An inverse problem of finding an absorption coefficient in a one-dimensional parabolic differential equation

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, University of Kurdistan, Sanandaj, Iran.

2 Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran.

10.22034/cmde.2026.64614.2931

Abstract

This paper addresses an inverse problem related to the one-dimensional heat equation, incorporating
the initial temperature and information from the heat flux and temperature on one of the boundaries
of the domain and a supplementary temperature measurement at an instant of time. To tackle this
problem, we utilize a discretization method, introducing approximations for both the temperature
distribution and absorption coeffcient functions. These approximations are established using Legendre
basis functions and the operational matrix of differentiation corresponding to the selected bases.
Subsequently, these estimations are incorporated into the residual function and then the least squares
technique is applied to transform the main problem into the solution of a nonlinear system of algebraic
equations. Notably, our proposed algorithm ensures accurate satisfaction of the given initial and
boundary conditions of the problem. We provide proof of the method’s convergence and showcase its
effectiveness through illustrative test examples.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 09 April 2026

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History

  • Receive Date: 20 November 2024
  • Revise Date: 03 April 2026
  • Accept Date: 07 April 2026

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