International Journal of Differential Equations

Solving Nonlinear Fourth-Order Boundary Value Problems Using a Numerical Approach: ( m + 1 ) th-Step Block Method

Oluwaseun Adeyeye and Zurni Omar

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Nonlinear boundary value problems (BVPs) are more tedious to solve than their linear counterparts. This is observed in the extra computation required when determining the missing conditions in transforming BVPs to initial value problems. Although a number of numerical approaches are already existent in literature to solve nonlinear BVPs, this article presents a new block method with improved accuracy to solve nonlinear BVPs. A m + 1 t h -step block method is developed using a modified Taylor series approach to directly solve fourth-order nonlinear boundary value problems (BVPs) where m is the order of the differential equation under consideration. The schemes obtained were combined to simultaneously produce solution to the fourth-order nonlinear BVPs at m + 1 points iteratively. The derived block method showed improved accuracy in comparison to previously existing authors when solving the same problems. In addition, the suitability of the m + 1 t h -step block method was displayed in the solution for magnetohydrodynamic squeezing flow in porous medium.

Article information

Source
Int. J. Differ. Equ., Volume 2017 (2017), Article ID 4925914, 9 pages.

Dates
Received: 15 June 2017
Revised: 29 August 2017
Accepted: 3 October 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1510801548

Digital Object Identifier
doi:10.1155/2017/4925914

Mathematical Reviews number (MathSciNet)
MR3722869

Zentralblatt MATH identifier
06915936

Citation

Adeyeye, Oluwaseun; Omar, Zurni. Solving Nonlinear Fourth-Order Boundary Value Problems Using a Numerical Approach: $(m+1)$ th-Step Block Method. Int. J. Differ. Equ. 2017 (2017), Article ID 4925914, 9 pages. doi:10.1155/2017/4925914. https://projecteuclid.org/euclid.ijde/1510801548


Export citation

References

  • M. Abbas, A. A. Majid, A. I. Ismail, and A. Rashid, “The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems,” Applied Mathematics and Com-putation, vol. 239, pp. 74–88, 2014.MR3213640
  • L. Ahmad Soltani, E. Shivanian, and R. Ezzati, “Shooting homo-topy analysis method: a fast method to find multiple solutions of nonlinear boundary value problems arising in fluid mechanics,” Engineering Computations, vol. 34, no. 2, 2017.
  • N. Freidoonimehr, M. M. Rashidi, and S. Mahmud, “Unsteady MHD free convective flow past a permeable stretching vertical surface in a nano-fluid,” International Journal of Thermal Sciences, vol. 87, pp. 136–145, 2015.
  • Z. Omar and J. O. Kuboye, “New seven-step numerical method for direct solution of fourth order ordinary differential equations,” Journal of Mathematical and Fundamental Sciences, vol. 48, no. 2, pp. 94–105, 2016.MR3550044
  • S. J. Kayode and O. Adeyeye, “A 3-Step hybrid method for direct solution of second order initial value problems,” Australian Journal of Basic and Applied Sciences, vol. 5, no. 12, pp. 2121–2126, 2011.
  • S. J. Kayode and F. O. Obarhua, “Continuous y-function hybrid methods for direct solution of differential equations,” International Journal of Differential Equations and Applications, vol.12, no. 1, pp. 37–48, 2013.
  • W. E. Milne, Numerical Solution of Ordinary Differential Equations, Wiley, New York, NY, USA, 1953.MR0068321
  • D. Sarafyan, “Multistep methods for the numerical solution of ordinary differential equations made self-starting,” Mathematics Research Center, MRC-TSR-495, 1965.
  • J. D. Lambert, Computational Methods in Ordinary Differential Equations, Wiley, New York, NY, USA, 1973.MR0423815
  • G. Mustafa, M. Abbas, S. T. Ejaz, A. I. M. Ismail, and F. Khan, “A numerical approach based on subdivision schemes for solving non-linear fourth order boundary value problems,” Journal of Computational Analysis and Applications, vol. 23, no. 4, pp. 607–623, 2017.MR3644331
  • P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, New York, NY, USA, 1962.MR0135729
  • J. C. Butcher, Numerical Methods for Ordinary Differential Equations, Wiley, West Sussex, 2nd edition, 2008.MR2401398
  • I. Ullah, M. T. Rahim, H. Khan, and M. Qayyum, “Homotopy analysis solution for magnetohydrodynamic squeezing flow in porous medium,” Advances in Mathematical Physics, vol. 2016, Article ID 3541512, 9 pages, 2016.MR3518490 \endinput