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![International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD45066 | Volume – 5 | Issue – 5 | Jul-Aug 2021 Page 1342 with ⋅ ⋅ ⋅ ⋅ ⋅ = , 2 , 1 , 0 n whereh is step-size. Later on, Euler’s method is also known as first order Runge-Kutta method. The local error occurred in this method is proportional to the square of step size and the global error occurred in this method is proportional to step- size. 2. Improved Euler Method Euler method gives best result when the functions are linear in nature. When functions are non-linear there occurs a truncation error. To remove this drawback Improved Euler’s method is introduced. In this method instead of point the arithmetic average of the slopes at xn and xn+1(that is, at the end points of each subinterval) is used. This method based on two values of dependent variable 1 + n y and predicted value by Euler method i.e., 1 + n y . ( ) n n n n y x f h y y , 1 • + = + ( ) ( ) [ ] 1 1 1 , , 2 + + + + + = n n n n n n y x f y x f h y y This method is also known as Heun’s method or second order Runge-Kutta method. The local error occurred in this method is proportional to the cube of step size and the global error occurred in this method is proportional to square of step size. 3. Third order Runge-Kutta method The third order Runge-Kutta formula is as follows: ( ) ( ) ( ) 2 1 3 1 2 1 3 2 1 1 2 , 2 , 2 , 4 2 1 k k y h x f h k k y h x f h k y x f h k where k k k y y n n n n n n n n + − + • = + + • = • = + • + + = + This method is also known as Heun’s method of order Three. The local error occurred in this method is proportional to the fourth power of step size and the global error occurred in this method is proportional to cube of step size. 4. Fourth order Runge-Kutta method It is most popular Runge-Kutta method and widely used in solving the first order D.E. The formula for this method is as follows: ( ) ( ) ( ) 3 4 2 3 1 2 1 4 3 2 1 1 , 2 , 2 2 , 2 , 2 2 6 1 k y h x f h k k y h x f h k k y h x f h k y x f h k where k k k k y y n n n n n n n n n n + + • = + + • = + + • = • = + • + • + + = + The local error occurred in this method is proportional to the fifth power of step size and the global error occurred in this method is proportional to fourth power of step size. Application and comparison: Consider the first order ordinary D.E.as ( ) 1 0 , = − = y with y x dx dy . We first solve it by classical method and then by numerical methods. 1. Classical method: This is linear D.E. Comparing with ( ) ( ) x Q y x P dx dy = + Here ( ) ( ) x x Q x P = = , 1 Integrating factor= ( ) x dx dx x P e e e I = ∫ = ∫ = 1 It’s solution is ∫ + = c Idx Q I y c e xe ye x x x + − = Using initial conditions ( ) 1 0 = y 2 = c The particular solution is x e x y − + − = 2 1 For different values of x we get different solutions for y. Now we apply four above numerical method to given D.E. one by one. Here we consider the step size 0.2 for each method. The following table gives the value of y for x from 0 to 2, taking h=0.2.](https://image.slidesharecdn.com/178comparativeanalysisofdifferentnumericalmethods-210918050454/75/Comparative-Analysis-of-Different-Numerical-Methods-for-the-Solution-of-Initial-Value-Problems-in-First-Order-Ordinary-Differential-Equations-2-2048.jpg)
![International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD45066 | Volume – 5 | Issue – 5 | Jul-Aug 2021 Page 1343 Sr. No. X Exact solution by classical method Euler’s method Improved Euler method Third order Runge- Kutta method Fourth order Runge- Kutta method 1 0 1 1 1 1 1 2 0.2 0.83746 0.80 0.84 0.82368 0.837467 3 0.4 0.74064 0.68 0.7448 0.721139 0.740649 4 0.6 0.69762 0.624 0.702736 0.677912 0.697634 5 0.8 0.69866 0.6192 0.704244 0.68237 0.698669 6 1 0.73576 0.65636 0.74148 0.725165 0.73577 7 1.2 0.80239 0.724288 0.808013 0.798781 0.8024 8 1.4 0.89319 0.8194304 0.898571 0.897175 0.893205 9 1.6 1.00379 0.9355443 1.00883 1.14982 1.0038 10 1.8 1.13060 1.0684355 1.13524 1.14982 1.13061 11 2 1.27067 1.2147484 1.2749 1.29702 1.27068 The above table shows that Euler’s method shows more variation with exact solution while Improved method and Fourth order Runge-kutta method gives solution which is closed to exact solution. Conclusion The main aim of this paper is to make the comparison between the numerical methods Euler method, Improved Euler method, Third order Runge-kutta method and Fourth order Runge-Kutta method and also with classical solution of first order ordinary D.E. In this paper we showed that the solution obtained by classical method is closed to solution obtained by numerical method. If in numerical method we reduce step size as small as possible to get more accurate solution. Among all these methods Fourth order Runge-kutta method gives more accurate and more efficient solution. Therefore, this method is widely used in solving first order ordinary D.E. References [1] Raisinghaniya, M.D., Ordinary and Partial Differential Equations, S. Chand&Comp. Ltd., 15th revised edition 2013,ISBN-81-219-0892-5. [2] S. S. Sastry., Introductory Methods of Numerical Analysis Fifth Edition, PHI Learning Private Limited, New-Delhi- 110001.2012. [3] Numerical Methods in Engineering with Python 3, Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA, ISBN 978-1-107-03385-6, 2013 [4] Scilab software. [5] Chapra, Steven C., & Canale Numerical Methods for Engineers, S Tata MacGraw Hill Education Pvt. L 25-902744-9. [6] Ince, E., L., Ordinary Differential Equations, First Edition, Dover Publication, Inc., New York. [7] Chandrajeet Singh Yadav, Comparative Analysis of Different Numerical Methods of Solving First Order Differential Equation., International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2, Issue-4, June 2018, pp.567-570,](https://image.slidesharecdn.com/178comparativeanalysisofdifferentnumericalmethods-210918050454/75/Comparative-Analysis-of-Different-Numerical-Methods-for-the-Solution-of-Initial-Value-Problems-in-First-Order-Ordinary-Differential-Equations-3-2048.jpg)
Comparative Analysis of Different Numerical Methods for the Solution of Initial Value Problems in First Order Ordinary Differential Equations
- 1. International Journal ofTrend in Scientific Research and Development (IJTSRD) Volume 5 Issue 5, July-August 2021 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470 @ IJTSRD | Unique Paper ID – IJTSRD45066 | Volume – 5 | Issue – 5 | Jul-Aug 2021 Page 1341 Comparative Analysis of Different Numerical Methods for the Solution of Initial Value Problems in First Order Ordinary Differential Equations Vibahvari Tukaram Dhokrat Assistant Professor, K.T.H.M. College, Nashik, Maharashtra, India ABSTRACT A mathematical equation which involves a function and its derivatives is called a differential equation. We consider a real-life situation, from this form a mathematical model, solve that model using some mathematical concepts and take interpretation of solution. It is a well-known and popular concept in mathematics because of its massive application in real world problems. Differential equations are one of the most important mathematical tools used in modeling problems in Physics, Biology, Economics, Chemistry, Engineering and medical Sciences. Differential equation can describe many situations viz: exponential growth and de-cay, the population growth of species, the change in investment return over time. We can solve differential equations using classical as well as numerical methods, In this paper we compare numerical methods of solving initial valued first order ordinary differential equations namely Euler method, Improved Euler method, Runge-Kutta method and their accuracy level. We use here Scilab Software to obtain direct solution for these methods. KEYWORDS: Differential Equations, Accuracy, local Error, Global Error Step-size How to cite this paper: Vibahvari Tukaram Dhokrat "Comparative Analysis of Different Numerical Methods for the Solution of Initial Value Problems in First Order Ordinary Differential Equations" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456- 6470, Volume-5 | Issue-5, August 2021, pp.1341- 1343, URL: www.ijtsrd.com/papers/ijtsrd45066.pdf Copyright © 2021 by author (s) and International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) (http://creativecommons.org/licenses/by/4.0) INTRODUCTION Numerical analysis is the continuation of Mathematics. The main aim of numerical analysis is to construct computational methods and provide software for the efficient approximate solution of mathematical problems of all kinds using computers as its main tool. In numerical analysis we design methods that gives accurate approximations which converge to exact solution. Non-linear problems and problems from analysis such as differential equation can be solved by approximate i.e. numerical methods. The techniques for solving differential equation using numerical approximations were developed before computer programming existed. In the Eighteenth century the mathematician Euler (1768),in the Nineteenth century the mathematician Carl (1895) and In the Twentieth century the mathematician Martin Kutta (1905) developed the formulae to describe the solution of initial value problem in first order ordinary differential equations. For comparative analysis we consider the standard differential equation of first order as follows: ( ) ( ) x y x f y , = ′ where ( ) . 0 0 y x y = 1. Euler Method This was the oldest and simplest method originated by Leonhard Euler in 1768.It is the first order numerical method for solving ordinary differential equations with given initial conditions. This method is use to analyze a differential equation(D.E.) which uses the idea of linear approximation, where we use small tangent lines over a short distance to approximate the solution to an initial value problem. Consider a D.E. ( ) ( ) x y x f y , = ′ where ( ) . 0 0 y x y = The formula to solve given D.E. using Euler’s method is given by IJTSRD45066
- 2. International Journal ofTrend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD45066 | Volume – 5 | Issue – 5 | Jul-Aug 2021 Page 1342 with ⋅ ⋅ ⋅ ⋅ ⋅ = , 2 , 1 , 0 n whereh is step-size. Later on, Euler’s method is also known as first order Runge-Kutta method. The local error occurred in this method is proportional to the square of step size and the global error occurred in this method is proportional to step- size. 2. Improved Euler Method Euler method gives best result when the functions are linear in nature. When functions are non-linear there occurs a truncation error. To remove this drawback Improved Euler’s method is introduced. In this method instead of point the arithmetic average of the slopes at xn and xn+1(that is, at the end points of each subinterval) is used. This method based on two values of dependent variable 1 + n y and predicted value by Euler method i.e., 1 + n y . ( ) n n n n y x f h y y , 1 • + = + ( ) ( ) [ ] 1 1 1 , , 2 + + + + + = n n n n n n y x f y x f h y y This method is also known as Heun’s method or second order Runge-Kutta method. The local error occurred in this method is proportional to the cube of step size and the global error occurred in this method is proportional to square of step size. 3. Third order Runge-Kutta method The third order Runge-Kutta formula is as follows: ( ) ( ) ( ) 2 1 3 1 2 1 3 2 1 1 2 , 2 , 2 , 4 2 1 k k y h x f h k k y h x f h k y x f h k where k k k y y n n n n n n n n + − + • = + + • = • = + • + + = + This method is also known as Heun’s method of order Three. The local error occurred in this method is proportional to the fourth power of step size and the global error occurred in this method is proportional to cube of step size. 4. Fourth order Runge-Kutta method It is most popular Runge-Kutta method and widely used in solving the first order D.E. The formula for this method is as follows: ( ) ( ) ( ) 3 4 2 3 1 2 1 4 3 2 1 1 , 2 , 2 2 , 2 , 2 2 6 1 k y h x f h k k y h x f h k k y h x f h k y x f h k where k k k k y y n n n n n n n n n n + + • = + + • = + + • = • = + • + • + + = + The local error occurred in this method is proportional to the fifth power of step size and the global error occurred in this method is proportional to fourth power of step size. Application and comparison: Consider the first order ordinary D.E.as ( ) 1 0 , = − = y with y x dx dy . We first solve it by classical method and then by numerical methods. 1. Classical method: This is linear D.E. Comparing with ( ) ( ) x Q y x P dx dy = + Here ( ) ( ) x x Q x P = = , 1 Integrating factor= ( ) x dx dx x P e e e I = ∫ = ∫ = 1 It’s solution is ∫ + = c Idx Q I y c e xe ye x x x + − = Using initial conditions ( ) 1 0 = y 2 = c The particular solution is x e x y − + − = 2 1 For different values of x we get different solutions for y. Now we apply four above numerical method to given D.E. one by one. Here we consider the step size 0.2 for each method. The following table gives the value of y for x from 0 to 2, taking h=0.2.
- 3. International Journal ofTrend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD45066 | Volume – 5 | Issue – 5 | Jul-Aug 2021 Page 1343 Sr. No. X Exact solution by classical method Euler’s method Improved Euler method Third order Runge- Kutta method Fourth order Runge- Kutta method 1 0 1 1 1 1 1 2 0.2 0.83746 0.80 0.84 0.82368 0.837467 3 0.4 0.74064 0.68 0.7448 0.721139 0.740649 4 0.6 0.69762 0.624 0.702736 0.677912 0.697634 5 0.8 0.69866 0.6192 0.704244 0.68237 0.698669 6 1 0.73576 0.65636 0.74148 0.725165 0.73577 7 1.2 0.80239 0.724288 0.808013 0.798781 0.8024 8 1.4 0.89319 0.8194304 0.898571 0.897175 0.893205 9 1.6 1.00379 0.9355443 1.00883 1.14982 1.0038 10 1.8 1.13060 1.0684355 1.13524 1.14982 1.13061 11 2 1.27067 1.2147484 1.2749 1.29702 1.27068 The above table shows that Euler’s method shows more variation with exact solution while Improved method and Fourth order Runge-kutta method gives solution which is closed to exact solution. Conclusion The main aim of this paper is to make the comparison between the numerical methods Euler method, Improved Euler method, Third order Runge-kutta method and Fourth order Runge-Kutta method and also with classical solution of first order ordinary D.E. In this paper we showed that the solution obtained by classical method is closed to solution obtained by numerical method. If in numerical method we reduce step size as small as possible to get more accurate solution. Among all these methods Fourth order Runge-kutta method gives more accurate and more efficient solution. Therefore, this method is widely used in solving first order ordinary D.E. References [1] Raisinghaniya, M.D., Ordinary and Partial Differential Equations, S. Chand&Comp. Ltd., 15th revised edition 2013,ISBN-81-219-0892-5. [2] S. S. Sastry., Introductory Methods of Numerical Analysis Fifth Edition, PHI Learning Private Limited, New-Delhi- 110001.2012. [3] Numerical Methods in Engineering with Python 3, Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA, ISBN 978-1-107-03385-6, 2013 [4] Scilab software. [5] Chapra, Steven C., & Canale Numerical Methods for Engineers, S Tata MacGraw Hill Education Pvt. L 25-902744-9. [6] Ince, E., L., Ordinary Differential Equations, First Edition, Dover Publication, Inc., New York. [7] Chandrajeet Singh Yadav, Comparative Analysis of Different Numerical Methods of Solving First Order Differential Equation., International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2, Issue-4, June 2018, pp.567-570,