Numerical Methods Applied To Integer And Fractional Differential Equations

The main purpose of the research is to use some effective methods of mathematics and physics to solve the differential equations of integer order and fractional order,and to restudy the fractional differential transformation method in the sense of local fractional derivatives,and to get some useful propositions and theorems.The details are as follows:First part.We mainly present the background of this study and describe some typical mathematical physics methods.Second part.Based on the generalized Hirota bilinear derivatives and linear su-perposition principle.The real number domain,the multi-solitary wave solutions of generalized(3+1)-dimensional KP equations are studied.The solitons and complex so-lutions of the(3+1)-dimensional BLMP equations are studied in the complex number domain.Under appropriate parameters,they exhibit a certain degree of resonance and provide a means for us to better understand and study the ocean waves.Third part.In the sense of Caputo and Riemann-Liouville fractional derivatives,we use the(N+1)-dimensional fractional reduced differential transformation method to obtain the numerical solutions of the two-coupled systems of the time fractional WBK-like equations,which are used to describe shallow water waves.At the same time,using the nonlinear self-adjoint property,we obtained the conservation law of the two-coupled systems of the time fractional WBK-like equation.Fourth part.In the sense of local fractional derivatives,we reexamine the(N+1)-dimensional local fractional reduced differential transformation method.As a result,we obtained new propositions and theorems.At the same time their proofs are also presented.Finally,an example is used to illustrate the effectiveness of this method.Fifth part.Some important contents of this paper and the prospects for future work were disscussed.