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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
LSODA; ODEPACK
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##### References:
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