Study On Shifted Chebyshev-tau Method For Solving The Space Fractional Diffusion Equations

It is known that, with the development of computer applications, as a mathematicaltool, fractional calculus has evolved from the purely mathematical gradually penetratinginto many branches of scientific and engineering. Anomalous diffusion and convectiondiffusion phenomena as complex problems which are most frequently studied in recentyear have been paid much attention in today’s society. Fractional differential equation isa powerful tool to describe the type of diffusion phenomena. Therefore, finding accurateand efficient methods for solving fractional differential equations has become an activeresearch undertaking. Many experts and scholars are committed to this aspect of theresearch.In order to deal with the diffusion processes, a new numerical algorithm isdeveloped in the paper. Because of Chebyshev polynomials having good properties inthe approximation of functions they have been used widely in the numerical solutions.The article is based on a shifited Chebyshev-tau method to solve a class of fractionaldiffusion equations with boundary conditions.Firstly, according to the approximation theory, functions are approximated byfinite shifted Chebyshev polynomials which are as a series expansion entries;Moreover, the shifted Chebyshev operational matrix of fractional derivatives isderived and used together with the properties of the Shifed Chebyshev polynomial forsolving the each part of the equation;Then, using the tau method and the boundary conditions, the problem can bereduced to a set of linear algebraic equations and the approximate solution of theproblem can be obtained.Finally, the numerical results are presented. From the computational point of view,the solution obtained by this method is in excellent agreement with those obtained byprevious work in the literature and also it is efficient to use.