Application of Third Order Taylor Method to Population Equations

Abstract

In this paper, we implement the third order Taylor method for three different population initial value problems. The Taylor method is derived from the Taylor series expansion. The Taylor series method is one of the earliest methods for the approximate solution for initial value problems for ordinary differential equations. Taylor method is implemented to linear population equation, non-linear population equation and non-linear population equation with an oscillation. The method of solving three initial value problems is implemented using Python Programming.

Introduction

I. INTRODUCTION

The Taylor polynomial series approximation method is well known and is used in variety of applications. Important application of Taylor method is that it can be executed using interval arithmetic and hence allows us to obtain validated numerical methods for differential equations [12]. Taylor series expansion is an amazing concept not only in Mathematics but also in Optimization theory, Function approximation and Machine Learning [11]. It is widely applied in numerical computations when estimate of function values at different points are required [9]. Georg Fuchs et al. presented the application of Taylor series method for a practical mechanical engineering application. The performance of Taylor method is demonstrated by comparison to standard fixed step numerical integration methods [1]. Okan Ozer et al. applied Taylor expansion to determine the analytical expression for eigenfunctions. The results are obtained by simple algorithm produces excellent numerical results for eigenvalues [2]. Robert Bario investigated Taylor series method by using an efficient variable step variable order scheme [3]. Atefeh Armand et al. have proposed Taylor expansion for fuzzy valued functions. The effectiveness of the proposed method is verified by examples [4].

Marija Milosevic et al. have investigated the application Taylor series method for solving stochastic differential equations with time-dependent delay [5]. Vazquez-Leal H. et al. proposed the application of Taylor series method for solving non-linear differential equations on finite intervals. Their result shows that the Taylor series method is capable to generate easily computable and highly accurate approximations for non-linear equations [6]. Suchismita Ghosh et al. have applied Taylor series method to solve states of control systems. They have analyzed the states of the control system by Taylor series method and compared with exact solutions [7]. Eduardo Pasquetti and Paulo B.G. have proposed the application of Taylor expansion to solve non-linear ordinary differential equations with non-polynomial non-linearities [8]. This paper proposes the application of third order Taylor method for three different population initial value problems. The paper is organized as follows: Section II presents Taylor Method, Section III discusses the Population Equation, Section IV focuses on Implementation and Results and finally the Conclusion is presented in Section V.

Conclusion

We first introduced third order Taylor series expansion to the first order differential equation to obtain the numerical approximation of y at time t. We have proposed three different population initial value problems for linear population equation, non-linear population equation and non-linear population equation with an oscillation. To obtain the exact solution for the population equations, we have presented specific third order Taylor difference equation for the initial value problem. The time interval is discretized into N points by a constant step size. The solution is obtained by implementing Python programming for three initial value problems. The results are shown in Figure 2 for linear population equation and Table 1 shows the exact solution. Figure 3 and Table 2 shows the solution for non-linear population equation. Figure 4 and Table 3 shows the solution for non-linear population equation with an oscillation. In all three initial value problems, we observe that for Taylor approximation at time t shows the different population. The difference between population for linear population equation and non-linear population equation is 0.216997 billion approximately and that of linear population equation and non-linear population equation with an oscillation is 0.261902 billion approximately. The difference between population for non-linear population equation and non-linear population equation with an oscillation is 0.044905 billion approximately.

References

[1] Georg Fuchs et al., “Application of the Modern Taylor Series Method to a multi-torsion Chain”, Simulation modeling practice and theory, Volume 33, April 2013, pp. 89-101. [2] Okan Ozer et al., “Application of Asymptotic Taylor expansion method to Bistable potentials”, Advances in Mathematical Physics, Hindawi Publication, Vol. 2013, pp.1-12. [3] Roberto Bario, “Performance of Taylor Series Method for ODEs/DAEs”, Applied Mathematics and Computation, Elsevier, Vol. 163, 2005, pp. 525-545. [4] Atefeh Armand et al., “The fuzzy generalized Taylor Expansion with Application in Fractional Differential Equations”, Iranian Journal of Fuzzy systems, Volume 16, Issue 2, April 2019, pp. 57-72. [5] Marija Milosevic and Miljana Jovanovic, “An Application of Taylor Series in the Approximation of Solutions to Stochastic Differential equations with Time-dependent delay”, Journal of Computational and Applied Mathematics, Volume 235,Issue 15, June 2011, pp. 4439-4451. [6] Vazquez-Leal H. et al., “Modified Taylor Series Method for Solving Non-linear Differential Equations with Mixed Boundary Conditions defined on Finite Intervals”, SpringerPlus3,160(2014), pp. 1-7. [7] Suchismita Ghosh et al., “A New Recursive Method for Solving State Equations Using Taylor Series”, International Journal of Electrical, Electronics and Computer Engineering, Vol.1 (2), 2012, pp. 22-27. [8] Eduardo Pasquetti and Paulo B.G., “Application of Taylor Expansion and Symmetry Concepts to Oscillations with non-polynomial non-linearities”, International Journal of Computational and Applied Mathematics, Vol. 6, Issue 1, 2011, pp. 57-69. [9] S. S. Sastry, “Introductory Methods of Numerical Analysis” Third Edition, Prentice Hall of India, New Delhi. [10] M. K. Jain et al., “Numerical Methods for Scientific and Engineering computation”, New Age International Publishers, Sixth Edition, 2014 [11] Kendall E. Atkinson, An Introduction to Numerical Analysis”, John Wiley & Sons.(1989) [12] Stoer, J., & Bulirsch, R., Introduction to Numerical Analysis. Springer-Verlag (1980).

Copyright

Copyright © 2022 Arunachalam Sundaram . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.