The Study Of The Exact And Numerical Solutions For Several Types Of Nonlinear Partial Differential Equations

Nonlinear differential equations play an important role in understanding natural phenomena and objective laws.Therefore,finding the solutions of these nonlinear differential equations is naturally regarded as an important means of researching these phenomena and laws.In this paper,we mainly study the exact or numerical solutions for the(2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation and the time fractional extended coupled KdV equations by some validity methods.In the first chapter,we mainly explain research background and meaning,then introduce the research contents and methods of this article.In the second chapter,based on the Hirota bilinear equation of the(2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation.We first take the per-turbation term as a function of a symmetric matrix.So,it's perfectly natural to get the general single Lump solutions and analyze its propagation path,amplitude and posi-tion at any time.Next,obtaining the Lumpoff solutions through adding an exponential function to the original perturbation term.By the rigorous analysis,we find it has the same movement track as the general single Lump solutions.Finally,according to the general lump solutions,we are also consider a particular rogue wave by introducing hyperbolic cosine function,and research its predictability which include the time of the rogue wave appearance,position at time,propagation path and the maximum value of wave height.In the third chapter,we construct complexiton solutions and resonant multiple wave solutions of the(2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov e-quation by its Hirota bilinear equation.Using pairs of conjugate wave variables in 2N-soliton solutions,we obtained a series of N-complexiton solutions.Then,consid-ering a linear superposition principle in real and complex field of exponential traveling waves to the bilinear equation,a sequence of resonant multiple wave solutions in real and complex field are presented.Furthermore,according to the two bases in the vec-tor space of solutions,we present the mixed type function solutions like Complexiton solutions.In the fourth chapter.To start with,we acquire the Lie point symmetries of the time fractional extended coupled KdV equations and similarity transformations.On the basis of the similarity transformations,we derive the ordinary differential equations with fractional order.Next,the identified ordinary differential equations are solved by the power series expansion method and discuss its convergence.Then utilizing the new conservation theorem and the Norther operators to construct Nonlinear self-adjointness and conservation laws.Lastly,in order to construct the numerical solutions to the time fractional extended coupled KdV equations,the residual power series method is employed.In the last chapter,the conclusions and prospects of our paper are introduced.