Types Of Pseudo-parabolic Equation Definite Solution Of The Problem And The Inverse Problem

The models of many physical phenomena such as the seepage of homogeneous fluids through a fissured rock, transfer of the damp in the earth, heat exchange between different media and so on can be described by pseudoparabolic partial differential equations, so researches on direct and inverse problems of the pseudoparabolic equations are all valuable for both theory and application.In this paper, firstly we transform the pseudoparabolic equation into an abstract partial differential equation by introducing the fundamental solution K(x) of some operator, and solve two kinds of direct problems of nonlinear pseudoparabolic equations, which are the bounded solution problem of a kind of nonlinear pseudoparabolic equation with time delay and the well-posedness problem of a kind of pseudoparabolic Cauchy problem.Secondly, based on the different structure characteristics and additional conditions, we study several kinds of inverse problems of pseudoparabolic equations. One is a kind of pseudoparabolic inverse problem of identifying a constant coefficient solved by combining the formal solution of the problem and the additional condition properly. The second is the pseudoparabolic inverse problems of identifying an unknown boundary function and an unknown source term solved by using the Riemann function method to get the formal solution of the problem and then using the additional condition to transform the problem into a Volterra integral equation of the second kind. The third is a kind of backward heat flow problem of nonlinear pseudoparabolic equation solved by combining the Riemann function method and the fixed point theory properly.Lastly, we study a model of the heat flow cryptosystem base on a linear pseudoparabolic equation. The C-N difference scheme arithmetic of the encryption and decryption problems is designed. Then the numerical simulation is done through computer programming. So by analysing the simulation data we deduce many regularly conclusions on the influence of the key functions to the encryption and decryption operations, which establish the theoretical base for the application of this cryptosystem.The main work of this paper also has important reference to enrich the theory and find new research methods for high partial differential equations.