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Optimal solution of a general class of nonlinear system of fractional partial differential equations using hybrid functions

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Abstract

This paper introduces a general class of nonlinear system of fractional partial differential equations with initial and boundary conditions. A hybrid method based on the transcendental Bernstein series and the generalized shifted Chebyshev polynomials is proposed for finding the optimal solution of the nonlinear system of fractional partial differential equations. The solution of the nonlinear system of fractional partial differential equations is expanded in terms of the transcendental Bernstein series and the generalized shifted Chebyshev polynomials, as basis functions with unknown free coefficients and control parameters. The corresponding operational matrices of fractional derivatives are then derived for the basis functions. These basis functions, with their operational matrices of fractional order derivatives and the Lagrange multipliers, transform the problem into a nonlinear system of algebraic equations. By means of Darbo’s fixed point theorem and Banach contraction principle, an existence result and a unique result for the solution of the nonlinear system of fractional partial differential equations are obtained, respectively. The convergence analysis is discussed and several illustrative experiments illustrate the efficiency and accuracy of the proposed method.

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Authors and Affiliations

  1. Department of Mathematics, Anand International College of Engineering, Jaipur, 303012, India

    H. Hassani

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

    J. A. Tenreiro Machado

  3. Department of Mathematics, Yasouj University, Yasouj, Iran

    E. Naraghirad

  4. Department of Applied Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, China

    Z. Avazzadeh

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

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Hassani, H., Machado, J.A.T., Naraghirad, E. et al. Optimal solution of a general class of nonlinear system of fractional partial differential equations using hybrid functions. Engineering with Computers 39, 2401–2431 (2023). https://doi.org/10.1007/s00366-022-01627-4

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  • DOI: https://doi.org/10.1007/s00366-022-01627-4

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