Research Papers

An Efficient Operational Matrix Technique for Multidimensional Variable-Order Time Fractional Diffusion Equations

[+] Author and Article Information
M. A. Zaky

Department of Applied Mathematics,
National Research Centre,
Giza 12622, Egypt;
Zewail City of Science and Technology,
University of Science and Technology,
Zayed City,
Giza 12588, Egypt
e-mail: ma.zaky@yahoo.com

S. S. Ezz-Eldien

Department of Mathematics,
Faculty of Science,
Assiut University,
New Valley Branch,
El-Kharja 72511, Egypt
e-mail: s_sezeldien@yahoo.com

E. H. Doha

Department of Mathematics,
Faculty of Science,
Cairo University,
Giza 12613, Egypt
e-mail: eiddoha@eun.eg

J. A. Tenreiro Machado

Department of Electrical Engineering,
Institute of Engineering,
Polytechnic of Porto,
Rua Dr. Antonio Bernardino de Almeida,
Porto 431 4249-015, Portugal
e-mail: jtenreiromachado@gmail.com

A. H. Bhrawy

Department of Mathematics,
Faculty of Science,
Beni-Suef University,
Beni-Suef 62511, Egypt
e-mail: alibhrawy@yahoo.co.uk

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.

J. Comput. Nonlinear Dynam 11(6), 061002 (Jun 20, 2016) (8 pages) Paper No: CND-15-1388; doi: 10.1115/1.4033723 History: Received November 20, 2015; Revised April 03, 2016

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.

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Grahic Jump Location
Fig. 2

Convergence of problem (6.1) with ζ(x,t)=(2+sin(xt))/4 (a) and ζ(x,t)=(6−x3+t5)/10 (b) at h=L=τ=K=1

Grahic Jump Location
Fig. 1

Convergence of problem (6.1) with K = 0.01 (a) and K = 1 (b) at h=L=τ=1 and various choices of ζ(x,t)

Grahic Jump Location
Fig. 3

Convergence of problem (6.2) at ζ(x,y,t)=05 (a) and ζ(x,y,t)=08 (b)

Grahic Jump Location
Fig. 4

Convergence of problem (6.2) at ζ(x,y,t)=(20−exyt)/30 (a) and ζ(x,y,t)=(9−(xt)2−sin (xy)3)/50 (b)



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