Variables Separating Method For Fractional Biological Population Model

This paper mainly studies the nonlinear time fractional biological population mod-el established under given environmental factors.Under certain initial boundary con-ditions,the variable separation method is used to solve the model.Then;the long time asymptotic behavior of the initial value problem of NLS equation is introduced,The first chapter introduces the history and development of biomathematics,the development of fractional calculus and its application in biological p opulation models,the definition and properties of fractional derivatives,the history of soliton theory and Riemann-Hilbert method,the topic of this paper and the main work.The second chapter first introduces the establishment process of the model in detail,and then expounds the main work of this paper.Using the variable separation method to solve the model,the original equation is decomposed into the product of linear homogeneous fractional equation and Helmholtz equation.Linear homogeneous fractional equation is obtained by the Laplace transform or the undetermined coeffi-cient method is us ed to find the series solution.Under certain initial boundary condi-tions,the Helmh-oltz equation can also be solved by the variable separation method.Finally,the solutions of the two equations are multiplied to abtain the solution of the original model.The third chapter introduces the long time asymptotic behavior of the NLS equa-tion.The NLS equation is transformed into its corresponding Lax equation,so that the nonlinear equation becomes a linear equation.Using the Lax pair of the NLS equar tion,the initial Riemann-Hilbert problem of the NLS equation is established.Then,it is transformed into the standard Riemann-Hilbert problem to solve the long-term asymptotic behavior of the NLS equation.