Applying the Ritz method to solve optimal control problems of parabolic equations using internal controller

Document Type : Original Paper

Authors

1 Department of Mathematics, University of science and Technology of Mazandaran

2 Department of Applied Mathematics University of Science and Technology of Mazandaran

3 Department of Mathematics, University of Science and Technology of Mazandaran, P.O. Box 48518-78195, Behshahr, Iran

Abstract

In this article, we solve control problems with the conditions of parabolic partial differential
equations. In order to solve this type of mentioned problems, we use an iterative method based on basic polynomials and generating function. In this method, we first consider the solution of the problem based on the given partial differential equation, and then we approximate this solution based on the basic polynomials and the generating function (the function that applies to the initial conditions of the problem). which apply precisely in the given boundary conditions. After that, by placing this solution in the given sub-function and optimality conditions, we reach a system of nonlinear equations. By solving this device of unknown coefficients, we get the approximate answer. In fact, an approximation of the answer is made in a finitedimensional space. We have also shown the convergence of the proposed method for this type of problem.
In order to check the effectiveness of the method, we have solved some examples using it and presented the obtained numerical results.

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Main Subjects

Journal of Advanced Mathematical Modeling
Volume 14, Issue 4 - Serial Number 36
November 2024
Pages 128-139

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History

  • Receive Date: 16 June 2024
  • Revise Date: 07 October 2024
  • Accept Date: 02 February 2025

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