| Splines are applied widely in function approximation, computational geometry, CAGD, finite element, wavelet, numerical solutions of differential equations and so on. Most of diff -erential equations arise from the practical engineering problems whose analytical solutions are difficult to obtain. therefore, the research of numerical solution of differential equations is very important and necessary. In this paper, we study the numerical solution methods bas -ed on various kind of spline functions for initial and boundary value problems of ordinary,partial differential equations which have a practical application background,give out some kinds of numerical methods of boundary problem. we analyse the precision and astringency of the numerical solution methods.Chapter 1 : Introduction Introduce various types of spline functions and the investigative ba -ckground of differential equations initial and boundary value problems of differential equa -tions.Chapter 2 : Discusses the application of the various types of splines to solve ordinary diff errential equations , sum up and compare the forepassed,recent existent numerical method, based on the results , the article do the work as follow : No1,the numerical method of a kind of second order singular-perturbed boundary value problems,and analyse the error and precision of the numerical solution, using the existent boundary equations,the article give out the numerical example.No2: The article solve a kind of fourth-order boundary problem, choosing different parameter value,can get different error precision, take the mid knots as the mesh point,in order to avoid the error accumulated at knots, at the same time , compute the numerical examples of the other articles,and compared with the other results. The article introduce numerical solution of matrix diferential model.Chapter 3: Discusses the application of the non- polynomial splines,B-splines to solve par -tial differential equations.In this chapter based on the anterior results ,we use the semidis -cretization method,and disperse time variable using finite difference method, disperse space variable using non- polynomial splines , take the knots as the mesh points. construct the new boundary equations with multistep method ,give out the numerical solution of a kind of fourth-order parabolic partial differential equations,discusses the error estimate and astringency analyse of the methods ,numerical examples are presented to illustrate the efficiency of the new methods. The article also introduce numerical method of a kind of hyperbolic partial differential equation using B-spline 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. |
