Approximate Analytical Solutions For Solving Linear And Nonlinear Second-order Initial And Boundary Value Problems

Second-order initial and boundary value problems occur in a wide variety of problems inscienti?c and engineering ?elds. It is well known that analytical solutions to these problemscannot be obtained generally except for very simple cases. Fortunately several good methodshave been proposed to give approximate analytical solutions to these problems in the lastdecade such as the homotopy analysis method by Liao and homotopy perturbation methodby He.When we use homotopy perturbation method to solve these problems analytically, thekey step is the homotopy construction. However, it is worth to note that, even for a sameproblem, the homotopy can be freely constructed in many ways and the initial approximatesolution can also be freely selected. As we will see, homotopies constructed in some way maybe impossible to conduct real computation or the computational cost is in fact quite expensiveeven if they are theoretically ?exible. Therefore it is signi?cant to determine or create anoptimal homotopy in the sense that whose logical structure is simple and computational costis cheap by comparing and analyzing various possible construction ways.In this thesis, for the two-point perturbed boundary value problem with Dirichlet andNeumann boundary conditions, nonlinear two-point boundary value problems and systemsof second-order nonlinear initial value problems, various possible homotopy constructionways are discussed and compared, then some e?cient homotopies for these problems areconstructed and so that accurate analytical solutions are given. Numerical examples aretested to illustrate the e?ciency of our construction methods of homotopies.