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Parameter determination in a partial differential equation from the overspecified data. (English) Zbl 1080.35174
Summary: Several schemes are presented for computing the unknown coefficient \(p(t)\) in the quasilinear equation \(u_t=u_{xx}+ p(t)u+\varphi\), in \(R\times (0,T]\), \(u(x,0)=f(x)\), \(x\in R=[0,1]\), \(u\) is known on the boundary of \(R\) and subject to the integral overspecification over the spatial domain \(\int^1+0k(x)u (x,t)dx=E(t)\), \(0\leq t\leq T\) or the overspecification at a point in the spatial domain \(u(x_0,t)=E(t)\), \(0\leq t\leq T\), where \(E(t)\) is known and \(x_0\) is a given point of \(R\). These numerical procedures are developed for identifying the unknown control parameter which produces, at any given time, a desired energy distribution in the spatial domain, or a desired temperature distribution at a given point in the spatial domain. Several finite-difference techniques are used to determine the solution. The accuracy and stability of the methods are discussed and compared. Numerical illustrations are given.

MSC:
35R30 Inverse problems for PDEs
35K55 Nonlinear parabolic equations
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
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