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We develop a new matrix-based approach to the numerical solution of partial differential equations (PDE) and apply it to the numerical solution of partial fractional differential equations (PFDE). The proposed method is to discretize a given PFDE as a Sylvester Equation, and parameterize the integral surface using matrix algebra. The combination of these two notions results in an algorithm which can solve a general class of PFDE efficiently and accurately by means of an O ( n 3 ) algorithm for solving the Sylvester Matrix Equation (over an m × n grid with m ∼ n ). The proposed parametrization of the integral surface allows for the solution with the more general Robin boundary conditions, and allows for high-order approximations to derivative boundary conditions. To achieve our ends, we also develop a new matrix-based approximation to fractional order derivatives. The proposed method is demonstrated by the numerical solution of the fractional diffusion equation with fractional derivatives in both the temporal and spatial directions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a wavelet operational method based on Haar wavelet is proposed to solve the fractional partial differential equations in the Caputo derivative sense. We give the Haar wavelet operational matrix of fractional order integration. A truncated Haar wavelet series together with the wavelet operational matrix are utilized to reduce the fractional partial differential equations to Sylvester equations. In addition, some examples are presented to show the efficiency and the accuracy of the approach. [Copyright &y& Elsevier]