Determination of control parameter in an inverse time fractional‎ ‎diffusion equation using a linearized fourth-order finite difference scheme

Mohebbi, Akbar; Salahi, Behnam

doi:10.22124/jmm.2026.32796.2986

Determination of control parameter in an inverse time fractional‎ ‎diffusion equation using a linearized fourth-order finite difference scheme

Articles in Press

Document Type : Research Article

Authors

Akbar Mohebbi ¹
Behnam Salahi ²

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

² Department of Mathematics‎, ‎Faculty of Mathematics‎, ‎Statistics and Computer, Semnan University‎, ‎Semnan‎, ‎Iran

10.22124/jmm.2026.32796.2986

Abstract

‎The problem of finding the space‎- ‎or time-dependent control parameter in partial differential equations has increasingly appeared in physical‎
‎phenomena‎, ‎for example‎, ‎in the study of control theory‎, ‎heat conduction process‎, ‎and‎
‎chemical diffusion‎. ‎This study aims to construct an efficient numerical method to determine a time-dependent source term in a time fractional diffusion‎
‎equation subject to over-specification at a point in the spatial domain‎.
‎We use a second order scheme to discretize the equation in the time direction‎, ‎then we replace the space derivative with a fourth-order compact finite difference approximation‎. ‎We will construct a linearized difference scheme and prove the solvability, and unconditional stability of the proposed method‎. ‎Due to the usually ill-posed nature of inverse problems‎, ‎we examine the stability of the method with respect to perturbations of the data‎. ‎We show that the proposed method achieves stable and accurate numerical‎ ‎approximations without using any regularization techniques‎. ‎Numerical experiments show satisfactory results for problems with smooth‎, ‎non-smooth‎, ‎and discontinuous initial conditions‎.

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