2},\) represent exact solutions and numerical solutions, respectively.

Example 1

Let us consider the time-fractional coupled Burgers’ equation:

$$\begin{aligned} \dfrac{\partial ^{\alpha _{1}}}{\partial t^{\alpha _{1}}} Z_{1}= & {} \dfrac{\partial ^{2}}{\partial x^{2}}Z_{1}+2Z_{1} \dfrac{\partial ^{}}{\partial x^{}}Z_{1}-\dfrac{\partial ^{}}{\partial x^{}}(Z_{1}Z_{2}),\qquad (0<\alpha _{1}\le 1),\end{aligned}$$
(6.1)
$$\begin{aligned} \dfrac{\partial ^{\beta _{1}}}{\partial t^{\beta _{1}}} Z_{2}= & {} \dfrac{\partial ^{2}}{\partial x^{2}}Z_{2}+2Z_{2} \dfrac{\partial ^{}}{\partial x^{}}Z_{2}-\dfrac{\partial ^{}}{\partial x^{}}(Z_{1}Z_{2}), \qquad (0<\beta _{1}\le 1) \end{aligned}$$
(6.2)

with initial condition:

$$\begin{aligned} Z_{1}(x,0) = \sin (x),~Z_{2}(x,0) = \sin (x),\qquad (x) \in \Omega , \end{aligned}$$

the exact solution for the problem is given: [1]

$$\begin{aligned} Z_{1}(x,t)= & {} \sin (x)-\sin (x)\frac{t^{\alpha _{1}}}{\Gamma (\alpha _{1}+1)}+(\sin (x)-2\sin (x)\cos (x))\frac{t^{2\alpha _{1}}}{\Gamma (2\alpha _{1}+1)}\\&2\sin (x)\cos (x)\frac{t^{\alpha _{1}+\beta _{1}}}{\Gamma (\alpha _{1}+\beta _{1}+1)}+\cdots \\ Z_{2}(x,t)= & {} \sin (x)-\sin (x)\frac{t^{\beta _{1}}}{\Gamma (\beta _{1}+1)}+(\sin (x)-2\sin (x)\cos (x))\frac{t^{2\beta _{1}}}{\Gamma (2\beta _{1}+1)}\\&2\sin (x)\cos (x)\frac{t^{\alpha _{1}+\beta _{1}}}{\Gamma (\alpha _{1}+\beta _{1}+1)}+\cdots \end{aligned}$$

When \( \alpha _{1}=\beta _{1}=1 \), then exact solution is:

$$\begin{aligned} Z_{1}(x,t)=Z_{2}(x,t)=e^{-t}\sin (x). \end{aligned}$$

In this example, numerical solutions of the proposed method have obtained over the domain \( \Omega =[- 10,10]^{},~~ t \in [0,0.005] \) and fractional-order derivative \( 0< \alpha _{1},\beta _{1} \le 1 \). The error norms for different grid points and different fractional-order derivatives are presented in Table 1. In Table 1, it can be seen that the accuracy of the numerical results is increased along with the number of grid points. We are also depicted the 3D graph of numerical solutions \( Z_{1} \) and \( Z_{2} \) at time \( T= 0.005 \) and \( \alpha _{1}=0.5,\beta _{1}=0.25 \) in Fig.  1. Figure 2 illustrates the 2D curve of the numerical and exact solutions with different fractional-order derivatives at time \( T=0.005 \). In this figure, the solid line represents the exact solutions; however, the block represents the numerical solutions at various \( \alpha _{1} \) and \( \beta _{1} \). Accuracy of numerical solutions is merely seen in Fig. 2. Authors have discussed the numerical solutions of the equation with the different set of data, and for comparison purpose, we refer to [1, 8, 28, 30]. Furthermore, it is found that the results obtained by the proposed method show very good agreement with published results. Moreover, proposed method has obtained the eighth order of accuracy.

Table 1 Numerical solutions of proposed method at time \( T=0.005 \) with different \( \alpha _{1} \)\( \beta _{1} \) and grids points M for example 1
Fig. 1
figure1

Numerical solutions of \( Z_{1}(x,t) \) and \( Z_{2}(x,t) \) for example 1 when different \( \alpha _{1}=050, \beta _{1}=0.25\) and time \( T=0.005 \)

Fig. 2
figure2

Numerical and exact solutions of \( Z_{1}(x,t) \) and \( Z_{2}(x,t) \) for example 1 with different \( \alpha _{1}\) and \( \beta _{1}\) and time \( T=0.005 \)

Example 2

Let us consider the space-fractional coupled Burgers’ equation:

$$\begin{aligned} \dfrac{\partial ^{}}{\partial t^{}} Z_{1}= & {} \dfrac{\partial ^{2}}{\partial x^{2}}Z_{1}+2Z_{1} \dfrac{\partial ^{\alpha _{2}}}{\partial x^{\alpha _{2}}}Z_{1}-\dfrac{\partial ^{}}{\partial x^{}}(Z_{1}Z_{2}), \qquad (0<\alpha _{2}\le 1),\end{aligned}$$
(6.3)
$$\begin{aligned} \dfrac{\partial ^{}}{\partial t^{}} Z_{2}= & {} \dfrac{\partial ^{2}}{\partial x^{2}}Z_{2}+2Z_{2} \dfrac{\partial ^{\beta _{2}}}{\partial x^{\beta _{2}}}Z_{2}-\dfrac{\partial ^{}}{\partial x^{}}(Z_{1}Z_{2}), \qquad (0<\beta _{2}\le 1) \end{aligned}$$
(6.4)

with initial condition:

$$\begin{aligned} Z_{1}(x,0) = x^{2},~Z_{2}(x,0) = x^{3},\qquad (x) \in \Omega , \end{aligned}$$

the exact solution for the problem is given as: [8, 30]

$$\begin{aligned}&Z_{1}(x,t)=x^{2}+\left( 2-5x^{4}+f_{1}x^{(4-\alpha _{2})} \right) t\\&\quad +\frac{t^{2}}{2}\left( f_{2}x^{(6-2\alpha _{2})}+ f_{3}x^{(6-\alpha _{2})}+f_{4}x^{(2-\alpha _{2})}+f_{5}x^{(7-\beta _{2})}+20x^{6}+10x^{5}-72x^{2}\right) ,\\&Z_{2}(x,t)=x^{3}+\left( 6x-5x^{4}+g_{1}x^{(6-\beta _{2})} \right) t\\&\quad +\frac{t^{2}}{2}\left( g_{2}x^{(9-2\beta _{2})}+ g_{3}x^{(7-\beta _{2})}+g_{4}x^{(4-\beta _{2})}+g_{5}x^{(6-\alpha _{2})}+20x^{6}+10x^{5}-72x^{2}\right) , \end{aligned}$$

where

$$\begin{aligned} f(x)= & {} x^{2},~~f_{1}(x)=\frac{4}{\Gamma (3-\alpha _{2})},~~f_{2}(x)=\left[ \frac{4}{\Gamma (3-\alpha _{2})}+\frac{2 \Gamma (5-\alpha _{2})}{\Gamma (5-2\alpha _{2})} \right] f_{1},\\ f_{3}(x)= & {} (\alpha _{2}-4)f_{1} -\frac{20}{\Gamma (3-\alpha _{2})} -\frac{240}{\Gamma (5-\alpha _{2})},~~ f_{4}(x)=(4-\alpha _{2})(3-\alpha _{2})f_{1}+\frac{8}{\Gamma (3-\alpha _{2})},\\~ f_{5}(x)= & {} -2g_{1}(x).\\ g(x)= & {} x^{3},~~g_{1}(x)=\frac{12}{\Gamma (4-\beta _{2})},~~g_{2}(x)=\left[ \frac{12}{\Gamma (4-\beta _{2})}+\frac{2 \Gamma (7-\beta _{2})}{\Gamma (7-2\beta _{2})} \right] g_{1},\\ g_{3}(x)= & {} -2g_{1} -\frac{60}{\Gamma (4-\beta _{2})} -\frac{240}{\Gamma (5-\beta _{2})},~~ g_{4}(x)=(5-\beta _{2})(6-\beta _{2})g_{1}+\frac{12}{\Gamma (2-\beta _{2})}+\frac{72}{\Gamma (4-\beta _{2})},\\~ g_{5}(x)= & {} (\alpha _{2}-4)f_{1}(x).\\ \end{aligned}$$
Table 2 Numerical solutions of proposed method at time \( T=1.00 \) with different \( \alpha _{2} \)\( \beta _{2} \) and grid points M for example 2
Fig. 3
figure3

Numerical solutions of \( Z_{1}(x,t) \) and \( Z_{2}(x,t) \) for example 2 when a\( \alpha _{1}=0.25, \beta _{1}=0.50\), b\( \alpha _{1}=0.75, \beta _{1}=0.75\), and c\( \alpha _{1}=1.00, \beta _{1}=1.00\) at time \( T=1 \)

In this example, numerical solutions of the proposed method have obtained over the domain \( \Omega =[0,10]^{},~~ t \in [0,1] \) and fractional-order derivative \( 0< \alpha _{2},\beta _{2} \le 1 \). The error norms for different grid points and different fractional order derivatives are presented in Table 2. In Table 2 , it can be seen that the accuracy of the numerical results is increased along with the number of grid points. In Fig.  3, we are also depicted the of numerical solutions \( Z_{1} \) and \( Z_{2} \) in the form of 3D graph for \( (a) \alpha _{1}=0.5, \beta _{1}=0.5\), (b) \( \alpha _{1}=0.75, \beta _{1}=0.75\) and (c) \( \alpha _{1}=1.00, \beta _{1}=1.00\) at time \( T=1 \). Authors have discussed the numerical solutions of the equation with the different set of data; for comparison purpose, we refer to [8, 28, 30]. Furthermore, it is found that the results obtained by the proposed method show very good agreement with published results. Moreover, proposed method has obtained the ninth order of accuracy.

Conclusion

In this paper, time–space Chebyshev pseudospectral method has been successfully employed to time- and space-fractional coupled Burgers’ equation. The fractional-order differentiation matrix has been established using Caputo–Riemann–Liouville derivative formula at CGL points for fractional-order derivative. It has shown that the numerical results of the proposed method are more close to exact solution. For the equation, convergence analysis of the proposed method has been presented. To demonstrate the performance, the method has been employed two test problems, first for time-fractional and other for space-fractional coupled Burgers’ equation. Reported numerical results are highly accurate which shows the efficiency of the proposed method.

References

  1. 1.

    Ahmed, H.F.; Bahgat, M.; Zaki, M.: Analytical approaches to space-and time-fractional coupled burgers’ equations. Pramana 92(3), 38 (2019)

    Article  Google Scholar 

  2. 2.

    Alam Khan, N.; Ara, A.; Mahmood, A.: Numerical solutions of time-fractional burgers equations: a comparison between generalized differential transformation technique and homotopy perturbation method. Int. J. Numer. Methods Heat Fluid Flow 22(2), 175–193 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Asgari, Z.; Hosseini, S.: Efficient numerical schemes for the solution of generalized time fractional burgers type equations. Numer. Algorithm 77(3), 763–792 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Balyan, L.K.; Mittal, A.K.; Kumar, M.; Choube, M.: Stability analysis and highly accurate numerical approximation of fisher’s equations using pseudospectral method. Math. Comput. Simul. 177, 86–104 (2020)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Benton, E.R.; Platzman, G.W.: A table of solutions of the one-dimensional burgers equation. Q. Appl. Math. 30(2), 195–212 (1972)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Cao, W.; Xu, Q.; Zheng, Z.: Solution of two-dimensional time-fractional burgers equation with high and low reynolds numbers. Adv. Differ. Equ. 2017(1), 338 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Cenesiz, Y.; Baleanu, D.; Kurt, A.; Tasbozan, O.: New exact solutions of burgers’ type equations with conformable derivative. Wave Random Compl. Media 27, 103–116 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Chen, Y.; An, H.-L.: Numerical solutions of coupled burgers equations with time-and space-fractional derivatives. Appl. Math. Comput. 200(1), 87–95 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    El-Danaf, T.S.; Hadhoud, A.R.: Parametric spline functions for the solution of the one time fractional burgers’ equation. Appl. Math. Model. 36(10), 4557–4564 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Emad, A.-B.; Hassan, G.F.: Multi-wave solutions of the space-time fractional burgers and sharma-tasso-olver equations. Ain Shams Eng. J. 7(1), 463–472 (2016)

    Article  Google Scholar 

  11. 11.

    Esen, A.; Tasbozan, O.: Numerical solution of time fractional burgers equation. Acta Univ. Sapientiae Math. 7(2), 167–185 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Esen, A.; Tasbozan, O.: Numerical solution of time fractional burgers equation by cubic b-spline finite elements. Mediterr. J. Math. 13(3), 1325–1337 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Guesmia, A.; Daili, N.: Numerical approximation of fractional burgers equation. Commun. Math. Appl. 1(2), 77–90 (2010)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Inc, M.: The approximate and exact solutions of the space-and time-fractional burgers equations with initial conditions by variational iteration method. J. Math. Anal. Appl. 345(1), 476–484 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Islam, M.T.; Akbar, M.A.; Azad, M.A.K.: Closed-form travelling wave solutions to the nonlinear space-time fractional coupled burgers’ equations. Arab J. Basic Appl. Sci. 26, 1–11 (2019)

    Article  Google Scholar 

  16. 16.

    Kurt, A.; Senol, M.; Tasbozan, O.; Chand, M.: Two reliable method for the solution of fractional coupled burgers’ equation arising as a model of polydispersive sedimentation. Appl. Math. Nonlinear Sci. 4(2), 523–534 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Li, C.; Zeng, F.: Finite difference methods for fractional differential equations. Int. J. Bifurc. Chaos 22(04), 1230014 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Li, D.; Zhang, C.; Ran, M.: A linear finite difference scheme for generalized time fractional burgers equation. Appl. Math. Model. 40(11–12), 6069–6081 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Li, L.; Zhou, B.; Chen, X.; Wang, Z.: Convergence and stability of compact finite difference method for nonlinear time fractional reaction-diffusion equations with delay. Appl. Math. Comput. 337, 144–152 (2018)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Mittal, A.: A stable time-space jacobi pseudospectral method for two-dimensional sine-gordon equation. J. Appl. Math. Comput. 63, 1–26 (2020)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Mittal, A.; Balyan, L.: A highly accurate time-space pseudospectral approximation and stability analysis of two dimensional brusselator model for chemical systems. Int. J. Appl. Comput. Math. 5(5), 140 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Mittal, A.; Balyan, L.; Tiger, D.: An improved pseudospectral approximation of generalized burger-huxley and fitzhugh-nagumo equations. Comput. Methods Differ. Equ. 6(3), 280–294 (2018)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Mohammadizadeh, F.; Tehrani, H.; Georgiev, S.; Noori Skandari, M.: An optimal control problem associated to a class of fractional burgers’ equations. Asian-Eur. J. Math. 2019, 2050079 (2019)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Momani, S.: Non-perturbative analytical solutions of the space-and time-fractional burgers equations. Chaos Solitons Fractals 28(4), 930–937 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Prakash, A.; Kumar, M.; Sharma, K.K.: Numerical method for solving fractional coupled burgers equations. Appl. Math. Comput. 260, 314–320 (2015)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Rossikhin, Y.A.; Shitikova, M.V.: Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl. Mech. Rev. 63(1), 010801 (2010)

    Article  Google Scholar 

  27. 27.

    Senol, M.; Tasbozan, O.; Kurt, A.: Numerical solutions of fractional burger’s type equations with conformable derivative. Chin. J. Phys. 58, 75–84 (2019)

    Article  Google Scholar 

  28. 28.

    Singh, J.; Kumar, D.; Swroop, R.: Numerical solution of time- and space-fractional coupled burgers’ equations via homotopy algorithm. Alex. Eng. J. 55, 1753–1763 (2016a)

    Article  Google Scholar 

  29. 29.

    Singh, J.; Kumar, D.; Swroop, R.: Numerical solution of time-and space-fractional coupled burgers’ equations via homotopy algorithm. Alex. Eng. J. 55(2), 1753–1763 (2016b)

    Article  Google Scholar 

  30. 30.

    Yıldırım, A.; Kelleci, A.: Homotopy perturbation method for numerical solutions of coupled burgers equations with time-and space-fractional derivatives. Int. J. Numer. Methods Heat Fluid Flow 20(8), 897–909 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Yokus, A.; Kaya, D.: Numerical and exact solutions for time fractional burgers’ equation. J. Nonlinear Sci. Appl. 10(7), 3419–3428 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Zeng, F.; Li, C.; Liu, F.; Turner, I.: The use of finite difference/element approaches for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35(6), A2976–A3000 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Zeng, F.; Li, C.; Liu, F.; Turner, I.: Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J. Sci. Comput. 37(1), A55–A78 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Zhang, J.; Wei, Z.; Yong, L.; Xiao, Y.: Analytical solution for the time fractional bbm-burger equation by using modified residual power series method. Complexity 2018, 8 (2018)

    MATH  Google Scholar 

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The author is highly grateful to the anonymous referee for his comments and suggestions which helped improve the quality of manuscript.

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Mittal, A.K. Spectrally accurate approximate solutions and convergence analysis of fractional Burgers’ equation. Arab. J. Math. (2020). https://doi.org/10.1007/s40065-020-00286-x

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