Certain Development Equation Pros And Cons Of The Problem Of Numerical Methods And Applications

In this thesis, we mainly discuss two types of evolution equations.The first type studied in this paper is pseudo-parabolic equation. It has application models in many technical fields, especially in an unconventional cryptosystem, the Heat Flow Cryptosystem.In chapter 1, a difference scheme of some two-dimensional linear pseudo-parabolic equation is concerned. The mixed partial differential terms of time variables and space variables in the pseudo-parabolic equation make the discussion of the difference scheme more difficult. In this chapter, a two-dimensional implicit difference scheme of the linear IBVP is designed, and the stability of the homogeneous scheme is proved by the method of separation of variable. At last, some numerical examples are presented. And the results are consistent with the theoretical result.It is important that the security of the Heat Flow System dictates the introduction of nonlinear model. In chapter 2, the difference scheme of two-dimensional semi-linear model is discussed. Since many methods used in the stability analysis are not suitable for nonlinear model, we should use special techniques. In this chapter, the existence and uniqueness of the solution to the scheme, the convergence and stability to the scheme are proved by using discrete functional analysis theory, the apriority estimate method and the Brouwer fixed-point theory. Moreover, numerical simulations are realized by the scheme. The pseudo-parabolic equation, which includes two space variables, has inherent superiority for encrypting higher dimensional information carrier, such as image, etc.In order to accelerate the encryption and decryption processing, a predictor-corrector difference scheme is designed in chapter 3. In every iteration, we only need solving two linear algebraic equation systems instead of solving nonlinear equation systems. So it improves the calculating efficiency. By using the scheme, the encryption and decryption tests of the continuous signal and the Chinese text are done. This method provides an effective way for practical application.The crypto-analysis of the Heat Flow Cryptosystem comes down to the parameter identification of the pseudo-parabolic equation. But it is difficult to do. So we begin with some inverse problem of the parabolic equation. In chapter 4, the identification of an unknown source term in a heat transfer equation is considered. Then by defining an object functional in sense of Tikhonov, the problem is transformed into an optimal control problem. It is solved by Fourier transform and its inverse transform in the frequency space, and it simplified the calculation process. In the end, some numerical examples are given to show the method is effective.