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.


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
Full Text: DOI


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