• editor.aipublications@gmail.com
  • Track Your Paper
  • Contact Us
  • ISSN: 2456-866X

International Journal Of Chemistry, Mathematics And Physics(IJCMP)

Derivation and Application of Multistep Methods to a Class of First-order Ordinary Differential Equations

Uwem Akai , Ubon Abasiekwere , Paul Udoh , Jonas Achuobi


International Journal of Chemistry, Mathematics And Physics(IJCMP), Vol-3,Issue-2, March - April 2019, Pages 30-52 , 10.22161/ijcmp.3.2.2

Download | Downloads : 7 | Total View : 1672

Share

Of concern in this work is the derivation and implementation of the multistep methods through Taylor’s expansion and numerical integration. For the Taylor’s expansion method, the series is truncated after some terms to give the needed approximations which allows for the necessary substitutions for the derivatives to be evaluated on the differential equations. For the numerical integration technique, an interpolating polynomial that is determined by some data points replaces the differential equation function and it is integrated over a specified interval. The methods show that they are only convergent if and only if they are consistent and stable. In our numerical examples, the methods are applied on non-stiff initial value problems of first-order ordinary differential equations, where it is established that the multistep methods show superiority over the single-step methods in terms of robustness, efficiency, stability and accuracy, the only setback being that the multi-step methods require more computational effort than the single-step methods.

linear multi-step method; numerical solution; ordinary differential equation; initial value problem; stability; convergence.

[1] Lambert, J. D., Computational Methods in Ordinary Differential Equations, John Wiley and Sons, New York, 1973.
[2] Awoyemi, D. O., Kayode, J. S. and Adoghe, L. O., A Four-Point Fully Implicit Method for the Numerical Integration of Third-Order Ordinary Differential Equations, International Journal of Physical Sciences, 2014;9(1), 7-12.
[3] Okunuga, S. A. and Ehigie, J., A New Derivation of Continuous Collocation Multistep Methods Using Power Series as Basis Function, Journal of Modern Mathematics and Statistics, 2009; 3(2), 43-50.
[4] Mathews, J. H.,Numerical Methods for Mathematics, Science and Engineering, Prentice Hall, India, 2005.
[5] Butcher, J. C., Numerical methods for ordinary differential equations, (2nd Edition) London: John Wiley & sons Ltd, 2008.
[6] Draux, A., On quasi-orthogonal polynomials of order r,Integral Transforms and Special Functions, 2016;27(9), 747-765.
[7] Turki, M. Y., Ismail, F., Senu, N. and Ibrahim,Z. B., Second derivative multistep method for solving first-order ordinary differential equations, In:AIP Conference Proceedings, 2016; 1739(1), 489-500.
[8] Burden, R. L. and Faires, J. D.,Numerical Analysis, (9th Edition), Canada: Brook/Cole, Cengage Learning, 2011.
[9] Karapinar, E. and Rakocevic, V., On cyclic generalized weakly-contractions on partial metric spaces, Journal of Applied Mathematics, 2013;48(1), 34-51.
[10] Stroud, K. A. and Booth,D. J., Advanced Engineering Mathematics, (7th Edition), Red Globe Press, New York, 2013.
[11] Suli, E. & Mayers, D., An introduction to numerical analysis, Cambridge: Cambridge University Press, 2003.
[12] Pavlovic, M. and Palaez, J. A., An equivalence for weighted integrals of an analytic function and its derivative, Journal of Mathematische Nachrichten, 2008; 281(11), 1612-1623.
[13] Griffiths, D. F. and Higham, D. J., Numerical methods for ordinary differential equations, (1st Edition) London: Springer-Verlag, 2010.