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On the stability and convergence of a difference scheme for an one-dimensional parabolic inverse problem. (English) Zbl 1119.65089
The article is devoted to the following inverse parabolic problem: given $E(t)$, $0\leq t \leq T$ find $p(t)$ and $u(x,t)$, $0<x < 1$, $0\leq t \leq T$ such that $\partial u/ \partial t= \partial^2u / \partial x^2+p(t)u+\phi(x,t)$, $0<x < 1$, $0\leq t \leq T$; $u(x,0)=f(x)$, $0<x < 1$; $u(0,t)=g_0(t)$, $u(1,t)=g_1(t)$, $u(x^*,t)=E(t)$, $0\leq t \leq T$. The original nonlinear problem is transformed into a linear one, and the backward Euler scheme is applied to the latter. The convergence orders of both $u$ and $p$ are $O(\tau+h^2)$.

65M32Inverse problems (IVP of PDE, numerical methods)
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35K55Nonlinear parabolic equations
35R30Inverse problems for PDE
Full Text: DOI
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