On the solution of inverse problems for generalized oxygen consumption. (Q1771822)

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.