# A FAMILY OF HYBRID LINEAR MULTI-STEP METHODS TYPE FOR SPECIAL THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS

## Abstract

In this paper, we derive a family of three step hy-brid linear multi-step (HLM) method type with one to three

o-step points. Orders and error constants and convergence

analysis of the proposed method are established. Numerical

experiments on special third order initial and boundary value

problems (IVPs, BVPs) are performed to show the eciency

and accuracy of the proposed methods over existing method

found in the literature.

## References

J. Canosa, J. Gazdag, The Korteweg-de Vries-Burgers equation, Journal of Com-

putational Physics 23 (4) 393{403, 1977.

W. C. Troy, Solutions of third-order dierential equations relevant to draining and

coating

ows, SIAM Journal on Mathematical Analysis 24 (1) 155{171, 1993.

E. Poisson, An introduction to the Lorentz-Dirac equation, ArXiv General Rela-

tivity and Quantum Cosmology e-prints http://arxiv.org/abs/gr-qc/9912045.

W. Nakpim, Third-order ordinary dierential equations equivalent to linear second-

order ordinary dierential equations via tangent transformations, Journal of Sym-

bolic Computation 77 63{77, 2016.

A. Sergyeyev, Coupling constant metamorphosis as an integrability-preserving trans-

formation for general nite-dimensional dynamical systems and ODEs, Physics

Letters A 376 2015{2022, 2012.

X. You, Z. Chen, Direct integrators of Runge-Kutta type for special third-order

ordinary dierential equations, Applied Numerical Mathematics 74 128{150, 2013.

K. Hussain, F. Ismail, N. Senu, Solving direct special fourth order ordinary dier-

ential equations using Runge-Kutta type method, Journal of Comp. and Applied

Mathematics 306 179{198, 2016.

D. O. Awoyemi, A p-stable linear multi-step method for solving general third order

ordinary dierential equations, International Journal of Computer Mathematics

(8) 987{993, 2003.

D. O. Awoyemi, O. M. Idowu, A class of hybrid collocation method for third order

ordinary dierential equations, International Journal of Computer Mathematics

(10) 1287{1293, 2005.

S. O. Fatunla, Block method for second order initial value problem (IVP), Inter-

national Journal of Computer Mathematics 41 55{63, 1991.

S. N. Jator, A sixth order linear multistep method for the direct solution of y00 =

f(x; y; y0), International Journal of Pure and Applied Mathematics 40 (1) 457{

, 2007.

S. N. Jator, J. Li, A self stationary linear multistep method for a direct solution of

the general second order initial value problem, International Journal of Computer

Mathematics 85 (5) 817{836, 2009.

U. Mohammed, A class of implicit ve step block method for general second order

ordinary dierential equations, Journal of Nigerian Mathematical Society 30 25{

, 2011.

U. Mohammed, R. B. Adeniyi, A class of implicit six step hybrid backward dier-

entiation formulas for the solution of second order dierential equations, British

Journal of Mathematics and Computer Science 6 (1) 41{52, 2015.

S. N. Jator, Solving sti second order initial value problem directly by backward

dierentiation formulas, in: Proceeding of the 2007 Int. Conference on computa-

tional and Mathematical Methods in Science and Engineering, Illinois, Chicago,

USA, 2007, pp. 223{232.

B. T. Olabode, Y. Yusuph, A new block method for special third order ordinary

dierential equation, Journal of Mathematics and Statistics 5 (3) 167{170, 2009.

L. Collatz, The Numerical Treatment of Dierential Equations, Berlin, Springer-

Verlag, 1960.

A. Khan, T. Aziz, The numerical solution of third order boundary value problem

using quintic splines, Appl. Math. Comput. 137 253{260, 2003.

S.-U. Islam, I. A. Tirmiz, A smooth approximation for the solution of special

non-linear third order boundary value problem based on non-polynomial splines,

International Journal of Computer Mathematics 83 (4) 397{407, 2006.

P.K. Pandey, Solving third-order Boundary Value Problems with Quartic Splines,

Springer Plus, 5 (1) 1{10, 2016.

A. A. Salama, A. A. Mansour, Fourth-order nite-dierence method for third order

BVP, Numerical heat transfer part B 47 383{401, 2005.

T. Biala, S. Jator, R. Adeniyi, Numerical approximations of second order PDEs

by boundary value methods and the method of lines, Afrika Matematika 1{8, 2016.

S. Jator, On the numerical integration of third order BVP by linear multi-step

methods. A sixth order linear multistep methods, International Journal of Pure

and Applied Mathematics 46 (3) 375{388, 2008.

R. K. Sahi, S. N. Jator, N. A. Khan, Contnuous fourth derivative method for

third order boundary value problems, International Journal of Pure and Applied

Mathematics 85 (2) 907{923, 2013.

Far-i-Hag, I. Hussain, A. Arshed, A Haar wavelets based numerical methods for

third order boundary and initial value problems, World Applied Sciences Journal

(10) 2244{2251, 2011.

J. D. Lambert, Computational method in ordinary dierential equation, John Wi-

ley and Sons, London, U.K., 1993.

P. Henrici, Discrete Variable Methods for ODEs, John Wiley and Sons, New York,

USA, 1962.

E. Al-Said, M. Noor, Cubic spline method for a system of third order BVP, Applied

## Published

## How to Cite

*Journal of the Nigerian Mathematical Society*,

*37*(1). Retrieved from https://ojs.ictp.it/jnms/index.php/jnms/article/view/186