An efficient finite element method for optimal control problems involving distributed-order time-fractional diffusion equations

Document Type : Original Article

Authors

1 Department of Applied Mathematics, Faculty of Mathematical Science, University of Kashan, Kashan, Iran

2 Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran

Abstract

In this paper, we present a finite element method to approximate the solution of optimal control problems involving distributed-order time-fractional diffusion equations. The dynamics of these problems involve distributed-order time-fractional derivatives, which are a generalization of fractional derivatives. Despite the importance of these problems, there exist few researches on solving them in the literature. Numerical methods for solving optimal control problems are classified into two main categories, namely, indirect methods and direct methods. In the indirect methods, by using the Pontryagin principle, the necessary conditions for optimality are derived and formulated as a nonlinear two-point boundary value problem. On the other hand, in the direct methods, by discretizing the control and state variables, the considered problem is reduced to a nonlinear programming problem. Due to the difficulties related to solving the system of equations resulting from the necessary conditions for optimality in optimal control problems involving distributed-order time-fractional diffusion equations, in this paper, we use the approach of direct methods to approximate the solution of these problems. To approximate the distributed-order time-fractional derivatives, we employ the Grünwald-Letnikov and L1 approximation methods and derive two approximation formulas for the derivative. Also, for spatial discretization, we utilize the piecewise linear finite element method. Therefore, we transform the original problem into a convex quadratic optimization problem, which can be efficiently solved using existing optimization algorithms. We consider two numerical examples to demonstrate the efficiency and accuracy of the proposed method.

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References

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Soft Computing Journal
Volume 14, Issue 1 - Serial Number 27
September 2025
Pages 108-121

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History

  • Receive Date: 31 January 2024
  • Revise Date: 26 May 2024
  • Accept Date: 09 July 2024

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