Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

Abstract

Abstract:In this paper, a class of fractional diffusion equations with variable coefficients is considered. An accurate and efficient spectral tau technique for solving the fractional diffusion equations numerically is proposed. This method is based upon Chebyshev tau approximation together with Chebyshev operational matrix of Caputo fractional differentiation. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. We apply this general method to solve four specific examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving the timedependent fractional diffusion equations.PACS (2008)

On the Sums of Powers of Chebyshev Polynomials and Their Divisible Properties

“…See [1][2][3][4][5][6][7][8][9][10][11]. For example, Li [1] proved some identities involving power sums of ( ) and ( ).…”

“…Ma and Lv [2] studied the computational problem of the reciprocal sums of Chebyshev polynomials and obtained some identities. Some theoretical results related to Chebyshev polynomials can be found in Ma and Zhang [3], Cesarano [4], Lee and Wong [5], Bhrawy and others (see [6][7][8][9]), and Wang and Zhang [10]. Bircan and Pommerenke [11] also obtained many important applications of the Chebyshev polynomials.…”

Some Identities Involving the Reciprocal Sums of One‐Kind Chebyshev Polynomials

“…Some theoretical results related to Chebyshev polynomials can be found in Ma and Zhang [4], Cesarano [5], Babusci et al [6][7][8], Lee and Wong [9], and Wang and Han [10]. Doha and others [11][12][13][14] and Bircan and Pommerenke [15] also obtained many important applications of the Chebyshev polynomials.…”

A finite difference method for fractional diffusion equations with Neumann boundary conditions

“…can be combined with tau methods [14,[17][18][19] or with Gauss-Lobatto collocation methods [20]. A furher alternative to solve fractional diffusion problems numerically is the application of fractional order Laplace transform [21], [22].…”