A new algorithm to determine the creation or depletion term of parabolic equations from boundary measurements. (15th November 2020)

• Record Type: Journal Article
• Authors: Nguyen, Loc Hoang
• Is Part Of: Computers & mathematics with applications. Volume 80:issue 10(2020)
• Page Start: 2135
• Page End: 2149
• Publication Date: 2020-11-15
• Subjects: Coefficient inverse problem -- Parabolic equations -- Approximation -- Fourier coefficients; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2020.09.010
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: We propose a robust numerical method to find the coefficient of the creation or depletion term of parabolic equations from the measurement of the lateral Cauchy information of their solutions. Most papers in the field study this nonlinear and severely ill-posed problem using optimal control. The main drawback of this widely used approach is the need of some advanced knowledge of the true solution. In this paper, we propose a new method that opens a door to solve nonlinear inverse problems for parabolic equations without any initial guess of the true coefficient. This claim is confirmed numerically. The key point of the method is to derive a system of nonlinear elliptic equations for the Fourier coefficients of the solution to the governing equation with respect to a special basis of L 2 . We then solve this system by a predictor–corrector process, in which our computation to obtain the first and second predictors is effective. The desired solution to the inverse problem under consideration follows. Highlights: Introduce a model to find desired coefficient. No need an initial guess. Fast convergence. Interesting numerical results.