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.


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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