This paper derives a new operational matrix of the variable-order (VO) time fractional partial derivative involved in anomalous diffusion for shifted Chebyshev polynomials. We then develop an accurate numerical algorithm to solve the 1 + 1 and 2 + 1 VO and constant-order fractional diffusion equation with Dirichlet conditions. The contraction of the present method is based on shifted Chebyshev collocation procedure in combination with the derived shifted Chebyshev operational matrix. The main advantage of the proposed method is to investigate a global approximation for spatial and temporal discretizations, and it reduces such problems to those of solving a system of algebraic equations, which greatly simplifies the solution process. In addition, we analyze the convergence of the present method graphically. Finally, comparisons between the algorithm derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones.
An Efficient Operational Matrix Technique for Multidimensional Variable-Order Time Fractional Diffusion Equations
National Research Centre,
Giza 12622, Egypt;
University of Science and Technology,
Giza 12588, Egypt
Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 20, 2015; final manuscript received April 3, 2016; published online June 20, 2016. Assoc. Editor: Dumitru Baleanu.
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Zaky, M. A., Ezz-Eldien, S. S., Doha, E. H., Tenreiro Machado, J. A., and Bhrawy, A. H. (June 20, 2016). "An Efficient Operational Matrix Technique for Multidimensional Variable-Order Time Fractional Diffusion Equations." ASME. J. Comput. Nonlinear Dynam. November 2016; 11(6): 061002. https://doi.org/10.1115/1.4033723
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