A filter method for inverse nonlinear sideways heat equation

Advances in Difference Equations

article

Anh Triet, Nguyen¹  O’Regan, Donal²  Baleanu, Dumitru³  Hoang Luc, Nguyen⁶  Can, Nguyen⁷ 

[1] Faculty of Natural Sciences, Thu Dau Mot University;School of Mathematics, National University of Ireland;Department of Mathematics, Cankaya University;Institute of Space Sciences;Department of Medical Research, China Medical University Hospital, China Medical University;Institute of Research and Development, Duy Tan University;Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University

关键词: Backward problem;    Nonlinear heat equation;    Ill-posed problem;    Cauchy problem;    Regularization method;    Error estimate;   

DOI  :  10.1186/s13662-020-02601-4

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

In this paper, we study a sideways heat equation with a nonlinear source in a bounded domain, in which the Cauchy data at $x = \mathcal {X}$ are given and the solution in $0 \le x < \mathcal {X}$ is sought. The problem is severely ill-posed in the sense of Hadamard. Based on the fundamental solution to the sideways heat equation, we propose to solve this problem by the filter method of degree α, which generates a well-posed integral equation. Moreover, we show that its solution converges to the exact solution uniformly and strongly in $\mathscr {L}^{p}(\omega,\mathcal {X};\mathscr {L}^{2}(\mathbb {R}))$, $\omega\in [0,\mathcal {X})$ under a priori assumptions on the exact solution. The proposed regularized method is illustrated by numerical results in the final section.

【 授权许可】

CC BY