Abstract

A direct explicit Runge-Kutta type (RKT) method via shooting technique to approximate analytical solutions to the third-order two-point boundary value problems (BVPs) with boundary condition type I and II are proposed. In this paper first, a three-stage fourth-order direct explicit Runge-Kutta type method denoted as RKT3s4 is constructed. A new algorithm of shooting technique for solving two-point BVPs for third-order ordinary differential equations (ODEs) is presented.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 32
Issue: 2
Year: 2019

DOI: 10.12732/ijam.v32i2.1

Download Section

Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

1. [1] R.L. Burden, J.D. Faires, Numerical Analysis, Brooks/Cole, Cengage Learning, Boston (2011).
2. [2] T.S. Mohamed, N. Senu, Z.B. Ibrahim and N.M.A. Nik Long, Efficient two-derivative Runge-Kutta-Nystr¨om methods for solving general secondorder ordinary differential equations, Discrete Dynamics in Nature and Society, 2018 (2018), 10 pages.
3. [3] N. Ghawadri, N. Senu, F. Ismail and Z.B. Ibrahim, Exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type methods for solving y′′′ = f(x, y, y′), J. of Applied Mathematics, 2018 (2018), 19 pages.
4. [4] T.Y. Na, Computational Methods in Engineering Boundary Value Problems, Academic Press, New York (1980).
5. [5] U.M. Ascher, R.M.M. Mattheij and R.D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, Philadelphia (1994).
6. [6] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York (1962).
7. [7] C. Kanittha and A. Dhamacharoen, Solving boundary value problems of ordinary differential equations with non-separated boundary conditions, Applied Mathematics and Computation, 217, No 24 (2011), 10355-10360. [8] A. Dhamacharoen and K. Chompuvised, An efficient method for solving multipoint equation boundary value problems, World Academy of Science, Engineering and Technology, 75 (2013), 61-65.
8. [9] I.A. Tirmizi and E.H. Twizell, Higher-order finite-difference methods for nonlinear second-order two-point boundary-value problems, Applied Mathematics Letters, 15, No 7 (2002), 897-902.
9. [10] A. Pierluigi and I. Sgura, High-order finite difference schemes for the solution of second-order BVPs, J. of Computational and Applied Mathematics, 176, No 1 (2005), 59-76.
10. [11] S.N. Ha, A nonlinear shooting method for two-point boundary value problems, Computers and Mathematics with Applications, 42, No 10 (2001), 1411-1420.
11. [12] B. Ahmad, J.J. Nieto and N. Shahzad, Generalized quasilinearization method for mixed boundary value problems, Applied Mathematics and Computation, 133, No 2 (2002), 423-429.
12. [13] R.A. Khan, The generalized quasilinearization technique for a second order differential equation with separated boundary conditions, Mathematical and Computer Modelling, 43, No 7 (2006), 727-742.
13. [14] M. Cherpion, C. De Coster and P. Habets, A constructive monotone iterative method for second-order BVP in the presence of lower and upper solutions, Applied Mathematics and Computation, 123, No 1 (2001), 75-91.
14. [15] P.W. Eloe and Y. Zhang, A quadratic monotone iteration scheme for twopoint boundary value problems for ordinary differential equations, Nonlinear Analysis: Theory, Methods and Applications, 33, No 5 (1998), 443-453.
15. [16] J. Lu, Variational iteration method for solving two-point boundary value problems, J. of Computational and Applied Mathematics, 207, No 1 (2007), 92-95. [17] Z.M. Odibat and S. Momani, Variational iteration method for solving nonlinear boundary value problems, Applied Mathematics and Computation, 183, No 2 (2006), 1351-1358.
16. [18] S.N. Jator, Numerical integrators for fourth order initial and boundary value problems, International J. of Pure and Applied Mathematics, 47, No 4 (2008), 563-576.
17. [19] K. Hussain, F. Ismail and N. Senu, Solving directly special fourth-order ordinary differential equations using Runge-Kutta type method, J. of Computational and Applied Mathematics, 306 (2016), 179-199.
18. [20] M. Mechee, N. Senu, F. Ismail and B. Nikouravan, A three-stage fifthorder Runge-Kutta method for directly solving special third-order differential equation with application to thin film flow problem, Mathematical Problems in Engineering, 2013 (2013), 7 pages.
19. [21] J.R. Dormand and P.J. Prince, A family of embedded Runge-Kutta formulae, J. of Computational and Applied Mathematics, 6, No 1 (1980), 19-26.
20. [22] J.C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley and Sons Ltd., England (2008).
21. [23] M. Mechee, F. Ismail, Z.M. Hussain and Z. Siri, Direct numerical methods for solving a class of third-order partial differential equations, Applied Mathematics and Computation, 247 (2014), 663-674.
22. [24] E. Hairer, S.P. Nrsett and G. Wanner, Solving Ordinary Differential Equations, Springer-Verlag, Berlin (1993).
23. [25] A. Khan and T. Aziz, The numerical solution of third-order boundaryvalue problems using quintic splines, Applied Mathematics and Computation, 137, No 2 (2003), 253-260.
24. [26] F.A.A. El-Salam, A.A. El-Sabbagh and Z.A. Zaki, The numerical solution of linear third order boundary value problems using nonpolynomial spline technique, J. of American Science, 6, No 12 (2010), 303-309.
25. [27] S. Saini and H.K. Mishra, A new quartic B-spline method for third order self-adjoint singularly perturbed boundary value problems, Applied Math ematical Sciences, 9, No 8 (2015), 399-408. [28] G. Akram, Quartic spline solution of a third order singularly perturbed boundary value problem, ANZIAM J., 53 (2012), 44-58.