Some Applications Of Spline Functions In Numerical Solutions Of Differential Equations

Splines are applied widely in function approximation, numerical solutions of differential equations, computational geometry, CAGD, finite element, wavelet and so on. Some applications of spline functions in numerical solutions of differential equations are discussed in this dissertation.Chapter 1 introduces univariate splines and multivariate splines roughly.Chapter 2 gives some applications of spline functions in numerical solutions of ordinary differential equations, and summarizes and compares some existing methods, especially recent methods. This chapter discusses numerical methods for two-point boundary-value problems and a class of second-order, boundary problems with univariate cubic splines and parametric cubic splines, and introduces some applications of spline functions in singularly perturbed boundary-value problems and numerical solution of matrix differential models .In Chapter 3, two series of symmetrical B-spline in S₄² (Δ_mn⁽²⁾) are applied in numericalsolutions of partial differential equations. By using these two series of B-spline, a numerical method for solutions of Poisson equations has been presented in this paper. The corresponding quasi-interpolation operators of these two series of B-spline are constructed, which provide error analysis of the numerical solutions. And an numerical example is presented to illustrate the efficiency of the new method. Similar method may be applied in other types of partial differential equations.Chapter 4 summarizes the whole dissertation and gives some expectation for future research.