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Numerical solution of arbitrary-order linear partial differential equations using an optimal control technique

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  • Zarepour, Mazyar
  • Loghmani, Ghasem Barid

Abstract

This study aims to present a new approach for finding the numerical solution of linear partial differential equations using an optimal control technique. First, a partial differential equation is transformed into an equivalent optimal control problem and then, control and state variables are approximated by Chebychev series. Therefore, the obtained problem is converted to an optimization problem subject to algebraic equality constraints. Finally, a suitable iterative optimization technique is implemented to approximate the unknown coefficients of Chebychev polynomials to find the numerical solution of the original problem. In this method, a new idea is used, enabling us to deal with almost all kinds of linear partial differential equations with different types of initial and boundary conditions. Several kinds of numerical examples are solved to illustrate the accuracy and efficiency of the proposed method.

Suggested Citation

  • Zarepour, Mazyar & Loghmani, Ghasem Barid, 2021. "Numerical solution of arbitrary-order linear partial differential equations using an optimal control technique," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 77-96.
    • Handle: RePEc:eee:matcom:v:187:y:2021:i:c:p:77-96
    DOI: 10.1016/j.matcom.2021.02.008
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    References listed on IDEAS

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    1. Pakdaman, M. & Ahmadian, A. & Effati, S. & Salahshour, S. & Baleanu, D., 2017. "Solving differential equations of fractional order using an optimization technique based on training artificial neural network," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 81-95.
    2. Lapin, Alexander & Zhang, Shuhua & Lapin, Sergey, 2019. "Numerical solution of a parabolic optimal control problem arising in economics and management," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 715-729.
    3. M. Zarepour & G. B. Loghmani, 2012. "Numerical Solution of Arbitrary-Order Ordinary Differential and Integro-Differential Equations with Separated Boundary Conditions Using Optimal Control Technique," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 933-948, September.
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