A NEW HIGHER ORDER SECOND DERIVATIVE BLENDED BLOCK LINEAR MULTISTEP METHODS FOR THE SOLUTIONS STIFF INITIAL VALUE PROBLEMS
Main Article Content
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.
Â
Downloads
Article Details
COPYRIGHT
Submission of a manuscript implies: that the work described has not been published before, that it is not under consideration for publication elsewhere; that if and when the manuscript is accepted for publication, the authors agree to automatic transfer of the copyright to the publisher.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work
- The journal allows the author(s) to retain publishing rights without restrictions.
- The journal allows the author(s) to hold the copyright without restrictions.
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.