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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 $O(\tau+h^2)$.

MSC:
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
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References:
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