Maple And Special Functions For Boundary Value Problems Of Differential Equations

In mathematical modeling we often need to solve partial differential equations, and these PDEs could be solved first by using generalized coordinate systems such as spherical coordi-nates or cylindrical coordinates, then the traditional separation of variables method could be applied to get a generalized Fourier series solution. In general, separation of variables method leads to boundary value problems of Sturm-Liouville type. These boundary value problems define some unbounded self-adjoint operators in certain Hilbert spaces. The eigenfunctions of these differential operators are identified as some special functions, for example, Legendre polynomial, Bessel function etc, and the corresponding eigenvalues are zeros of some special functions, for example, zeros of certain Bessel functions. Maple, with its powerful symbolic computation support and impressive computer graphics capability, has become one of three most popular softwares in mathematics. In this work, we present a introduction on how to use Maple to study the properties of special functions, solve boundary value problems of partial differential equations.This thesis is organized into five chapters. In chapter one we present a short introduction to the history of the series solutions to partial differential equations and list some background knowledge necessary for this thesis; in chapter two we introduce some essential commands of Maple for elementary mathematics, calculus, linear algebra, differential equations and other branches of mathematics; in chapter three we use the Function Advisor command of Maple to search for properties of special functions, apply differentiate command diff to derive differential recurrence relations for some special functions; in chapter four we demonstrate how to take advantage of Maple’s powerful capabilities. We mainly focus on how to use Maple to carry out the method of separation of variables in spherical coordinates and cylindrical coordinates, we first apply the pdsolve to separate variables of partial differential equations directly, then we use Maple to solve the obtained Sturm-Liouville systems, finally we construct the solutions of the partial differential equations in terms of special functions and plot approximate solutions. In our final chapter, chapter five we summarize this thesis and pose some questions for future research.