Invariant Subspace Method And Exact Solutions Of Partial Differential Equations

The exact solutions of nonlinear evolution equations are of great significance to the study of nonlinear phenomena in practical problems,so it has been widely paid attention in many fields such as mathematics,mechanics,biology,finance,etc.Nonlinear diffusion equations with source terms are important biological and physical models,which can describe many important biophysical phenomena such as the diffusion of impurities in semiconductors and the contact inhibition between cell populations.Therefore,it is of great importance to study the exact solution of these partial differential equations.In this thesis,we mainly used the invariant subspace method to obtain the exact solution of the nonlinear diffusion equation with source term,and considered the two nonlinear diffusion equations with Riemann-Liouville time fractional derivative and classical integer derivative respectively.The key idea is to get invariant subspaces of the nonlinear partial equation to be solved,through the differential constraints defined by the linear ordinary differential equations.And the problem of solving a partial differential equation will be changed as the question of solving ordinary differential equations,thus exact solutions of nonlinear diffusion equations can be obtained as long as the corresponding fractional ordinary differential equations or ordinary differential equations with integer order are solved.Specific research contents include:Firstly,the fractional-order nonlinear diffusion equation or the integer-order nonlinear diffusion equation are classified according to the invariant subspace corresponding to the equation,so as to solve the exact solution of the equation in five-,four-,three-and two-dimension invariant subspace respectively.Secondly,the concrete functional expressions of the diffusion term and source term in the equation are obtained from the invariant conditions,and the(fractional order)ordinary differential equations with the solution coefficients relative to time are further obtained.Finally,the(fractional)ordinary differential equations are solved to obtain the exact solutions of the corresponding nonlinear diffusion equations.