偏微分方程 PARTIAL DIFFIERENTIAL EQUATION (PDE).PPT

Orthogonal Function Expansion 正交函數展開 Introduction of the Eigenfunction Expansion Abstract Space Function Sapce Linear Operator and Orthogonal Function The Sturm-Liouville Boundary Value Problem A differential equation defined on the interval ???????? having the form of and the boundary conditions ?????????????????????????????????????? is called as Sturm-Liouville boundary value problem or Sturm-Liouville system, where , ??????? ???? ; the weighting function r(x)0 are given functions; a1?, a2 , b1?, b2 ? are given constants; and the eigenvalue is an unspecified parameter. The Regular Sturm-Liouville Equation It is a special kind of boundary value problem which consists of a second-order homogeneous linear differential equation and linear homogeneous boundary conditions of the form where the p, q and r are real and continuous functions such that p has a continuous derivative, and p(x) 0, r(x) 0 for all x on a real interval a ? x ? b; and ? is a parameter independent of x. L is the linear homogeneous differential operator defined by L(y) = [p(x)y′]′+q(x)y.And two supplementary boundary conditions where A1 , A2 , B1 and B2 are real constants such that A1 and A2 not both zero and B1 and B2 are not both zero. A1y(a)+A2y′(a) = 0 B1y(b)+B2y′(b) = 0 . Definition 1.1 : Consider the Sturm-Liouville problem consisting of the differ entail equation and supplementary conditions. The value of the parameter in for which there exists nontrivial solution of the problem is called the eigenvalue of the problem. The corresponding nontrivial solution is called the eigenfunction of the problem. The Sturm-Liouville problem is also called an eigenvalue problem. The Nonhomogeneous Sturm-Liouville Problems And as in regular Sturm-Liouville problems we assume that p, p?, q, and r are continuous on a ? x ? b and p(x) 0, r(x) 0 there.We solve the problem by making use of the eigenfunctions of the corresponding homogene