Hölder-type approximation for the spatial source term of a backward heat equation
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Hölder-type approximation for the spatial source term of a backward heat equation. / Dang, Duc Trong; Mach, Minh Nguyet; Pham, Ngoc Dinh Alain; Phan, Thanh Nam.
I: Numerical Functional Analysis and Optimization, Bind 31, Nr. 12, 2010, s. 1386-1405.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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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 -
ID: 33906477