The Inverse Source Problem In A Degenerate Parabolic Equation

The degenerate parabolic equation is a type of problem in the equation of mathematicalphysics, compared with the traditional parabolic equation, it can be unencumbered by theboundary conditions, in other words, it is still well posed without boundary conditions insome special conditions. However, in some practical problems, the solution of originalproblem is always known, while the operator of equation, the right hand item and boundaryconditions are unknown, which constitute the inverse problem. It is more difficult toinvestigate the theory and numerical method of the inverse problem than that of the directproblem, since the inverse problem is ill-posed and nonlinear. In recent years, with thedevelopment of the financial mathematics, the hydromechanics of porous media andscattering geometrical morphology, to meet the practical need, many inverse problems ofdegenerate parabolic equations are proposed, hence have widely drawn attention of domesticand foreign scholars, which makes the inverse problem develop quickly and become a popularsubject spanning the computational mathematics, applied mathematics and system science.This paper deals with an inverse problem of identifying the source coefficient indegenerate parabolic equations. The framework and main contents are arranged as follows,(1) The research situation at home and abroad of inverse problem is introduced, andbackground, purpose and significance of the topic in this paper and the present problem in theresearch of inverse problem are summarized.(2) The maximum principle and some prior estimates of the proposed problem used todiscuss uniqueness of solution are given, and uniqueness is obtained in sense of partial order.(3) Because of the degeneracy of boundary condition of the proposed problem, finitevolume method is applied to construct finite difference scheme with good nature. The generalFourier method can not directly be used to discuss the stability of the difference scheme, sincethe diffusion coefficient is variable and the energy inequality is constructed to prove it. Theconclusion shows that the implicit finite difference scheme is unconditionally stable andexplicit finite difference scheme is conditionally stable. Finally numerical experiments areperformed to test the validity of the theory.(4) Based on the study of direct problem, iterative algorithm is designed on the basis ofLandeweber iterative algorithm to reconstruct the original item, for continuous original term itcan be reconstructed perfectly whatever the boundary conditions are or the case with noisydata. For some original items with poor nature, the proposed iterative algorithm is also stableand effective, may appear error only at some discontinuous and non-differential points.Generally speaking, the proposed iterative algorithm is stable and can construct the originalitem well when iterations and iteration parameter match appropriately.