One step hybrid block method for solving nonlinear second order Dirichlet value problems of ordinary differential equations directly

Document Type : Research Paper

Author

Department of Mathematics, College of Art and Sciences-Tabarjal, Jouf University, Saudi Arabia

10.22075/ijnaa.2023.24666.2794

Abstract

The aim of this article is to approximate the solution of nonlinear second-order Dirichlet boundary value problems of ordinary differential equations directly using the hybrid block method. To derive this method, we first transform the boundary value problem to its corresponding second-order initial value problem via the nonlinear shooting method. Then, a direct one-step hybrid block with three off-step points is derived using a collocation and interpolation approach. The numerical results clearly show that the developed method is able to generate good results when it is compared with the existing method in terms of error.

Keywords

[1] R. Abdelrahim, and Z. Omar Direct solution of second-order ordinary differential equation using a single-step hybrid block method of order five, Math. Comput. Appl. 21 (2016), no. 2, 12.
[2] R. Abdelrahim and Z.Omar Solving third order ordinary differential equations using hybrid block method of order five, Int. J. Appl. Engin. Res. 10 (2016), no. 24, 44307–44310.
[3] L.R. Burden Numerical Analysis, Brooks/Cole Cengage Learning, 2011.
[4] U. Erdogan and T.A. Ozis, Smart nonstandard finite difference scheme for second order nonlinear boundary value problems, J. Comput. Phys. 230 (2011), no. 17, 6464–6474.
[5] S. N. Ha, A nonlinear shooting method for two point boundary value problems, Comput. Math. Appl.  42 (2001), 1411–1420.
[6] M.D. Jafri, M. Suleiman, Z.A. Majid, and Z.B. Ibrahim Solving directly two point boundary value problems using direct multistep method, Sains Malay. 38 (2009), no. 5, 723–728.
[7] S.P. Phang, A.Z. Majid, K.I. Othman, F. Ismail, and M. Suleman, New algorithm of two-point block method for solving boundary value problem with Dirichlet and Neumann boundary conditions, Math. Prob. Engin. 2013 (2013), 1–10.
[8] S.P. Phang, A.Z. Majid, and M. Suleman, Solving nonlinear two point boundary value problem using two step direct method, J. Qual. Measur. Anal. 7 (2011), no. 1, 129–140.
[9] S.M. Roberts, J.S. Shipman, and M. Suleman On the closed form solution of Troesch’s problem, J. Comput. Phys. 21 (1976), 291–304..
[10] J. Yun, A note on three-step iterative method for nonlinear equations, Appl. Math. Comput. 202 (2008), 401–405.

Articles in Press, Corrected Proof
Available Online from 29 January 2024
  • Receive Date: 26 September 2021
  • Revise Date: 30 September 2021
  • Accept Date: 07 December 2023