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Numerical approach for solving variable-order space–time fractional telegraph equation using transcendental Bernstein series

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Abstract

This paper presents the transcendental Bernstein series (TBS) as a generalization of the classical Bernstein polynomials for solving the variable-order space–time fractional telegraph equation (V-STFTE). An approximation method using optimization techniques and the TBS is introduced. The solution of the problem under consideration is expanded in terms of TBS with unknown free coefficients and control parameters. The new corresponding operational matrices of variable-order fractional derivatives, in the Caputo type, are derived. The proposed approach reduces the V-STFTE to a system of algebraic equations and, subsequently, to find the free coefficients and control parameters using the Lagrange multipliers technique. The convergence analysis of the method is guranteed by means of a new theorem concerning the TBS. The experimental results confirm the high accuracy and computational efficiency of the TBS method.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, Iran

    H. Hassani

  2. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Z. Avazzadeh

  3. Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, R. Dr. António Bernardino de Almeida, 4249-015, Porto 431, Portugal

    J. A. Tenreiro Machado

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  1. H. Hassani
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  2. Z. Avazzadeh
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  3. J. A. Tenreiro Machado
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Correspondence to H. Hassani.

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Hassani, H., Avazzadeh, Z. & Machado, J.A.T. Numerical approach for solving variable-order space–time fractional telegraph equation using transcendental Bernstein series. Engineering with Computers 36, 867–878 (2020). https://doi.org/10.1007/s00366-019-00736-x

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  • DOI: https://doi.org/10.1007/s00366-019-00736-x

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