### Numerical Scheme based on Non-polynomial Spline Functions for ‎the System of Second Order Boundary Value Problems arising in ‎Various Engineering Applications

Document Type : Research Paper

Authors

1 Department of Mathematics, Birla Institute of Technology, Allahabad-211010 (U.P.), India‎

2 Department of Mathematics, Jaypee Institute of Information Technology, Noida-201309 (U.P.), India‎

3 Department of Mathematics, Birla Institute of Technology, Patna-800014 (Bihar), India‎

Abstract

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.

Keywords

Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1]   Chang, C.C., Gambhir, A., Humble, T.S., Sota, S., Quantum Annealing for Systems of Polynomial Equations, Scientific Reports, 9(1), 2019, 10258.
[2]   Gupta, Y., Numerical Solution of System of Boundary Value Problems using B-Spline with Free Parameter, AIP Conference Proceedings, 1802(1), 2017, 020006.
[3]   Geng, F., Cui, M., Solving a Non-linear System of Second Order Boundary Value Problems, Journal of Mathematical Analysis and Applications, 327, 2007, 1167-1181.
[4]   Dehgan, M., Saadatmandi, A., The Numerical Solution of a Nonlinear System of Second Order Boundary Value Problems Using Sinc-Collocation Method, Mathematical and Computer Modelling, 46, 2007, 1434-1441.
[5]   Gamel,E. M., Sinc-collocation Method for Solving Linear and Non-Linear System of Second-Order Boundary Value Problems, Applied Mathematics, 3(11), 2012, 1627-1633.
[6]   Lu, J. F., Variational Iteration Method for Solving a Nonlinear System of Second-Order Boundary Value Problems, Computers and Mathematics with Applications, 54, 2007, 1133-1138.
[7]   Bataineh, M., Noorani, S. M., Hashim, I., Modified Homotopy Analysis Method for solving system of second-order BVPs, Communications in Nonlinear Science and Numerical Simulation, 14(2), 2009, 430- 442.
[8]   Ogunlaran, O.M., Ademola, A.T., On the Laplace Homotopy Analysis Method for a Nonlinear System of Second-Order Boundary Value Problems, General Mathematics Notes, 26(2), 2015, 11–22.
[9]   Chaurasia, A., Srivastava, P.C., Gupta, Y., Solution of Higher Order Boundary Value Problems by Spline Methods, AIP Conference Proceedings, 1897, 2017, 020018.
[10] Dehghan, M., Lakestani, M., Numerical Solution of Non-linear System of Second-Order Boundary Value Problems using Cubic B-Spline Scaling Functions, International Journal of Computer Mathematics, 85(9), 2008, 1455–1461.
[11] Caglar, N., Caglar, H., B-spline Method for Solving Linear System of Second Order Boundary Value Problems, Computers and Mathematics with Applications, 57, 2009, 757-762.
[12] Khuri, S. A., Sayfy, A., Collocation Approach for the Numerical Solution of a Generalized System of Second-Order Boundary-Value Problems, Applied Mathematical Sciences, 3(45), 2009, 2227–2239.
[13] Heilat, A., Hamid, N., Ismail, A.I.M., Extended Cubic B‑spline Method for Solving a Linear System of Second‑Order Boundary Value Problems, Springer Plus, 5(1), 2016, 1-18.
[14] Goh, J., Majid, A.A., Ismail, A.I.M., Extended Cubic Uniform B-spline for a Class of Singular Boundary Value Problems, Science Asia, 37(1), 2011, 79–82.
[15] Fay, T. H., Coupled Spring Equations, International Journal of Mathematical Education in Science and Technology, 34(1), 2003, 65–79.
[16] Chen, S., Hu, J., Chen, L., Wang, C., Existence Results for N-Point Boundary Value Problem of Second Order Ordinary Differential Equations, Journal of Computational and Applied Mathematics, 180(2), 2005, 425–432.
[17] Cheng, X., Zhong, C., Existence of Positive Solutions for a Second-Order Ordinary Differential System, Journal of Mathematical Analysis and Applications, 312(1), 2005, 14–23.
[18] Thompson, H.B., Tisdell, C., Boundary Value Problems for Systems of Difference Equations Associated with Systems of Second-Order Ordinary Differential Equations, Applied Mathematics Letters, 15(6), 2002, 761–766.
[19] Thompson, H.B., Tisdell, C., The Non-existence of Spurious Solutions to Discrete Two-Point Boundary Value Problems, Applied Mathematics Letters, 16(1), 2003, 79-84.
[20] Chaurasia, A., Srivastava, P.C., Gupta, Y., Exponential Spline Approximation for the Solution of Third-Order Boundary Value Problems, V. E. Balas et al. (eds.), Data Management, Analytics and Innovation, Advances in Intelligent Systems and Computing 808, Springer Nature Singapore Pte Ltd. 2019.
[21] Chaurasia, A., Srivastava, P. C., Gupta, Y., Bhardwaj, A., Composite Non-polynomial Spline Solution of Boundary Value Problems in Plate Deflection Theory, International Journal for Computational Methods in Engineering Science and Mechanics, 20(5), 2019, 372-379.
[22] Albasiny, E.L., Hoskins, W.D., Cubic Spline Solutions to Two-Point Boundary Value Problems, Computer Journal, 12, 1969, 151–153.
[23] Jain, M.K., Numerical Solution of Differential Equations, Wiley Eastern, New Delhi, 1984.
[24] Gil, M.I., Invertibility Conditions for Block Matrices and Estimates for Norms of Inverse Matrices, Journal of Mathematics, 33(4), 2003, 1323-1335.