||In this thesis, we demonstrate the capability of power series, combined with numerical methods, to solve boundary value problems and Sturm-Liouville eigenvalue problems of ordinary differential equations. This kind of schemes is usually called the numerical-symbolic, numerical-analytic or semi-analytic method. |
In the first chapter, we develop an adaptive algorithm, which automatically decides the terms of power series to reach desired accuracy. The expansion point of power series can be chosen freely. It is also possible to combine several power series piecewisely. We test it on several models, including the second and higher order linear or nonlinear differential equations. For nonlinear problems, the same procedure works similarly to linear problems. The only differences are the nonlinear recurrence of the coefficients and a nonlinear equation, instead of linear, to be solved.
In the second chapter, we use our semi-analytic method to solve singularly perturbed problems. These problems arise frequently in fluid mechanics and other branches of applied mathematics. Due to the existence of boundary or interior layers, its solution is very steep at certain point. So the terms of series need to be large in order to reach the desired accuracy. To improve its efficiency, we have a strategy to select only a few required basis from the whole polynomial family. Our method is shown to be a parameter diminishing method.
A specific type of boundary value problem, called the Sturm-Liouville eigenvalue problem, is very important in science and engineering. They can also be solved by our semi-analytic method. This is our focus in the third chapter. Our adaptive method works very well to compute its eigenvalues and eigenfunctions with desired accuracy. The numerical results are very satisfactory.