Numerical solution of the one-dimensional fractional convection diffusion equations based on Chebyshev operational matrix

In this paper, we are concerned with nonlinear one-dimensional fractional convection diffusion equations. An effective approach based on Chebyshev operational matrix is constructed to obtain the numerical solution of fractional convection diffusion equations with variable coefficients. The principal characteristic of the approach is the new orthogonal functions based on Chebyshev polynomials to the fractional calculus. The corresponding fractional differential operational matrix is derived. Then the matrix with the Tau method is utilized to transform the solution of this problem into the solution of a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via examples. It is shown that the proposed algorithm yields better results. Finally, error analysis shows that the algorithm is convergent.

methods are new on this area which have made great success presently. Especially in recent 20 years, the articles about operational matrix methods are springing up. In Ref. Abbasbandy et al. (2015), the authors proposed the operational matrix method of fractional-order Legendre functions for solving time-fractional convection-diffusion equations.
In the following we mention some real-world applications of convection diffusion equations. In this paper, we take an example of a diffusion kinetic model, the flow rate is u = u(x, t), dye concentration is c(x, t) and diffusion flux of dyes is q = q(x, t). We also set up a variety of biological, chemical and other factors to control the production rate of dye (Production of unit volume per unit time) F c . In the flow field, a system (its volume can be arbitrary), as shown in Fig. 1, is obtained. The space occupied by the system during the flow of V(t) is controlled at the time t.
According to the model, and the related initial-boundary conditions, we obtain the one-dimensional convection diffusion equations (Chen and Jin (2007)).
In this paper, a numerical approach based on Chebyshev operational matrix is proposed for solving one-dimensional fractional convection diffusion equations with variable coefficients of the following form: the parameter ν refers to the fractional order of spatial derivative with 1 < ν≤2. In the proposed method, the operational matrices of fractional-order are employed to obtain the numerical solutions of Eq. (2). (1) The schematic diagram of dye diffusion in the system The paper is organized as follows: In "Preliminaries and notations" section, some basic definitions and mathematical preliminaries of fractional calculus are introduced. The fractional differential operational matrix is given in "The fractional derivative operational matrix P (ν) " section. We mainly illustrate the proposed method in "Description of the proposed method" section. In "Error analysis" section, the convergence of the proposed approach is proved. In "Numerical simulation" section, the proposed approach is applied to test several numerical examples. Finally, a conclusion is given in "Conclusion" section.

Preliminaries and notations
The basic definitions of fractional integral and differential operator Definition 1 The Riemann-Liouville fractional integral operator I ν of order ν is defined as

Definition 2
The Riemann-Liouville fractional differential operator D ν of order ν is defined as Definition 3 The Caputo fractional differential operator is defined as For the Caputo derivative, we have

Properties of the Chebyshev polynomials
The well-known Chebyshev polynomials are defined on the interval (−1, 1) and can be determined with the aid of the following recurrence formula (Doha et al. 2011): where T 0 (t) = 1 and T 1 (t) = t. In order to use these polynomials on the interval x ∈ (0, 1), we define the Chebyshev polynomials by introducing the change of variable t = 2x − 1. Let the Chebyshev polynomials T i (t) = 2x − 1 are denoted by T i (x), then T i (x) can be obtained as follows: where T 0 (x) = 1 and T 1 (x) = 2x − 1. The analytic form of the Chebyshev polynomials T i (x) of degree i is given by for γ ∈ N 0 and γ < ⌈ν⌉; Γ (γ +1−ν) t γ −ν , for γ ∈ N 0 and γ ≥ ⌈ν⌉ or γ / ∈ N 0 and γ > ⌊ν⌋.
The orthogonally condition is

Function approximation
Suppose u(x) ∈ L 2 (0, 1), it may be expressed in terms of the Chebyshev polynomials as where the coefficients c i is given by If we consider truncated series in Eq. (10), then we have: where Then the derivative of vector Φ can be expressed by where P (1) is the (M + 1) × (M + 1) operational matrix of derivative given by For example for even M, we have Based on the function approximation theory which the solution function is expressed as orthogonal polynomials. For arbitrary function u(x, t) ∈ L 2 ((0, 1) × (0, 1)), it can be expanded as the following way where If we consider truncated series in Eq. (16), then we have: where In this paper, we use the Tau method Dehghan and Saadatmandi 2006) to compute the coefficients u ij .

The fractional derivative operational matrix P (ν)
The main objective of this section is to prove the following theorem for the fractional derivatives of the Chebyshev polynomials.
Lemma 1 Let T i (x) be a Chebyshev polynomial; then (Doha et al. 2011) Proof This Lemma can be easily proved by making use of relation (5) and (7).
Theorem 1 Let Φ(x) be the Chebyshev vector defined in Eq. (12) and suppose ν > 0; then where P (ν) is the (M + 1) × (M + 1) differential operational matrix of order ν in the Caputo sense and it is defined as follows: where Proof The analytical form of the Chebyshev polynomials T i (x) of degree i is given by Eq. (8), using Eqs. (6) and (8)  where S ν i, j = i k=⌈ν⌉ θ ijk , and x k−ν , i = ⌈ν⌉, ⌈ν⌉ + 1, . . . , M. (23) After some lengthy manipulation, θ i,j,k may be put in the form Accordingly, Eq. (21) can be written in a vector form as follows: Also, according to Lemma 1, we can write A combination of Eqs. (27) and (28) leads to the desired result.

Description of the proposed method
In the section, we will use the Chebyshev polynomials operational matrix of fractional derivative to obtain the numerical solutions of one-dimensional fractional convection diffusion equations with variable coefficients.
Here, for simplicity we consider the convection diffusion equations of the following form: where ∂ ν ∂x ν denote fractional derivatives in the Caputo's sense, with the initial-boundary conditions In order to use the Chebyshev polynomials, we first approximate where U = u ij (M+1)×(N +1) is an unknown matrix.
The following is the product of two vectors based on Chebyshev operational matrix method. Let (Bhrawy et al. 2015) where Q is the (M + 1) × (M + 1) operational matrix with the element q kj . In virtue of Eq. (12) and above relation, enable us to write Multiplying both sides of the above equation by T k (x)ω(x), k = 0, 1, . . . , M and integrating the result from 0 to 1, we obtain Equation (34)  We can also approximate the function h 1 (x), g 1 (t) and g 2 (t) as  46) gives (M + 1)(N + 1) equations, which can be solved for u ij , (i = 0, 1, …, M, j = 0, 1, …, N). So the unknown function u(x, t) can be find out.

Error analysis
In real problems, we often tend to solve some equations with unknown exact solutions. Hence, when we apply our method to these kinds of problems, it is necessary to introduce a process for estimating the error function (Chen et al. 2014).
We consider e n (x, t) = u(x, t) − u mn (x, t) as the error function of the approximate solution u mn (x, t) for u(x, t), where u(x, t) is the exact solution of Eq. (2) Therefore, u n (x, t) satisfies the following problem The perturbation term R mn (x, t) can be obtained by substituting the estimated solution u mn (x, t) into the equations: Subtracting Eq. (47) from Eq. (2), we get the following equations: Obviously the above equation is one-dimensional fractional convection diffusion equation in which the error function e mn (x, t), is the unknown function. We can easily apply our method to the above equation to find an approximation of the error function e ′ mn (x, t).

Numerical simulation
In this section, we apply the proposed algorithm in the previous section to obtain numerical solutions of some convection diffusion equations with variable coefficients.
Example 1 Consider the two-dimensional fractional convection diffusion equations with homogeneous initial-boundary conditions √ π , and the initial-boundary conditions: The exact solution of this problem is u(x, t) = xt(x − 1)(t − 1). The graphs of numerical solution for M = N = 4 is shown in Fig. 2. Absolute error between the numerical and analytical solutions is shown in Fig. 3. The graphs of analytical and approximate solutions for some nodes in (0, 1) × (0, 1) are presented in Fig. 4. Absolute error between the numerical and analytical solutions are also shown at different times in Fig. 5. Figures 2 and 4 show that the numerical solutions are very close to the analytical solutions. Figures 3 and 5 show that the proposed algorithm has a high convergence precision.
Example 2 Consider the fractional convection diffusion equations with variable coefficients with the initial-boundary conditions: where a(x) = x 3 , b(x) = x 2 6 ; f (x, t) = x 3 cosh(t). The exact solution of this problem is u(x, t) = x 3 sinh(t). The graph of the numerical solution for M = N=7 is shown in Fig. 6. Absolute error between the numerical and analytical solutions is shown in Fig. 7. The graphs of analytical and numerical solution for different M and N in some nodes are shown in Fig. 8. Absolute error between the numerical and analytical solution are also shown at different times in Fig. 9.
From Figs. 6 and 8, we can conclude that the numerical solutions converge to the exact solutions very well. Figures 7 and 9 show that the proposed algorithm can get a high u(x, 0) = 0, u(0, t) = 0, u(1, t) = sinh(t), t > 0 convergence precision for one-dimensional convection diffusion equations with variable coefficients. Table 1 and Fig. 11 show that the convergence and accuracy of the proposed algorithm is very good, with M and N increase. Moreover, a small M and N can achieve high precision.

Conclusion
Here a new operational method to approximate the numerical solution of one-dimensional fractional convection diffusion equations have been introduced. To this end, the new operational matrix of fractional-order differentiation is obtained. It appears that using Chebyshev operational matrix algorithm will give more accurate solutions than other existing methods. The approach is computationally efficient and the algorithm can be implemented easily on a computer. The advantage of the methods is that only small size operational matrix is required to provide the solutions at high accuracy. Numerical examples are given to show that the proposed algorithm is robust, efficient and applicable.