Recovery of coefficients of a heat equation by Ritz collocation method

Kuwait Journal of Science

article

Kamal Rashedi¹ 

[1] Dept. of Mathematics, University of Science and Technology of Mazandaran

关键词: Inverse coefficient problem;    mollification method;    parabolic equation;    Ritz approximation;    Tikhonov regularization;   

DOI  :  10.48129/kjs.18581

学科分类：社会科学、人文和艺术（综合）

来源: Kuwait University * Academic Publication Council

【 摘 要 】

In this work, we discuss a one dimensional inverse problem for the heat equation where the unknown functions are solely time-dependent lower order coefficient and multiplicative source term. We use as data two integral overdetermination conditions along with the initial and Dirichlet boundary conditions. In the first step, the lower order term is eliminated by applying a transformation and the problem is converted to an equivalent inverse problem of determining a heat source with initial and boundary conditions, as well as a nonlocal energy over-specification. Then, we propose a Ritz approximation as the solution of the unknown temperature distribution and consider a truncated series as the approximation of unknown time-dependent coefficient in the heat source. The collocation method is utilized to reduce the inverse problem to the solution of a linear system of algebraic equations. Since the problem is ill-posed, numerical discretization of the reformulated problem may produce ill-conditioned system of equations. Therefore, the Tikhonov regularization technique is employed in order to obtain stable solutions. For the perturbed measurements, we employ the mollification method to derive stable numerical derivatives. Numerical simulations while solving two test examples are presented to show the applicability of theproposed method.

【 授权许可】

Unknown