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On the solution of inverse problems for generalized oxygen consumption. (English) Zbl 1059.65086
The direct problem consists in the determination of functions \(u(x,t)\) and \(s(t)\) satisfying for positive \(t>0\) the equation \( u_t = \partial _x \big ( a(u) u_x + b(x)u \big ) - F(u) \,,\, x\in (0,s(t)) \,, \) the interface boundary conditions \(u(s(t),t) = u_x(s(t),t) = 0\) and initial-boundary conditions \( -\big ( a(u) u_x + b(x)u \big )_{x=0} = q(t), \;u(x,0) = u_0(x), \;s(0) = s_0. \) Investigating some inverse aspects of the above problem the authors try to reconstruct the data \(a(u)\) or \(F(u)\) from measurements of \(u(0,t)\) or \(s(t)\), and \(q(t)\) from \(u(x,T)\), \(T\) being a fixed time. The data are supposed to depend on a finite number of parameters, which naturally appear in the respective objective functionals. The problem is reduced to \(x\in (0,1)\) and discretized in this variable. In dealing with the obtained system of ODE the LSODA solver is used.

MSC:
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35R35 Free boundary problems for PDEs
49M15 Newton-type methods
49N45 Inverse problems in optimal control
35R30 Inverse problems for PDEs
35K55 Nonlinear parabolic equations
Software:
LSODA; ODEPACK
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References:
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