Research On Some Problems Of Analytic Approximation Methods

Many phenomena in the real world can be described by nonlinear differential equations.Solving these equations has become the key issues for many researchers.A lot of physical prob-lems dealt with by engineers,physicists and applied mathematicians exhibit certain essential features which preclude exact analytic solutions.The development of science and technolo-gy and the appearance of symbolic computation software promote the development of analytic approximation methods for nonlinear differential equations.The homotopy analysis method(HAM)and the Adomian decomposition method(ADM)are two popular methods among them.In this thesis,some significant theoretical improvements in these methods and nontrivial appli-cations of these improvements have been achieved.More specifically,we have accomplished the following four projects.1.A new hybrid analytic approach is proposed for solving nonlinear initial value problems.It is based on the HAM and the Laplace transform method.First,we transform an initial value problem into a new problem with initial point at zero,and the standard HAM is used to trans-form the new nonlinear differential equation into a system of linear differential equations.Then the Laplace transform and the inverse Laplace transform are applied to solve the correspond-ing linear initial value problems.The analytic solutions to these linear initial value problems are employed to form a convergent series solution to the given problem.Compared with the standard HAM,the new method is more powerful in solving higher order deformation equa-tions,as demonstrated by some nontrivial examples.Therefore,it can be applied to solve more complicated nonlinear real world problems.2.One of the outstanding features of the HAM is the convergence-control parameter which provides us with a convenient way to adjust and control the convergence region and rate of the resulting series solutions.However,from a rigorous mathematical point of view,how can the convergence-control parameter achieve this goal?We obtain complete theoretical results for higher order linear differential equations.In other words,we present a rigorous convergence proof for the series solution given by the HAM in a certain interval in terms of the convergence-control parameter,and obtain an upper bound for the absolute error of the approximation in that interval.Moreover,we also give an approach to determining the valid region of the convergence-control parameter that ensures convergence of the series solution.3.Based on the HAM,higher order parametric linear boundary value problems are solved.By establishing an explicit formula for the resulting series solution and the relationship between the large parameter and the convergence-control parameter,we have gained more insight into the solution structure of the given problem.For large parameter values,accurate approxima-tions can be obtained by properly choosing the values of the convergence-control parameter.Compared with other analytic methods,our method is more powerful in solving higher order linear boundary value problems with large parameters.4.Fractional partial differential equations,as generalizations of classical partial differential equations,have been increasingly used in different fields of science.Compared with classical partial differential equations,fractional partial differential equations are better tools for model-ing real world problems.A new approach is proposed for solving nonlinear fractional partial differential equations.The key idea of the new approach is to introduce two parameters to the traditional Adomian decomposition method,called the two-parameter ADM.It has been shown that the two-parameter ADM approximations are more accurate than the traditional AD-M approximations.To demonstrate the applicability and effectiveness of the new approach,two nonlinear fractional partial differential equations are solved.