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Mollified hyperbolic method for coefficient identification problems. (English) Zbl 0789.65090

The following parameter identification problem in connection with the one-dimensional parabolic equation is analyzed: \(u_ t = (a u_ x)_ x + f(x,t)\), \(0 < x < 1\), \(0 < t\); \(u(x,0)=g(x)\), \(0<x<1\), \(u(0,t) = \psi(t)\); \(u_ x(0,t) = \phi(t)\), \(0 < t\).
The regularization procedure used by R. E. Ewing and T. Lin [ISNM 91, 85-108 (1989; Zbl 0686.93016)] and by R. E. Ewing, T. Lin and R. Falk [Notes Rep. Math. Sci. Eng. 4, 483-497 (1987; Zbl 0667.35066)] is introduced to stabilize the identification problem. A combination of the mollification method with a space marching implementation of the hyperbolic regularization procedure is presented in a new form. An effectual numerical scheme and two stability estimates are also given. Three examples illustrate the application of the presented method.

MSC:

65Z05 Applications to the sciences
65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
35R30 Inverse problems for PDEs
35K15 Initial value problems for second-order parabolic equations
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