TY - JOUR

T1 - A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation

AU - Hào, Dinh Nho

AU - Thu Giang, Le Thi

AU - Kabanikhin, Sergey

AU - Shishlenin, Maxim

PY - 2018/12/1

Y1 - 2018/12/1

N2 - We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

AB - We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

KW - Cauchy problem

KW - elliptic equation

KW - finite difference scheme

KW - ill-posed problems

KW - non-local boundary value problems

KW - regularization

KW - very weak solution

KW - INVERSE

UR - http://www.scopus.com/inward/record.url?scp=85055007477&partnerID=8YFLogxK

U2 - 10.1515/jiip-2018-0060

DO - 10.1515/jiip-2018-0060

M3 - Article

AN - SCOPUS:85055007477

VL - 26

SP - 835

EP - 857

JO - Journal of Inverse and Ill-Posed Problems

JF - Journal of Inverse and Ill-Posed Problems

SN - 0928-0219

IS - 6

ER -