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Regularized solution for backward heat equation in Sobolev space
Author(s) -
Hapuarachchi Sujeewa,
Xu Yongzhi S.
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5851
Subject(s) - mathematics , sobolev space , regularization (linguistics) , heat equation , mathematical analysis , well posed problem , inverse problem , boundary value problem , hadamard transform , initial value problem , space (punctuation) , linguistics , philosophy , artificial intelligence , computer science
Inverse problems in partial deferential equations are severely ill posed in the sense of Hadamard. So the heat equation with a terminal condition problem is ill posed even in the sobolev space so regularization is needed. In this paper, we discuss about the convergence result of the approximation problem in the Sobolev spaceH 2 ( R n ) , which is well posed. By using a small parameter, we construct an approximation problem and use a quasi‐boundary value method to regularize nonlinear heat equation. Finally, we prove the approximation solution converges to the original solution whenever the parameter goes to zero inH 2 ( R n ) .