A Source Inversion Model And Numerical Algorithms For A Parabolic Equation On The Cylinder Domain

This paper focuses on a class of computable modeling and parameter inversion for the mesoscale behavior of iron-based catalysts prepared in a cylindrical reactor.The purpose of this study is to reveal the underlying mechanism of mesoscale behavior,establish a kind of computable mathematical model for a parabolic equation with source term,put forward a complete definite problem,and determine the source term and its parameters of nucleation rate and crystal growth rate by measurements.Source term inversion and parameter deter-mination is a kind of inverse problems,which is full of challenges,because of its ill-posedness,nonlinearity and high dimension.In the first chapter,the background and significance of a parabolic equation source inversion problem from the catalyst preparation process are described,and the related re-search results on source inversion problems for parabolic equation axe reviewed.The inverse problem discussed in this paper is divided into a source term inversion problem of parabolic equation and a parameter inversion problem of integral equationIn the second chapter,a source term inversion problem of parabolic equation on cylinder domain is studied.According to the axial symmetry of the cylinder domain,the solution of direct problem is presented by variable separation method.By means of the finite difference method,a numerical algorithm of the direct problem for high dimensional parabolic equations is established.In the light of the adjoint method,a cost function was established and its the Frechet derivative are derived.By this way,a numerical algorithm of the source term inversion problem is established.In the third chapter,the first type of Volterra integral equation on the time cone is solved based on the source term result in the second chapter,and the inverse problem of nucleation rate and crystal growth rate simultaneously was studied.A reasonable prior information assumption is proposed by combining with homogeneous nucleation theory.Under this assumption,the uniqueness of Volterra integral equation is obtained with respect to the simultaneous inversion of nucleation rate and crystal growth rate.Due to the instability of numerical calculating for the integral equation,a regularization method of the inverse problem was proposed and the numerical algorithm is constructed.The numerical results verifies the effectiveness of the numerical algorithm In the fourth chapter,the solution methodology for the source inversion problem of parabolic equation and the Volterra integral equation are summarized.Finally,the forth-coming researches are proposed.