A New Algorithm for Solving Generalized Systems of Second-Order Boundary Value Problems using Nonpolynomial Spline Techn
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RESEARCH ARTICLE
A New Algorithm for Solving Generalized Systems of SecondOrder Boundary Value Problems using Nonpolynomial Spline Technique Shahna1 • Arshad Khan1
Received: 12 April 2018 / Revised: 6 May 2020 / Accepted: 9 July 2020 Ó The National Academy of Sciences, India 2020
Abstract In this paper, a nonpolynomial spline technique is used for developing a new algorithm for solving systems of second-order boundary value problems. Convergence analysis of the algorithm has been carried out which shows that the algorithm is second- as well as fourth-order convergent. In addition to that, four numerical problems including linear, nonlinear and singularly perturbed systems are given to prove the effectiveness and applicability of the developed algorithm. The numerical results show that the developed algorithm is better than the existing methods for solving systems of second-order boundary value problems. Keywords Systems of second order Linear Nonlinear Nonpolynomial spline Convergence analysis Mathematics Subject Classification 65L10 65D07
1 Introduction We consider the systems of second-order boundary value problems (BVPs) which occur very frequently in theoretical physics, applied sciences, engineering and mathematical modeling of real-world problems. In fact, approximate solution for systems of second-order BVPs is of great importance due to their various applications in scientific & Arshad Khan [email protected] Shahna [email protected] 1
Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India
research. Numerical algorithm is becoming more useful in mathematical applications not only because of the difficulties occurred in finding exact solutions, but also because numerical techniques can be used easily in conjunction with modern high-speed digital computers. There are a large number of such systems, whose solutions cannot easily obtained by direct methods. There are various methods available in the literature to solve the systems of second-order BVPs. For example, Cheng and Zhong [1] discussed the existence of positive solutions for a secondorder ordinary differential system, Dehghan and Saadatmandi [2] developed the sinc-collocation method, Thompson and Tisdell [3] gave the systems of difference equations, and Caglar and Caglar [4] used B-spline method for solving linear system of second-order BVPs. For a nonlinear system, there are few methods available in the literature to obtain approximate solutions. El-Gamel [5] used sinc-collocation method to solve nonlinear systems and also singularly perturbed systems of second-order BVPs. Geng and Cui [6] have developed reproducing kernel space method, and Lu [7] developed the variational iteration method for obtaining the solution of a nonlinear system. Bataineh et al. [8] presented modified homotopy method for solving systems of second-order BVPs. Srivastava and Kumar [9] used quintic nonpolynomial spline for solving third-order BVPs associated with draining and coating flows. Srivastava et al. [10, 11] solved second-
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