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

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  • Duc Trong Dang
  • Minh Nguyet Mach
  • Ngoc Dinh Alain Pham
  • Thanh Nam Phan
We consider the problem of determining a pair of functions $(u,f)$ satisfying the two-dimensional backward heat equation
\bqq
u_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),
\eqq
together 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.
OriginalsprogEngelsk
TidsskriftNumerical Functional Analysis and Optimization
Vol/bind31
Udgave nummer12
Sider (fra-til)1386-1405
Antal sider20
ISSN0163-0563
DOI
StatusUdgivet - 2010

ID: 33906477