Research On Inverse Problems Of Fractional Differential Equations By Variational Iteration-Homotopy Perturbation Methods

In recent decades,with the rapid development of science and technology,the fractional order differential equations and the related inverse problems have been paid much attentions in cybernetics and intelligent robot,the system processing and signal recognition,viscoelastic mechanics,hydrogeology and environmental science,finance and other fields.In the study of inverse problems for the fractional order differential equations,it is much difficult to identify and determine time-dependent coefficients of the model using appropriate additional data.On the other hand,the variational iteration method and the homotopy perturbation method have been studied and applied for solving nonlinear differential/algebraic equations and the fractional differential equations.This paper is devoted to the research on utilization of the variational iteration method and the homotopy perturbation method for some inverse problems of determining the time-dependent coefficients in the fractional order diffusion equations.The main contents are arranged as follows:In Chapter 1,we introduce the research significance,the research situation and development trend of this thesis,and give the research motive and the main work.In Chapter 2,we mainly introduce the Riemann-Liouville and Caputo fractional order derivative and their properties,and the Laplace transformation of the fractional calculus.In Chapter 3,we utilize the variational iteration method to study inverse problems of determining the time-dependent diffusion coefficient in the time fractional diffusion equation,and the average flow velocity in the integer-fractional two-region solute transport model.By the observation at one space point in the studied region,the solutions of the inverse problem and the forward problem can be obtained exactly in some cases by the variational iteration method,and the series of the analytical solutions are also convergent to the exact solutions.In Chapter 4,we at first consider numerical inversion for the two inverse problems discussed in Chapter 3 by using the homotopy perturbation method.Also by the additional observations at one space point,numerical solutions of the time-dependent coefficients are worked out which coincide with the exact solutions.Furthermore,the homotopy perturbation method is applied to study an inverse problem of determining two time-dependent diffusion coefficient in the fractional Burgers equations.The inverse problem is solved successfully by the homotopy perturbation method and numerical solutions converges to the exact solution.In Chapter 5,we give summary and the main results of this thesis,and point out some problems related with the current study,and propose some research directions in the future.