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

Jiaquan Xie,Xia Yang,Qingxue Huang

doi:10.1186/s40064-016-2832-y

Jiaquan Xie, Xia Yang + Show 1 more

Open Access

https://doi.org/10.1186/s40064-016-2832-y

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Journal: SpringerPlus	Publication Date: Jul 22, 2016
Citations: 12	License type: CC BY 4.0

Affiliation: Taiyuan University of Science and Technology

Abstract

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.

Highlights

Convection diffusion equations are regarded as a kind of basic equations of motion, which have been applied in describing water flow movement (Hu et al 2016; Colla et al 2015; Su 2014), material transport and diffusion (Liu et al 2016; Calo et al 2015; Karalashvili et al 2015; Fang and Deng 2014)
C0, c(l, t) = 0, t > 0, 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:
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

Summary

Background

Convection diffusion equations are regarded as a kind of basic equations of motion, which have been applied in describing water flow movement (Hu et al 2016; Colla et al 2015; Su 2014), material transport and diffusion (Liu et al 2016; Calo et al 2015; Karalashvili et al 2015; Fang and Deng 2014). C0, c(l, t) = 0, t > 0, 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 operational matrices of fractional-order are employed to obtain the numerical solutions of Eq (2). 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. Numerical simulation we apply the proposed algorithm in the previous section to obtain numerical solutions of some convection diffusion equations with variable coefficients. Examples 2 and 3 show that the absolute error can reaches to 10−6 for general one-dimensional fractional convection diffusion equations with variable coefficients.

Consider the convection diffusion

Conclusion