Approximate Analytical Solutions For Several Lane-Emden Type Equations

There are five chapters in this thesis.We mainly introduce the background and present situation of the research in the first chapter.Some basic principles of fractional calculus and the basic principles of Adomian decomposition,homotopy analysis and homotopy perturbation method are given in the second chapterIn the third chapter,we study the initial value problem of Lane-Emden type equa-tion in the following form (?) where f(x)and g(y)are known functions of x and y,respectively.First we transform the higher-order Lane-Emden type equation into first order equations,then combine them with the Adomian decomposition method or homotopy analysis method or homotopy perturbation technique,thus we slove the question.This method can be also applied to the case of fractional differential equation.In the fourth chapter,we study the following Caputo fractional Lane-Emden type equation,which depends on the time t.(?)This kind of fractional equation is difficult to solve because the Caputo fractional deriva-tive does not satisfy the law of semigroups.In the existing literature there are just approximate numerical solutions.In this thesie,we first overcome the difficulty of the Caputo type fractional derivative not satisfying the law of semigroups,and convert the above equation into a set of equations.And then,we combine the equations with the tra-ditional homotopy analysis method or the homotopy perturbation technique,to obtain the approximate analytic solution of the fractional order equation.The idea provides a new way for the approximate analytical solution of the Lane-Emden type equation with Caputo derivativeIn the fifth chapter,we summarize the existing work and look forward to the work of future.