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The boundary-element method for the determination of a heat source dependent on one variable. (English) Zbl 1146.80007

The authors investigate from a numerical point of view two inverse problems associated to the linear 1D heat equation.
In the first one, they consider the equation \(\partial u/\partial t=\partial ^{2}u/\partial x^{2}+f(t)\), posed in \((0,l)\times (0,T)\), with the Dirichlet boundary conditions \( u(0,t)=h_{0}(t)\) and \(u(l,t)=h_{l}(t)\). The solution starts from the initial condition \(u(x,0)=u_{0}(x)\). Compatibility conditions are imposed to the data of the problem. Adding an overspecified condition \(u(x_{0},t)=\chi (t)\), which corresponds to the measure of the temperature through a thermocouple located at some \(x_{0}\in (0,l)\), the authors refer to a paper by A. I. Prilepko and V. V. Solov’ev [Differ. Equations 23, No. 10, 1230–1237 (1987; Zbl 0683.35089)] for an existence result concerning \((u,f)\in H^{2+\alpha ,1+\alpha /2}([0,l]\times [0,T])\times H^{\alpha /2}([0,T])\).
In the second inverse problem, the authors change the source and the overspecified condition respectively as \( g(x)\) and \(u(x,T)=\Psi (x)\). They prove an existence result for the couple \( (u,g)\in H^{2+\alpha ,1+\alpha /2}([0,l]\times [0,T])\times H^{\alpha }([0,l])\). The authors then give an expression of the solution of these problems in terms of Green’s function of the heat problem and of the data. This allows to introduce a boundary-element method which discretises this problem. In order to overcome the ill-conditioning difficulty of these inverse problems, the authors introduce a Tikhonov regularization as described in [S. Twomey, J. Assoc. Comput. Mach. 10, 97–101 (1963; Zbl 0125.36102)].
The last part of the paper presents examples (one for each inverse problem), where this BEM is invoked with noisy data. The authors describe the efficiency of the Tikhonov regularization within this context.

MSC:

80M15 Boundary element methods applied to problems in thermodynamics and heat transfer
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
80A23 Inverse problems in thermodynamics and heat transfer
80A20 Heat and mass transfer, heat flow (MSC2010)
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