TY - JOUR

T1 - Hölder-type approximation for the spatial source term of a backward heat equation

AU - Dang, Duc Trong

AU - Mach, Minh Nguyet

AU - Pham, Ngoc Dinh Alain

AU - Phan, Thanh Nam

PY - 2010

Y1 - 2010

N2 - We consider the problem of determining a pair of functions $(u,f)$ satisfying the two-dimensional backward heat equation \bqqu_t -\Delta u &=&\varphi(t)f (x,y), ~~t\in (0,T), (x,y)\in (0,1)\times (0,1),\hfill\\u(x,y,T)&=&g(x,y),\eqqtogether with the homogeneous boundary conditions, where the function $\varphi$ and the final temperature $g(x,y)$ are given approximately. The problem is severely ill-posed. Using an interpolation method and the truncated Fourier series, we construct a regularized solution for the source term $f(x,y)$. Our approximation gives the H\"older-type error estimates not only in $L^2$ but also in $H^1$. Some numerical experiments are given.

AB - We consider the problem of determining a pair of functions $(u,f)$ satisfying the two-dimensional backward heat equation \bqqu_t -\Delta u &=&\varphi(t)f (x,y), ~~t\in (0,T), (x,y)\in (0,1)\times (0,1),\hfill\\u(x,y,T)&=&g(x,y),\eqqtogether with the homogeneous boundary conditions, where the function $\varphi$ and the final temperature $g(x,y)$ are given approximately. The problem is severely ill-posed. Using an interpolation method and the truncated Fourier series, we construct a regularized solution for the source term $f(x,y)$. Our approximation gives the H\"older-type error estimates not only in $L^2$ but also in $H^1$. Some numerical experiments are given.

U2 - 10.1080/01630563.2010.528568

DO - 10.1080/01630563.2010.528568

M3 - Journal article

VL - 31

SP - 1386

EP - 1405

JO - Numerical Functional Analysis and Optimization

JF - Numerical Functional Analysis and Optimization

SN - 0163-0563

IS - 12

ER -