An Algorithm for Computing Spline Function

Applied Mathematics, 2013

In this communication we have used Bickley's method for the construction of a sixth order spline function and apply it to solve the linear fifth order differential equations of the form         v y x g x y x r x   where   g x and   r x are given functions with the two different problems of different boundary conditions. The method is illustrated by applying it to solve some problems to demonstrate the application of the methods discussed.

Applied Mathematics Letters, 1999

Collocation method with sixth degree B-splines as basis functions has been developed to solve a fifth order special case boundary value problem. To get an accurate solution by the collocation method with sixth degree B-splines, the original sixth degree B-splines are redefined into a new set of basis functions which in number match with the number of collocation points. The method is tested for solving both linear and nonlinear boundary value problems. The proposed method is giving better results when compared with the methods available in literature.

Our paper dedicated to find approximate solution of second order initial value problem by seven degree lacunary spline function of type (0, 1, 6). The convergence analysis of given method has studied. Numerical illustrations have given with example for calculating absolute error between spline functions and exact solution of second order initial value problem with their derivatives.

International Journal of Engineering Mathematics, 2013

In the present work a nonpolynomial spline function is used to approximate the solution of the second order two point boundary value problems. The classes of numerical methods of second order, for a specific choice of parameters involved in nonpolynomial spline, have been developed. Numerical examples are presented to illustrate the applications of this method. The solutions of these examples are found at the nodal points with various step sizes and with various parameters (α, β). The absolute errors in each example are estimated, and the comparison of approximate values, exact values, and absolute errors of at the nodal points are shown graphically. Further, shown that nonpolynomial spline produces accurate results in comparison with the results obtained by the B-spline method and finite difference method.

International Journal of …

In this paper, a new quintic spline method developed for computing approximate solution of differential equations. It is shown that the present method is of the order three and four derivatives and gives approximations which are better. The numerical result obtained by the present method has been compared with the exact solution using C++ programming and also illustrate graphically the applicability of the new method. By getting the advantages of the mathematical building functions like pow (for power), exp (for exponential),…etc. are provided in C++ programming library, all processing steps are done efficiently and illustrated as Pseudocode model. This method enables us to approximate the solution as well as its first and third derivatives at every point of the range of integration. We proved that this new method gives better numerical results than the previous known results. In recent years, Al-Said and Noor [2, 3], Khalifa and Noor [4] and Noor and Al-Said [5, 6] have used such types of penalty function in solving a class of contact problems in elasticity in conjunction with collocation, finite difference and spline techniques. The general fourth order initial value problem considered is of the form functions, Pseudocode.

European Journal of Pure and Applied Mathematics, 2021

In this research, second order linear two-point boundary value problems are treated using new method based on hybrid cubic B-spline. The values of the free parameter,Gamma , are chosen via optimization. The value of the free parameter plays an important role in giving accurate results. Optimization of this parameter is carried out. This method is tested on four examples and a comparison with cubic B-spline, trigonometric cubic B-spline and extended cubic B-spline methods has been carried out. The examples suggest that this method produces more accurate results than the other three methods. The numerical results are presented to illustrate the efficiency of our method.

In this paper numerical solutions of general linear boundary value problem of order eight are considered. Eleventh degree spline approximations are developed following Cubic Spline Bickley's procedure and applied. Approximate numerical solutions are computed at varying step lengths. Approximate and exact solutions are compared. Also absolute errors are calculated. The results are tabulated and pictorially illustrated. Further, the results of the current method are compared with those of other popular ones. Keywords: Spline functions, eighth order boundary value problems, Eleventh degree spline, Numerical results, two point boundary value problem.

Applied and Computational Mechanics, 2020

Several applications of computational science and engineering, including population dynamics, optimal control, and physics, reduce to the study of a system of second-order boundary value problems. To achieve the improved solution of these problems, an efficient numerical method is developed by using spline functions. A non-polynomial cubic spline-based method is proposed for the first time to solve a linear system of second-order differential equations. Convergence and stability of the proposed method are also investigated. A mathematical procedure is described in detail, and several examples are solved with numerical and graphical illustrations. It is shown that our method yields improved results when compared to the results available in the literature.

BIT, 1979

A numerical method, using spline functions of degree five, for obtaining approximate solutions to initial value problems is presented. It is shown that the method is stable and the convergence is analysed. Some numerical experiments are included. Introduction. Recently we presented in [4] a method for construction of global approximations to the initial value problems in ordinary differential equations, using interpolate, piecewise polynomial functions of degree three, where we achieve convergence of order four. Now, we work on the same initial value problem with piecewise polynomial functions of degree five, belonging to C2[a, b] and with the collocation method. We prove the convergence and stability of the method achieving approximations of order six.

Journal of Applied Mathematics and Physics, 2016

In this article, we develop numerical method by constructing ninth degree spline function using extended cubic spline Bickley's method to find the approximate solution of seventh order linear boundary value problems at different step lengths. The approximate solution is compared with the solution obtained by eighth degree splines and exact solution. It has been observed that the approximate solution is an excellent agreement with exact solution. Low absolute error indicates that our numerical method is effective for solving high order linear boundary value problems.