### Analysis of the Properties of a Third Order Convergence Numerical Method Derived via the Transcendental Function of Exponential Form

Received: 14 September 2019     Accepted: 18 October 2019     Published: 4 November 2019
Abstract

This paper proposes a new numerical method for the solution of the Initial Value Problems (IVPs) of first order ordinary differential equations. The new scheme has been derived via the transcendental function of exponential type. The analysis of the properties of the method such as local truncation error, order of accuracy, consistency, stability and convergence were investigated. Two illustrative examples/test problems were solved successfully to test the accuracy, performance and suitability of the method in terms of the absolute relative errors computed at the final nodal point of the associated integration interval via MATLAB codes. It is observed that the method is found to be of third order convergence, consistent and stable. The numerical results obtained via the method agree with the exact solution. Moreover, it is also observed that the method is an improvement on Fadugba-Falodun scheme. Hence, the proposed numerical method is a good approach for solving the IVPs of various nature and characteristics in diverse areas of Ordinary Differential Equations (ODEs).

Keywords

Accuracy, Consistency, Convergence, Initial Value Problem, Local Truncation Error, Order of Accuracy, Region of Stability, Stability

References
 [1] N. Ahmad, S. Charan, and V. P. Singh, Study of numerical accuracy of Runge-Kutta second, third and fourth order method, 2015. [2] 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. [3] J. C. Butcher, Numerical methods for ordinary differential equation, West Sussex: John Wiley & Sons Ltd, 2003. [4] M. E. Davis, Numerical methods and modeling for chemical engineers, Courier Corporation, 2013. [5] S. E. Fadugba, Numerical technique via interpolating function for solving second order ordinary differential equations, Journal of Mathematics and Statistics, 1: 1-6, 2019. [6] 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. [7] 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. [8] 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. [9] 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. [10] J. D. Lambert, Numerical methods for ordinary differential systems: the initial value problem, John Wiley & Sons, Inc., New York, 1991. [11] 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. [12] J. D. Lambert, Computational methods in ordinary differential equations, John Wiley & Sons Inc, 1973. [13] 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. [14] J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley & Sons, 2016. [15] 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.
• APA Style

Sunday Emmanuel Fadugba, Jethro Olorunfemi Idowu. (2019). Analysis of the Properties of a Third Order Convergence Numerical Method Derived via the Transcendental Function of Exponential Form. International Journal of Applied Mathematics and Theoretical Physics, 5(4), 97-103. https://doi.org/10.11648/j.ijamtp.20190504.11

ACS Style

Sunday Emmanuel Fadugba; Jethro Olorunfemi Idowu. Analysis of the Properties of a Third Order Convergence Numerical Method Derived via the Transcendental Function of Exponential Form. Int. J. Appl. Math. Theor. Phys. 2019, 5(4), 97-103. doi: 10.11648/j.ijamtp.20190504.11

AMA Style

Sunday Emmanuel Fadugba, Jethro Olorunfemi Idowu. Analysis of the Properties of a Third Order Convergence Numerical Method Derived via the Transcendental Function of Exponential Form. Int J Appl Math Theor Phys. 2019;5(4):97-103. doi: 10.11648/j.ijamtp.20190504.11

• ```@article{10.11648/j.ijamtp.20190504.11,
author = {Sunday Emmanuel Fadugba and Jethro Olorunfemi Idowu},
title = {Analysis of the Properties of a Third Order Convergence Numerical Method Derived via the Transcendental Function of Exponential Form},
journal = {International Journal of Applied Mathematics and Theoretical Physics},
volume = {5},
number = {4},
pages = {97-103},
doi = {10.11648/j.ijamtp.20190504.11},
url = {https://doi.org/10.11648/j.ijamtp.20190504.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20190504.11},
abstract = {This paper proposes a new numerical method for the solution of the Initial Value Problems (IVPs) of first order ordinary differential equations. The new scheme has been derived via the transcendental function of exponential type. The analysis of the properties of the method such as local truncation error, order of accuracy, consistency, stability and convergence were investigated. Two illustrative examples/test problems were solved successfully to test the accuracy, performance and suitability of the method in terms of the absolute relative errors computed at the final nodal point of the associated integration interval via MATLAB codes. It is observed that the method is found to be of third order convergence, consistent and stable. The numerical results obtained via the method agree with the exact solution. Moreover, it is also observed that the method is an improvement on Fadugba-Falodun scheme. Hence, the proposed numerical method is a good approach for solving the IVPs of various nature and characteristics in diverse areas of Ordinary Differential Equations (ODEs).},
year = {2019}
}
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T2  - International Journal of Applied Mathematics and Theoretical Physics
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UR  - https://doi.org/10.11648/j.ijamtp.20190504.11
AB  - This paper proposes a new numerical method for the solution of the Initial Value Problems (IVPs) of first order ordinary differential equations. The new scheme has been derived via the transcendental function of exponential type. The analysis of the properties of the method such as local truncation error, order of accuracy, consistency, stability and convergence were investigated. Two illustrative examples/test problems were solved successfully to test the accuracy, performance and suitability of the method in terms of the absolute relative errors computed at the final nodal point of the associated integration interval via MATLAB codes. It is observed that the method is found to be of third order convergence, consistent and stable. The numerical results obtained via the method agree with the exact solution. Moreover, it is also observed that the method is an improvement on Fadugba-Falodun scheme. Hence, the proposed numerical method is a good approach for solving the IVPs of various nature and characteristics in diverse areas of Ordinary Differential Equations (ODEs).
VL  - 5
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Author Information
• Department of Mathematics, Faculty of Science, Ekiti State University, Ado Ekiti, Nigeria

• Department of Mathematics, Faculty of Science, Ekiti State University, Ado Ekiti, Nigeria

• Sections