A Spectral Approach for Time-Fractional Diﬀusion and Subdiﬀusion Equations in a Large Interval

. In this paper, we concentrate on a class of time-fractional diﬀusion and subdiﬀusion equations. To solve the mentioned problems, we construct two-dimensional Genocchi-fractional Laguerre functions (G-FLFs). Then, the pseudo-operational matrices are used to convert the proposed equations to systems of algebraic equations. The properties of pseudo-operational matrices have reﬂected well in the process of the numerical technique and create an approximate solution with high precision. Finally, several examples are presented to illustrate the accuracy and eﬀectiveness of the technique.


Introduction
In recent years, fractional partial differential equations (FPDEs) have attracted considerable attention because of their application in the modeling of various phenomena. The fractional diffusion equation is an important type of FPDEs which has been studied widely. The physical application of fractional diffusion equations (FDEs) is related to the phenomena of anomalous diffusion in transport processes through complex and/or disordered systems including fractal media and fractional kinetic equations have proved particularly useful in the context of anomalous slow diffusion [32].
Moreover, the time-fractional subdiffusion equations (FSDEs) are created by replacing the time derivative in ordinary diffusion by a fractional derivative of order 0 < γ < 1. This class of equation is achieved from anomalous diffusive systems. The FSDEs are considered as the following form [7,17,36]: where κ γ generalized diffusion constant, γ is the anomalous diffusion exponent and D γ t is the γth-order Caputo derivative operator [46]. Since analytic techniques for finding the solutions of FDEs and FSDEs cannot work well. Therefore, the researchers seek to establish efficient and reliable numerical techniques. Note that various numerical techniques have existed to solve these problems that we are presenting some of them. Mustapha [34] solved sub-diffusion equations by an implicit finite-difference time-stepping method, Jiang et al. [23] derived and utilized high-order finite element methods for timefractional partial differential equations, Si et al. [40] presented Haar wavelet operational matrix for solving fractional partial differential equations with time-space variable coefficients, Gao et al. [17] have discussed on the fractional sub-diffusion equations with the help of the compact finite difference scheme, Wang et al. [43] investigated the solution of the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation by compact difference schemes, Yaseen et al. [44] derived a cubic trigonometric B-spline collocation approach for the fractional sub-diffusion equations, Dehestani et al. [8] established the fractional-order Legendre-Laguerre functions for approximate solution of fractional partial differential equations. For more details about the proposed equation, the reader is advised to see [4,11,13,18,20,24,25,45].
The main purpose of the present numerical scheme is solving FDEs and FSDEs based on the G-FLFs and collocation method. First, by combining the Genocchi polynomials (GPs) [21,22,29] and fractional-order Laguerre functions (FLFs), we introduce the bivariate G-FLFs. Then, by means of pseudooperational matrices (POMs), FDEs and FSDEs are reduced to a system of algebraic equations.
In this work, we apply FLFs to approximate the time variable because the behavior of the physical problems for large time is important. The Laguerre polynomials are orthogonal in the semi-infinite interval and also these polynomials are appropriate for approximating problems in semi-infinite intervals. Also, we utilize GPs for approximating variable x because these polynomials are superior to other polynomials. So that the advantages of these polynomials are mentioned in [9,10].
The paper is structured as follows: In the next section, we present the framework of numerical technique, which contains the GPs, FLFs and G-FLFs. Pseudo-operational matrices of integration, derivative and some necessary formulas of G-FLFs are introduced in Section 3. In Section 4, the formulation of the method based on G-FLFs is explained. In Section 5, we discuss error estimation. To verify the accuracy and efficiency of the proposed approach, we examine several test problems in Section 6. Finally, the conclusions of the proposed numerical scheme are discussed in last section.

Combination of GPs and FLFs
This section introduces the Genocchi-fractional Laguerre functions and their features.

Genocchi polynomials
The Genocchi polynomials of degree m, which is defined as follows [9,10]: where g k denotes the Genocchi numbers. Now, we mention some significant properties of Genocchi polynomials [12]: Any function g defined over the interval [0, 1] can be expanded in terms of GPs which is described in detail in [12].

Fractional-order Laguerre functions
The analytic form of FLFs is given by [8]: The orthogonal property of the FLFs with respect to the weight function w β (t) = t β−1 exp(−t β ) are presented in [8], which the following formula is gotten: where δ mn is the Kronecker symbol. In addition, any function h defined over the interval [0, ∞), can be approximated by FLFs which is described in detail in [8].

POMs and some necessary formulas
The present section is devoted to the technique of computing the POMs.

POM of integration with respect to x
The integration of the G-FLFs, GL β (x, t) with respect to the variable x is determined by the following relation: where P r denotes M (N + 1) × M (N + 1) POM of integration with respect to x.
To introduce the proposed matrix, we introduce the POM of GPs as follows: x 0 ξ r G(ξ)dξ x r+1P r G(x). Now, to compute each element of the POM for GPs, we follow the below process: Also, it can be written Then, by substituting the aforesaid equation in previous equation, we achieve where κ mki = m k g m−k k + r + 1 b ki . Therefore, we obtain the integral POM of G-FLFs as

POM of fractional integration with respect to t
In this part, we present the pseudo-operational matrix of fractional integration of order ν > 0 for G-FLFs vector GL β (x, t) with respect to t as Here Υ β,ν l denotes the pseudo-operational matrix of fractional integration of order ν and Υ β,ν l = I ⊗ Θ β,ν l . So that and Θ β,ν l is the fractional integral pseudo-operational matrix of FLFs, which is defined in [8].

POM of fractional derivative with respect to t
In the current section, we consider the pseudo-operational matrix of the fractional derivative of order ν > 0 with respect to t. Then, we have Here Φ β,ν s denotes the pseudo-operational matrix of fractional derivative for G-FLFs. To get the desired goal, we obtain the pseudo-operational matrix of the fractional derivative for FLFs as Also, it can be written t βk Consequently, the POM of the fractional derivative for G-FLFs is obtained by using the pseudo-operational matrix of the fractional derivative of FLFs as Φ β,ν s = I ⊗ Q β,ν s .

Necessary relations
In the proceeding of the numerical algorithm, we need to calculate the following formula: To achieve the sought goal, we use the below relation Thus, the matrix Λ is obtained as According to the Genocchi polynomials properties, we get the vector S as

Construction of the numerical method
This article is devoted to investigating the numerical solution of the following form of the time-fractional diffusion equation: and the time-fractional subdiffusion equation where F is a nonlinear function and the functions ω 0 (x), ω 1 (x), ϕ 0 (t) and ϕ 1 (t) are known in the domain Ω, and also u is an unknown function to be calculated. The involved fractional derivatives are described in the Caputo sense. In the present approach, we approximate the higher order derivative by G-FLFs as By integrating from Equation (4.3) with respect to t and utilizing the initial conditions and the POM of integration introduced in Section 3.2, we obtain integrating from Equation (4.4), to obtain In the following, by integrating Equation (4.5) with respect to x and using the POM of integration in Equation (3.1), we have To achieve the approximate solution, it is necessary to determine unknown function ∂u(0,t) ∂x . Thus, integrate Equation (4.6) with respect to x from 0 to 1 as Then, by substituting the above relation in Equations (4.6) and (4.7), the desired results are obtained. In a similar way, in order to express the approximation of the other unknown functions, we take integration from Equation (4.3) and follow the process below: and also, In order to calculate the unknown statement ∂ 3 u(0,t) ∂x∂t 2 , integrating Equation (4.8) from 0 to 1 as Then, from Equations (4.9) and (4.10), we get By applying the properties of Caputo fractional derivative operator of order 0 < ν ≤ 1 in Equation (4.11), we have Also, to approximate the fractional differential part of subdiffusion equation, we use Equation (4.5) as follows: Next, we substitute the approximations results in Equations (4.1) and (4.2) to get an algebraic equation. Then, we utilize M nodal points of Newton-Cotes [8] for variable x and N + 1 zeros of LPs for variable t, to collocate the proposed equations. Consequently, we apply Newtons method or Broydens method or any other iterative solver to get the solution of the nonlinear system.

Error bound
This section investigates the error bound of the integral POM and the error of partial derivative of the approximate solution. To compute the upper bound of error for POM of integration with respect to variable x, we use Equation (3.1), Lemmas 1 and 2 in [12], and the following relation: On the other hand, according to the above relation, we obtain Then, we get Each component ofR Therefore, due to the orthogonal property of fractional-order Laguerre functions in Equation (2.1), we have Moreover, from Equation (3.1) the following result is achieved: The above discussion indicates that with increasing the number of G-FLFs bases, each element ofR r 1,x tends to zero quickly.
Theorem 1. Suppose that u M N (x, t) is the approximate of the exact solution u(x, t) obtained by the proposed method over the region Ω, then the upper bound of error is given by Proof. From the algorithm of the method mentioned in Section 4, the second derivative of u(x, t) with respect to t is expressed by Considering Equation (4.9), the following expression is established: From the above theorem, it can be inferred that for large values of M and N the upper bound of error tends to zero. In a similar manner, it can be deduced that

Numerical experiments
In this section to illustrate the applicability and validity of the numerical scheme performed in the previous sections, we give seven examples. The calculations were performed on a personal computer and the codes were written in MATLAB 2016.

Example 1. First, we consider the one-dimensional linear inhomogeneous time fractional Burgers equation [31, 42]
subject to the initial condition u(x, 0) = x 2 , x ∈ [0, 1] and the boundary condition u(0, t) = t 2 , u(1, t) = 1 + t 2 , t > 0. The exact solution is u(x, t) = x 2 + t 2 . From the algorithm of the proposed approach, we obtain Then, by replacing the obtained results in Equation (6.1) and considering M = N = 2 for various values of β, ν, we get u(x, t) = x 2 + t 2 , which is the exact solution of Equation ( subject to the initial conditions u(x, 0) = 1 + sin(x), u t (x, 0) = 0, x ∈ [0, 1], and the boundary conditions u(0, t) = cosh(t), u(1, t) = sin(1) + cosh(t), t > 0. The exact solution of this problem is u(x, t) = sin(x) + cosh(t) for special value of ν = 2. According to the mentioned approach, we obtain By taking ν = 2, M = N = 2 and replacing the aforesaid relation in Equation (6.2), we get u(x, t) = sin(x) + cosh(t), which is the exact solution of the  Example 3. Let us consider the following two-term wave-diffusion equation with damping [5,28,38]: subject to the initial conditions u(x, 0) = 0, u t (x, 0) = 0, x ∈ [0, 1] and the boundary conditions Table 2 represents the absolute errors by the present technique and methods in [5,28,38]. From Table 2, we conclude that the results produced by our technique with a few terms of G-FLFs are more accurate than methods in [5,28,38]. Also, the absolute errors for ν = 1.5 and β = 1 on interval [0, 1] × [0, 50] are plotted in Figure 2. It illustrates that the proposed method has high precision in the large interval.
Present method Method in [14] Method in [16]  In Figure 3, the absolute errors are illustrated by considering the points in the interval [0, 1] × [0, 100] for different values of ν = 1.75, 1.95 and β = 1, M = 2, N = 8. Due to the obtained results and results of the method in [8], it can be concluded that the present method is more accurate than the mentioned method.
Also, Table 6 observed that the proposed method is more accurate than the Crank-Nicolson finite element method [45]. Example 6. Consider the following subdiffusion equation [7,17,20]: subject to the initial conditions u(x, 0) = 0, x ∈ [0, 1] and the boundary conditions u(0, t) = t 1+γ , u(1, t) = exp(1)t 1+γ , t > 0, The exact solution  Table 7.   The results of the present approach with wavelet Galerkin method [20] are compared in Table 7. It is clear that our method with a few numbers of G-FLFs is more accurate than wavelets Galerkin method. The author in [7] presented the maximum absolute errors for different cases γ by high-order compact finite difference method. Therefore, γ = 0.25 and γ = 0.75 are obtained 1.8928×10 −5 and 5.6363×10 −5 , respectively. Also, Gao and Sun [17] presented the maximum absolute errors by compact finite difference scheme for γ = 0.25 and γ = 0.75 are 1.4338 × 10 −8 and 7.1085 × 10 −6 , respectively. In addition, the graph of absolute error for different values of M and N with β = 0.5, γ = 0.25 are plotted in Figure 5. From the obtained results, we can conclude that the mentioned approach is in excellent agreement with the exact solution. Example 7. We consider the following nonlinear generalized Benjamin-Bona-Mahony-Burgers equation:  Table 8.  Table 9. Table 10

Conclusions
In this work, a Genocchi-fractional Laguerre collocation method was applied successfully for the numerical solution of FDEs and FSDEs. Then, we introduced the pseudo-operational matrices in implementation of the method. The great advantages of the presented scheme are the novel G-FLFs, the pseudooperational matrices with high accuracy, and quick reduction of FDEs and FSDEs to systems of nonlinear algebraic equations. The outcomes illustrate that the computed solutions for various values of β have good agreement with the exact solution. From the tables and graphs plotted, it can be inferred that by applying a small value of G-FLFs the satisfactory results with high accuracy is obtained.