Review of Some Numerical Methods for Solving Initial Value Problems for Ordinary Differential Equations
International Journal of Applied Mathematics and Theoretical Physics
Volume 6, Issue 1, March 2020, Pages: 7-13
Received: Apr. 21, 2020;
Accepted: Apr. 30, 2020;
Published: May 19, 2020
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Fadugba Sunday Emmanuel, Department of Mathematics, Ekti State University, Ado Ekiti, Nigeria
Adebayo Kayode James, Department of Mathematics, Ekti State University, Ado Ekiti, Nigeria
Ogunyebi Segun Nathaniel, Department of Mathematics, Ekti State University, Ado Ekiti, Nigeria
Okunlola Joseph Temitayo, Department of Mathematical and Physical Sciences, Afe Babalola University, Ado Ekiti, Nigeria
Numerical analysis is a subject that is concerned with how to solve real life problems numerically. Numerical methods form an important part of solving differential equations emanated from real life situations, most especially in cases where there is no closed-form solution or difficult to obtain exact solutions. The main aim of this paper is to review some numerical methods for solving initial value problems of ordinary differential equations. The comparative study of the Third Order Convergence Numerical Method (FS), Adomian Decomposition Method (ADM) and Successive Approximation Method (SAM) in the context of the exact solution is presented. The methods will be compared in terms of convergence, accuracy and efficiency. Five illustrative examples/test problems were solved successfully. The results obtained show that the three methods are approximately the same in terms of accuracy and convergence in the case of first order linear ordinary differential equations. It is also observed that FS, ADM and SAM were found to be computationally efficient for the linear ordinary differential equations. In the case of the non-linear ordinary differential equations, SAM is found to be more accurate and converges faster to the exact solution than the FS and ADM. Hence, It is clearly seen that the ADM is found to be better than the FS and SAM in the case of non-linear differential equations in terms of computational efficiency.
Fadugba Sunday Emmanuel,
Adebayo Kayode James,
Ogunyebi Segun Nathaniel,
Okunlola Joseph Temitayo,
Review of Some Numerical Methods for Solving Initial Value Problems for Ordinary Differential Equations, International Journal of Applied Mathematics and Theoretical Physics. Special Issue: Computational Mathematics.
Vol. 6, No. 1,
2020, pp. 7-13.
N. Ahmad, S. Charan, and V. P. Singh, Study of numerical accuracy of Runge-Kutta second, third and fourth order method, 2015.
Y. Ansari, A. Shaikh, and S. Qureshi. Error bounds for a numerical scheme with reduced slope evaluations, J. Appl. Environ. Biol. Sci., 8 (7), 2018.
J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley & Sons, 2016.
M. E. Davis, Numerical methods and modeling for chemical engineers, Courier Corporation, 2013.
S. E. Fadugba, Numerical technique via interpolating function for solving second order ordinary differential equations, Journal of Mathematics and Statistics, 1 (2): 1-6, 2019.
S. E. Fadugba and A. O. Ajayi, Comparative study of a new scheme and some existing methods for the solution of initial value problems in ordinary differential equations, International Journal of Engineering and Future Technology, 14: 47-56, 2017.
S. Fadugba and B. Falodun, Development of a new one-step scheme for the solution of initial value problem (IVP) in ordinary differential equations, International Journal of Theoretical and Applied Mathematics, 3: 58–63, 2017.
S. E. Fadugba and J. T. Okunlola, Performance measure of a new one-step numerical technique via interpolating function for the solution of initial value problem of first order differential equation, World Scientific News, 90: 77–87, 2017.
S. E. Fadugba and T. E. Olaosebikan, Comparative study of a class of one-step methods for the numerical solution of some initial value problems in ordinary differential equations, Research Journal of Mathematics and Computer Science, 2: 1-11, 2018, DOI: 10.28933/rjmcs-2017-12-1801.
R. B. Ogunrinde and S. E. Fadugba, Development of the New Scheme for the solution of Initial Value Problems in Ordinary Differential Equations, International Organization of Scientific Research Journal of Mathematics (IOSRJM), 2: 24-29, 2012.
S. E. Fadugba and S. Qureshi, Convergent numerical method using transcendental function of exponential type to solve continuous dynamical systems, Punjab University Journal of Mathematics, 51: 45-56. 2019.
G. Adomian, A review of the decomposition in applied mathematics, Mathematical Analysis and Applications, 135: 501-544, 1988.
G. Adomian, A review of the decomposition method and some recent results for nonlinear equation. Mathematical Computational Model, 13 (7), 1992.
A. M. Wazwaz, A reliable modification of Adomian decomposition method. Applied Mathematics and Computation, 102: 77-86, 1999.
A. M. Wazwaz, A new method for solving singular initial value problems in the second order ordinary differential equations, Applied Mathematics and Computation, 128: 45-57, 2002.
A. H. M. Abdelrazec, Adomian decomposition method convergence analysis and numerical approximations, M. Sc., McMaster University, 2008.
S. M. Holmquist, An examination of the effectiveness of the Adomian decomposition method in fluid dynamic applications, Ph. D., University of Central Florida, 2007.
G. Adomian and R. Rach, Modified Adomian polynomials, Mathematical Computational Model., 24 (11): 39-46, 1996.
Y. Q. Hasan and L. M. Zhu, Modified Adomian decomposition method for singular initial value problems in the second order ordinary differential equations. Surveys in Mathematics and its Applications, 3: 183-193, 2008.
S. E. Fadugba and J. O. Idowu, Analysis of the properties of a third order convergence numerical method derived via transcendental function of exponential form, International Journal of Applied Mathematics and Theoretical Physics, 5: 97-103, 2019.
J. D. Lambert, Numerical methods for ordinary differential systems: the initial value problem, John Wiley & Sons, Inc., New York, 1991.
S. Qureshi and S. E. Fadugba, Convergence of a numerical technique via interpolating function to approximate physical dynamical systems, Journal of Advanced Physics, 7: 446-450, 2018.
S. S. Motsa and S. Shateyi, New analytic solution to the lane-emden equation of index 2, Mathematical problems in Engineering, pp. 1-20, 2012.
Kent Nagle, Edward B. Saff, and Arthur David Snider, Fundamentals of differential equations and boundary value problems, Fourth edition, Pearson Addison Wesley, 2004.
Y. Cherruault, G. Saccomandi, and B. Some, New results for convergence of Adomian method applied to integral equations, Mathematical Computational Modelling., 16: 85-93, 1992.
Y. O. Fashomi, On the comparison between Picard’s iteration method and Adomain decomposition method in solving non-linear differential equations, Structured Masters, African Institute of Mathematical Sciences (AIMS), 2013.