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

Articles in Press

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 their solving in the literatures. 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 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 Grunwald-Letnikov and L1 approximation methods and derive two approximation formulas for 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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Soft Computing Journal

Articles in Press, Accepted Manuscript
Available Online from 30 September 2024

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History

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

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