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A NONLINEAR BACKWARD PARABOLIC PROBLEM: REGULARIZATION BY QUASI-REVERSIBILITY AND ERROR ESTIMATES

Nguyen Huy Tuan

Department of Applied Mathematics, SaiGon University, 273 An Duong Vuong street, HoChiMinh City, Vietnam

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Pham Hoang Quan

Department of Applied Mathematics, SaiGon University, 273 An Duong Vuong street, HoChiMinh city, Vietnam

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Dang Duc Trong

Department of Mathematics, University of Natural science, 227 Nguyen Van Cu street, HoChiMinh City, Vietnam

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Nguyen Do Minh Nhat

Department of Mathematics, Wayne State University, 351 Manoogian Hall Detroit, MI 48202, USA

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https://doi.org/10.1142/S1793557111000125Cited by:1 (Source: Crossref)

Abstract

In this paper, we consider an inverse time problem for a nonlinear parabolic equation in the form u_t + Au(t) = f(t, u(t)), u(T) = φ, where A is a positive self-adjoint unbounded operator and f is a Lipschitz function. As known, it is ill-posed. Using a quasi-reversibility method, we shall construct regularization solutions depended on a small parameter ϵ. We show that the regularized problem is well-posed and that their solution u^ϵ(t) converges on [0, T] to the exact solution u(t). This paper extends the work by Dinh Nho Hao et al. [8] to nonlinear ill-posed problems. Some numerical tests illustrate that the proposed method is feasible and effective.

Keywords:

AMSC: 35K05, 35K99, 47J06, 47H10

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