A NEW HIGHER ORDER SECOND DERIVATIVE BLENDED BLOCK LINEAR MULTISTEP METHODS FOR THE SOLUTIONS STIFF INITIAL VALUE PROBLEMS

Main Article Content

Omagwu Samson
Joshua Kyaharnan Victor
Muhammad Shakur Ndayawo

Abstract

:This paper is concerned with the accuracy and efficiency of a higher order second derivative blended block linear multistep method for the approximate solution of stiff initial value problems. The main methods were derived by blending of two linear multistep methods using continuous collocation approach. These methods are of uniform order ten. The stability analysis of the block methods indicates that the methods are A–stable, consistent and zero stable hence convergent. Numerical results obtained using the proposed new block methods were compared with those obtained by the well known ODE solver ODE 15s to illustrate the accuracy and effectiveness. The proposed block method is found to be efficient and accurate hence recommended for the solution of stiff initial value problems.

 

Downloads

Download data is not yet available.

Article Details

Section
Articles

References

. Butcher, J.C., (1966). On the Convergence of Numerical Solutions to Ordinary Differential

Equations. Math. Comp. 20, 1 -10.

. Chollom J.P., (2005): A study of Block Hybrid Methods with link of two-step Runge Kutta Methods for first order Ordinary Differential Equations. PhD Thesis (Unpublished) University of Jos,Jos Nigeria.

. Chollom J.P., Ndam, J.N.and Kumleng G.M., (2007):some properties of block linear multistep methods. Science world journal, 2(3), 11-17.

. Chollom J.P., Olatunbusin I.O. and Omagwu S., (2012): A Class of A-Stable Block Explicit Methods for the Solution of Ordinary Differential Equations. Research Journal of Mathematics and Statistics, 4(2), 52-56.

. Ehigie, J.O., Okunuga, S.A., Sofoluwe, A.B. & Akanbi, M. A. (2010). Generalized 2-step

Continuous Linear Multistep Method of Hybrid Type for the Integration of Second

Order Ordinary Differential Equations. Scholars Research Library (Archives of Applied

Science Research), 2(6), 362–372.

. Enright, W. H. (1972). Numerical Solution of Stiff Differential Equations (pp. 321–331). Dept of Computer Science, University of Toronto, Toronto, Canada,.

Enright, W.H (1974). Second Derivative Multistep Methods for Stiff Ordinary Differential

Equations. SIAM Journals on Numerical Analysis, 11 (2), 376-391.

Ezzeddine, A.K. and Hojjati, G., (2012). Third derivative multistep methods for stiff

systems,International journal of nonlinear science, 14, 443-450.

Gamal, A.F., Ismail, K. & Iman, H. I. (1999). A New Efficient Second Derivative multistep

Method for Stiff System. Applied Mathematical Modeling, 23, 279–288.

Henrici, P. (1962). Discrete variable methods in Ordinary Differential Equations (p. 407).John Willey, New York.

. Kumleng, G.M., Sirisena, U.W., & Chollon, J. P. (2012). A Class of A-Stable Order Four and

Six linear Multistep Methods for Stiff Initial Value Problems. Mathematical Theory and

Modeling, 3(11), 94–102.

. Kumleng, G.M., Sirisena, U.W.W, Dang, B. C. (2013). A Ten Step Block Generalized Adams

Method for the Solution of the Holling Tanner and Lorenz Models. African Journal of

Natural Sciences, 16, 63–70.

. Lie, I. & Norset, R. (1989). Super convergence for multistep collocation. Mathematics of

Computation, 52(185), 65–79.

. Lotka a. j. (1925) element of physical biology. Baltimore, Williams and wilkins company

. Mehdizadeh, M., Khalsarai, N., Nasehi, O. & Hojjati, G. (2012). A Class of Second

Derivative multistep methods for stiff systems. Journal of Acta, 15(2012), 209–222.

. Onumanyi, P., Awoyemi, D.O., Jator, S.N.& Sirisena, U. W. (1994). New Linear Multistep Methods with continuous coefficients for first order IVPs. Journal of NMS, 31(1), 37–51.

. Onumanyi, P., Sirisena, U.W. & Jator, S. N. (1999). Continuous finite difference approximations for solving differential equations. International Journal of Computer and Mathematics, 72(1), 15–27.

. Rosenzweng M. L. and MacArthur R. H. (1963) Graphical representatation and stability conditions of pradator-prey interactions. A.M. Nat. 97, 209-223

. Sahi, R.K., Jator, S.N. & Khan, N. A. (2012). A Simpson’s Type Second Derivative Method for Stiff Systems. International Journal of Pure and Applied Mathematics, 81(4), 619–633.