 Recursive Form of B-spline Based Collocation Solution to Homogeneous Differential Equations with Neumann's Boundary Value Problems

# Y. Rajashekhar Reddy1

1Department of Mathematics, JNT University College of Engineering, Jagitial, Karimnagar, Telangana, India.

Abstract: The area of differential equations is a very broad field of study. The versatility of differential equations allows the area to be applied to a variety of topics from physics to population growth to the stock market. They are a useful tool for modeling and studying naturally occurring phenomena such as determining when beams may break as well as predicting future outcomes such as the spread of disease or the changes in populations of different species over time. Anytime an unknown phenomenon is changing with respect to time or space, a differential equation is involved Such differential equations have been solved by using many numerical methods. One numerical method which is developed by using simplified form of Recursive B-spline function in collocation method as basis function. The present method is used to solve second order and third order homogeneous differential equations with Neumann's boundary conditions. Solutions of Numerical examples for non -uniform length of the points show the efficiency of the method and easiness. Stability of B-spline based collocation method and accuracy of numerical solution is constantly improved by decreasing the nodal space.
Keywords: Neumann's boundary conditions, Collocation method, Homogeneous differential equations, B-splines.

Cite this article as: Y. Rajashekhar Reddy, Recursive Form of B-spline Based Collocation Solution to Homogeneous Differential Equations with Neumann's Boundary Value Problems, Int. J. Math. And Appl., vol. 10, no. 1, 2022, pp. 119-124.

References
1. T. J. R. Hughes, J. A. Cottrell and Y. Bazilevs, Isgeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engg., 194(39-41)(2005), 4135-4195.
2. David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, 2nd Ed., Tata McGraw-Hill Edition, New Delh.
3. C. de Boor and K. Hollig, B-splines from parallelepipeds, J. Analyse Math., 42(1982), 99-115.
4. R. K. Pandey and Arvind K. Singh, On the convergence of a finite difference method for a class of singular boundary value problems arising in physiology, J. Comput. Appl. Math., 166(2004), 553-564.
5. F. Z. Geng and M. G. Cui, Solving Singular Nonlinear Second-Order Periodic Boundary Value Problems in the Reproducing Kernel Space, Applied Mathematics and Computation, 192(2007), 389-398.
6. Z. Y. Li, Y. L. Wang, F. G. Tan, X. H. Wan and T. F. Nie, The Solution of a Class of Singularly Perturbed Two-Point Boundary Value Problems by the Iterative Reproducing Kernel Method, Abstract and Applied Analysis, (2012), 1-7.
7. A. Mohsen and M. El-Gamel, On the Galerkin and Collocation Methods for Two Point Boundary Value Problems Using Sinc Bases, Computers and Mathematics with Allications, 56(2008), 930-941.
8. Abdalkaleg Hamad, M. Tadi and Miloje Radenkovic, A Numerical Method for Singular Boundary-Value Problems, Journal of Applied Mathematics and Physics, 2(2014), 882-887.
9. S. Joan Goh, Ahmad Abd. Majid and Ahmad Izani Md. Ismail, Extended cubic uniform B-spline for a class of singular boundary value problems, Science Asia, 37(2011), 79-82.
10. I. J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math., 4(1946), 45-99.
11. H. B. Currya and I. J. Schoenberg, On Polya frequency functions IV: The fundamental spline functions and their limits, J. Anal. Math., 17(1966), 71-107.
12. Carl De Boor, On Calculating with B-plines, Journal of Approximation Theory, 6(1972).
13. Hikmet Caglar, Nazan Caglar and Khaled Elfaituri, B-spline interpolation compared with finite difference, finite element and finite volume methods which applied to two-point boundary value problems, Applied Mathematics and Computation, 175(2006), 72-79.
14. Y. Rajashekhar Reddy, Numerical Solution to Euler--Cauchy Equation with Neumann Boundary Conditions Using B-Spline Collocation Method, Annals of Pure and Applied Mathematics, 11(2)(2016), 1-6.

Back