A widely convergent method for determining the distributed parameters of the elliptical equation
The authors study identification problems in elliptic partial differential equations. More precisely, the following equation defined on the time-space domain is considered \(L(a(x),t)u(x,t) = 0\) where \(a(x_ 1,\dots,x_ p)\) is the unknown operator to be identified. Of course initial and boundary conditions are given: \(Eu(x,0) = g(x)\); \(Bu(x,t) = f(x,t)\) on the boundary; \(Au(x,t) = \varphi(x,t)\) on a part of the boundary. A nonlinear operator \(Q\) is introduced: \(Q(a,t) = Au(a;x,t) - \varphi(x,t)\) and the inverse problem is stated as the following nonlinear operator equation: (1) \(Q(a,t) = 0\). The principle of continuation homotopy is used for solving (1) associated to a Newton-like method. Numerical results are given. Reviewer's remark: For this kind of problem I suggest to use a decomposition method (Adomian's technique) allowing to obtain \(u(x,t)\) in function of \(a(x)\) explicitly.
scientific article; zbMATH DE number 519651
scientific article