Abstract
In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.
Highlights
The theory of fractional calculus is an ancient topic that has many applications
Khader [20] and his co-author applied the finite difference method coupled with the Hermite formula for solutions of fractional diffusion wave equations
We proposed a hybrid method based on finite difference and Haar wavelets approximations
Summary
The theory of fractional calculus is an ancient topic that has many applications. practical work in this direction has been recently started (see References [1,2,3]). Zhou and Xu [17] applied the Chebyshev wavelets collocation method for the solution of time fractional diffusion wave equations. Bhrawya [18] used the spectral Tau algorithm based on the Jacobi operational matrix for the numerical solution of time fractional diffusion-wave equations. Khader [20] and his co-author applied the finite difference method coupled with the Hermite formula for solutions of fractional diffusion wave equations. We propose a hybrid numerical scheme, based on Haar wavelets and finite differences, to solve (1 + 1)- and (1 + 2)-dimensional TFDWEs. The stability of the proposed method is discussed with the matrix method which is an essential part of the manuscript.
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