IJPAM: Volume 119, No. 1 (2018)

Title

HYBRID ONE-STEP BLOCK FOURTH DERIVATIVE METHOD
FOR THE DIRECT SOLUTION OF THIRD ORDER INITIAL
VALUE PROBLEMS OF ORDINARY DIFFERENTIAL
EQUATIONS

Authors

Mohammad Alkasassbeh$^1$, Zurni Omar$^2$
$^{1,2}$Department of Mathematics
School of Quantitative Sciences
Universiti Utara Malaysia
MALAYSIA

Abstract

An efficient one step block method with generalized two-point-hybrid is developed for solving initial value problems of third order ordinary differential equations directly. In driving this algorithm, a power series approximate function is interpolated at $\{x_n,x_{n+r},x_{n+s}\}$ while its third and fourth derivatives are collocated at all points $\{x_n,x_{n+r},x_{n+s},x_{n+1}\}$ in the given interval. The proposed method is then tested for initial value problems of third order ordinary differential equations solved previously by other methods. The numerical results confirm the superiority of the new method to the existing methods regarding accuracy.

History

Received: April 1, 2017
Revised: June 18, 2018
Published: June 20, 2018

AMS Classification, Key Words

AMS Subject Classification: 65L05, 65L06
Key Words and Phrases: direct solution, third order ordinary differential equation, interpolation and collocation, hybrid block methods, fourth derivative

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Bibliography

1
S. Fatunla,Numerical Methods for Fatunla Initial Value Problems in Ordinary Differential Equations,Academic Press, USA(1988).

2
J. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley & Sons, USA(1973).

3
S. Jator, Block third derivative method based on trigonometric polynomials for periodic initial-value problems, Afrika Matematika, 27, No.3-4(2016), 365-377.

4
W. Enright, Second derivative multistep methods for stiff ordinary differential equations, SIAM Journal on Numerical Analysis, 11, No.2(1974), 321-331.

5
L. Yap, ,F. Ismail, , & N. Senu, An accurate block hybrid collocation method for third order ordinary differential equations, Journal of Applied Mathematics, 2014,2014,1-9.

6
M. Mechee, , N. Senu, F. Ismail, B. Nikouravan, & Z. Siri, A three-stage fourth-order Runge-Kutta method for directly solving special third-order differential equation with application to thin film problem. Mathematical Problems in Engineering, 2013,2013,1-7.

7
T . Myers, Thin films with high surface tension, SIAM Review, 40, No. 3(1998), 441-462.

8
E. Momoniat, Symmetries first integrals and phase planes of a third-order ordinary differential equation from thin film flow, Mathematical and Computer Modelling, 49,No. 1-2(2009), 215-225.

9
A. James, A. Adesanya, S. Joshua, Continuous block method for the solution of second order initial value problems of ordinary differential equation,International Journal of Pure and Applied Mathematics, 83, No.3(2013), 405-416.

10
J. Kuboye and Z. Omar. Numerical solution of third order ordinary differential equations using a seven-step block method, International Journal of Mathematical Analysis,9, No.15(2015),743-754.

11
A. Adesanya, M. Udoh, A. Alkali, A new block-predictor corrector algorithm for the solution of y'''= f (x, y, y', y''),American J. of Computational Mathematics, 2, (2012), 341-344.

12
N. Waeleh, Z. A Majid, F. Ismail, A new algoriths for solving higher order IVPs of ODEs,Applied Mathematical Sciences, 5, No. 56 (2011). 2795-2805.

13
R. Sahi, S. Jator, N. Khan, A Simpson's-type second derivative method for stiff systems, International journal of pure and applied mathematics, 81, No. 4(2012), 619-633.

14
F. Ngwane, S. Jator, Block hybrid-second derivative method for stiff systems, International Journal of Pure and Applied Mathematics , 80, No. 4(2012), 543-559.

15
G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations,Mathematica Scandinavica, (1956), 33-53.

How to Cite?

DOI: 10.12732/ijpam.v119i1.17 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 119
Issue: 1
Pages: 207 - 224


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