| Fractional partial differential equations are obtained by substituting the derivatives of the integer partial differential equations into the fractional derivative.Fractional cal-culus is a generalization of classic integer calculus,but it is not just In terms of fractions,it should be accurately any order calculus.In recent years,as the modeling of fractional differential equations has played an increasingly important role in certain natural phe-nomena and physical processes,scholars have It has conducted a lot of research and created various solving methods such as decomposition method,iterative method,ho-motopy method and integral transformation method.However,qualitative analysis of fractional differential equations rarely has systematic results.Most fractional differential equations cannot find accurate analytical solutions,and only approximate methods can be used to obtain their approximate analytical solutions,so as to facilitate theoretical analysis and engineering calculations.This paper mainly studies the Cauchy problem of a class of nonlinear partial differ-ential equations with Caputo-type time fractional derivatives,and discusses the solution.where x ∈Rn,n≥1,u(x,t)=(u1(x,t),u2(x,t),…,un(x,t))T.Dtαu(x,t)=diag(Dtα1u1,Dtα2u2,…,Dtαnun);f=(f1,f2,…,fn)T.Dtαiui represents the αi order Caputo fractional derivative of the time variable t.0<αi≤1,i=1,2,…,n.Dxu represents the first or higher order partial derivative of u with respect to the variable x.First,in the introduction,the research background and significance of this paper are given,and a brief introduction to the approximate analytical solutions of fractional nonlinear partial differential equations at home and abroad Research status.Secondly,describe the relevant knowledge used in functional analysis and fractional calculus,and use the fixed point theorem to prove the existence of solutions of such non-linear fractional partial differential equations.Then,based on the traditional Adomian decomposition method,LDGJ method,fractional power series method and variational iteration method,combined with prac-tical problems to improve,proposed a new Adomian decomposition method,LDGJ method,fractional power level Number method and Laplace transform iteration method to solve the system of nonlinear partial differential equations with Caputo type frac-tional derivative Dtαu Dxu And give approximate analytical solutions to specific prob-lems with strong application background.Finally,a summary of the various solving methods used,and further prospects for future research. |
