Two Nonpolynomial Splines And Their Applications In Numerically Solving Differential Equations

The traditional spline functions are piecewise polynomials with some glob-ally smoothness, generally they are piecewise polynomial functions with low de-grees. They have been widely applied in many fields because of their excellentproperties such as simplicity in computation, stability and good smoothness. Inthis thesis, we are going to study non-polynomial spline functions, which are ex-tensions of the traditional spline functions.Firstly, two kinds of non-polynomial spline spaces have been studied, whichare spaces of exponential spline functions and hyperbolic trigonometric splinefunctions. Their dimensions are given and bases with local supports are con-structed.Further, various non-polynomial spline relationships on the nodal points arederived, including the relation between function values and values of the secondorder derivatives of non-polynomial splines, the relation between function valuesand values of the first order derivatives of non-polynomial splines, and the relationbetween function values and values of the first and second order derivatives ofnon-polynomial splines.Then based on the two kinds of non-polynomial splines, we proposed sev-eral numerical methods to solve the boundary value problems of second orderordinary di?erential equations, including finite di?erence method based on expo-nential spline functions, collocation method based on exponential spline functions,finite di?erence method based on hyperbolic trigonometric spline functions andcollocation method based on hyperbolic trigonometric spline functions.In addition, based on the non-polynomial spline relationships on the nodalpoints, we proposed a fast Hermite interpolation algorithm with the time complex-ity being O(n).Since numerical solutions obtained by using nonuniform partitions are oftenmore accurate than those obtained by using uniform partitions, several methods toconstruct nonuniform partitions are given in Chapter 5.