| In physics, mechanics, engineering technology and other science fields, lots of practical problems can be modelled by partial diffential equations (PDEs for short), which are well-posed under some initial and/or boundary conditions. Solving such systems is very important in applications. Finding an analytic solution of PDEs is ususlly very difficult, therefore a numerical approximation of the solution becomes very important.Since Schocnberg proposed the definition of "spline function" in 194G, the corresponding theory has rapidly developed. In the mean time, the spline function theory turns to bo a main subject in the studies of function approximation theory. In the 80's of last century, a group of mathematicians, physical scientists, geographers set up the robust foundation of wavelet .theory. Wavelets have vast applications in the numerical approximation for differential and integral equations, digital signal (image) processing and other fields. Because of the locality of wavelets in the space and frequency domain, it is a effective tool to describe and test the function's singularity. The spirits of spline and wavelets have brought huge influence on the development of numerical analysis.In this paper, we study the applications of spline functions and wavelets on numerical approximation of PDEs, respectively.1. In chapter 2, we apply a boundary -type quadrature technique to derive a type of boundary element scheme, which will be used to solve the boundary-value problems of PDEs. As the numerical method, the boundary element method (BEM for short) only needs to discretize the boundary of the domain, and this requires very simple data input and storage techniques. If the BEM is used to solve an exterior problem with the domain such that Rn\m is bounded, it is not necessary to deal with the boundary at infinity, since the corresponding fundamental solution chosen in the BEM satisfies the radiationcondition. Thus, exterior problems with unbounded domains can be handled as easily as interior problems, which means the BEM is much more suitable for problems over unbounded domains than the traditional finite element method.We use 1) to denote a bounded and closed region in R". Suppose that the boundary of (denoted by d) can be described by a system of parametric equations.We begin with the dimension reducing expansions (DRE for short) related to the second order differential operator L defined bywhere a is the collection of all functions that have continuous partial derivatives It is well-known that the adjoint operatorM of L can be defined byThe purpose of this chapter is to sutdy the numerical approximation for solutions of the following boundary-value problem:where differential operator L is defined by (1) and .In ?2.4, We derive a method for solving the boundary value problem (3) by using the DRE. The main result is:Theorem 1 Let C R" be an n-dimensional bounded closed domain with the boundary being a piecewise smooth surface or a simple closed curve with finite length when n=2. Let u = u(X) be a function in C2 that satisfies Lu ?0, and let v = v(X) be the fundamental solution of Mv = 6(X ?X0), where X0 is an arbitrarily given fixed point in . then the solution of the boundary value problem (3) at the point XQ isFor the points where dii/dn denotes the outer normal derivative of xi on the surface {X: \X\ =e}, thenand d is the solid angle of with respect to X0. Here is a smalt ball centered at X0.Next, we consider the exterior boundary value problem (3) over n with its complement being bounded. In order to obtain the unique solution of the problem, we also require for u(x) to satisfy the radiation conditionswhere v(X) is the fundamental solution of Mv(X) = 0.If u(X) satisfies the radiation conditions (6), similar to the Theorem 1, we obtain a corresponding Theorem on the exterior boundary-value problem. Then we derive an application of the two theorems to the boundary value problems of the harmonic equations and the Helmholtz's equation, respectively.In ?2.5, we di...
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