Numerical Solution of the Two-Sided Space–Time Fractional Telegraph Equation Via Chebyshev Tau Approximation

The operational matrices of left Caputo fractional derivative, right Caputo fractional derivative, and Riemann–Liouville fractional integral, for shifted Chebyshev polynomials, are presented and derived. We propose an accurate and efficient spectral algorithm for the numerical solution of the two-si...

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Bibliographic Details
Published in:Journal of optimization theory and applications Vol. 174; no. 1; pp. 321 - 341
Main Authors: Bhrawy, Ali H., Zaky, Mahmoud A., Machado, José A. Tenreiro
Format: Journal Article
Language:English
Published: New York Springer US 01.07.2017
Springer Nature B.V
Subjects:
Accuracy
Algebra
Algorithms
Applications of Mathematics
Boundary conditions
Calculus of Variations and Optimal Control; Optimization
Chebyshev approximation
Dirichlet problem
Engineering
Iterative methods
Mathematics
Mathematics and Statistics
Matrices (mathematics)
Operations Research/Decision Theory
Optimization
Polynomials
Simplification
Spacetime
Theory of Computation
Operational matrix
Fractional telegraph equation
Riesz fractional derivative
Fractional Klein–Gordon equation
Shifted Chebyshev Tau method
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Summary:The operational matrices of left Caputo fractional derivative, right Caputo fractional derivative, and Riemann–Liouville fractional integral, for shifted Chebyshev polynomials, are presented and derived. We propose an accurate and efficient spectral algorithm for the numerical solution of the two-sided space–time Caputo fractional-order telegraph equation with three types of non-homogeneous boundary conditions, namely, Dirichlet, Robin, and non-local conditions. The proposed algorithm is based on shifted Chebyshev tau technique combined with the derived shifted Chebyshev operational matrices. We focus primarily on implementing the novel algorithm both in temporal and spatial discretizations. This algorithm reduces the problem to a system of algebraic equations greatly simplifying the problem. This system can be solved by any standard iteration method. For confirming the efficiency and accuracy of the proposed scheme, we introduce some numerical examples with their approximate solutions and compare our results with those achieved using other methods.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-016-0863-8