Study On Numerical Solutions Of A Class Of Inverse Problems Of Ordinary Differential Equations

The inverse problem of ordinary differential equations widely exists in the physical sciences,biosciences,chemical sciences,engineering and many other fields It's solution has always been the focus of theoretical and applied research.This paper takes the inverse problems existing in the field of ordinary differential equations as the main research background,and takes the thinking methods of optimization theory as the research basis.From a general meaning,a systematic analysis and research on a class of inverse problems of ordinary differential equations are carried out.Three feasible and effective numerical methods,namely,the improved Gauss-Newton method and Levenberg-Marquardt method,optimal perturbation algorithm,variational adjoint method are given.The main researches as follows:(1)Two typical iterative methods,i.e.Gauss-Newton method and Levenberg-Marquardt method,are combined with the sensitivity equation respectively.By using the sensitivity calculation,the iterative equation and normal equation are obtained more accurately.Therefore,we get the improved Gauss-Newton method and Levenberg-Marquardt method.(2)By using operator identification perturbation method,regularization theory and lineari-zation technique,the optimal perturbation algorithm for solving the parameter inversion of ordinary differential equations is established.It is applied to the mathematical model of many practical problems and compared with other methods.(3)Based on the adjoint operator theory and the Lagrange multiplier method,the adjoint equation and functional gradient corresponding to ordinary differential equation are derived.Then combined with the Broyden class algorithm in optimization method,the variational adjoint method for the inversion of parameters and unknown function about ordinary differential equation is given.(4)The viability and validity of the proposed method are verified by the numerical simulation on some specific examples.