New Efficient Numerical Model for Solving Second, Third and Fourth Order Ordinary Differential Equations Directly

Abstract

This article presents a two-step hybrid linear multistep block method for solving second, third and fourth order initial value problems of ordinary differential equations directly. The derivation of the method was done using collocation and interpolation techniques while approximated power series was used as an interpolating polynomial. The fourth derivative of the power series is collocated at the entire grid and off grid points while the fifth and sixth derivatives of the polynomial are collocated at the end point only. The basic properties of the developed method, that is, order, error constant, zero stability, region of absolute stability, convergence and consistence of the method are properly investigated. The numerical results demonstrated that the scheme developed handles second, third and fourth order ordinary differential equations efficiently and also better in accuracy when compared with existing methods. The proposed method takes away the burden of developing separate method for the solution of second, third and fourth order initial value problem of ordinary differential equations. 

Keywords

• Hybrid Block Method
• Collocation
• interpolation
• Higher Order Ordinary Differential Equations
• Power Series

References

1. [1] Lambert, J. D.: Computational methods in ordinary differential equation, John Wiley & Sons Inc. New York. 1973.
2. [2] Awoyemi D. O.: A class of continuous linear multistep methods for general second order initial value problems in ordinary differential equations, int.,J. Compt, Math., 72, 29-37, 1999.
3. [3] Brugnano L and Trigiante D.: Solving Differential Problems by Multistep IVPs and BVP methods. Gordon and Breach Science Publishers 1998.
4. [4] Gear, C. W.: The numerical integration of ordinary differential equations. Math. Comp., 21, 146 – 156, 1966.
5. [5] Gear, C. W.: Numerical initial value problems in ordinary differential equations. New jersey; prentice Hall. 1971.
6. [6] Gear, C. W.: The stability of numerical methods for second order ordinary differential equations. SIAM J. Numer. Anal., 15(1), 1187 – 197, 1978.
7. [7] Hall, G and Suleiman, M. B. Stability of Adams – type formulae for second order ordinary differential equations. IMA J. Numer. Anal., 1, 427 – 428. 1981.
8. [8] Mohammed, U.: A Six Step Block Method for Solution of Fourth Order Ordinary Differential Equations. The Pacific Journal of Science and Technology. 11(1):259-265, 2010.

9. [9] Badmus A. M and Yahaya Y. A.: A class of collocation methods for general second order ordinary differential equations. African Journal of Mathematics and Computer science research, , 2(4), 69-71, 2009.
10. [10] Ogunware B. G., Omole E. O. and Olanegan O. O.: Hybrid and Non-hybrid Implicit Schemes for Solving Third Order ODEs Using Block Method as Predictors. Journal of Mathematical Theory and Modeling (IISTE),5(3) , 2015.
11. [11] Omole E. O and Ogunware B. G.; 3- Point Single Hybrid Block Method (3PSHBM) for Direct Solution of General Second Order Initial Value Problem of Ordinary Differential Equations: JSRR, 20(3): 1-11; Article no.JSRR.19862, 2018.
12. [12] Areo E. A and Omole E. O. HALF-Step symmetric continuous hybrid block method for the numerical solutions of fourth order ordinary differential equations. Archives of Applied Science Research, 7 (10):39-49, 2015.
13. [13] Kuboye J.O, Omar Z, Abolarin O. E, Abdelrahim R.: Generalized Hybrid Block Method for Solving Second Order Ordinary Differential Equations Directly. Res Rep Math , 2:2, 2018.
14. [14] Omar, Z.B. and Suleiman, M.B.. Solving Higher Order ODEs Directly Using Parallel 2-point Explicit Block Method. Matematika, 21(1), 15-23 Pengintegrasian Teknologi Dalam Sains Matematik, Universiti Sains Malaysia, 2005.
15. [15] Majid Z. A.: Parallel block methods for solving ordinary differential equations. PhD Thesis, University Putra Malaysia. 2004.
16. [16] Fatunla, S.O.: Block methods for second order IVPs, Intern. J. Computer Math., 41(9), 55-63. 1991.
17. [17]. Awoyemi D. O.: A p-stable linear multistep method for solving third order ordinary differential equation int., J. Compt, Math., 80(8), 85, 991. 2003
18. [18] Adeniran A.O., Odejide S. A and Ogundare S. B.: one step hybrid numerical scheme for the direct solution of general second order ordinary differential equations. International Journal of Applied Mathematics, 2015, Volume 28 No. 3 197-212, 2015,
19. [19] Adoghe L. O., Ogunware B. G and Omole E. O.: A family of symmetric implicit higher order methods for the solution of third order initial value problems in ordinary differential equations Theoretical Mathematics & Applications, vol. 6, no. 3, 67-84, 2016
20. [20] Duromola, M.K. : An Accurate Five Off-Step Points Implicit Block Method for Direct Solution of Fourth Order Differential Equations. Open Access Library Journal, 3: e2667. http://dx.doi.org/10.4236/oalib.1102667, 2016.