scholarly journals A COMPARATIVE EXPLORATION ON DIFFERENT NUMERICAL METHODS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS

2020 ◽  
Vol 15 (12) ◽  
Author(s):  
Mohammad Asif Arefin
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Euler Method ◽  
Euler’S Method ◽  
Runge Kutta Method ◽  
Accuracy Level ◽  
Runge Kutta ◽  
Matlab Programming

In this paper, the initial value problem of Ordinary Differential Equations has been solved by using different Numerical Methods namely Euler’s method, Modified Euler method, and Runge-Kutta method. Here all of the three proposed methods have to be analyzed to determine the accuracy level of each method. By using MATLAB Programming language first we find out the approximate numerical solution of some ordinary differential equations and then to determine the accuracy level of the proposed methods we compare all these solutions with the exact solution. It is observed that numerical solutions are in good agreement with the exact solutions and numerical solutions become more accurate when taken step sizes are very much small. Lastly, the error of each proposed method is determined and represents them graphically which reveals the superiority among all the three methods. We fund that, among the proposed methods Runge-Kutta 4th order method gives the accurate result and minimum amount of error.

scholarly journals Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method

2019 ◽  
Vol 17 ◽  
pp. 147-154
Author(s):  
Abhinandan Chowdhury ◽  
Sammie Clayton ◽  
Mulatu Lemma
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Adaptive Methods ◽  
Integration Step ◽  
Nonlinear Ordinary Differential Equations ◽  
Kutta Method ◽  
Step Size ◽  
Runge Kutta Method ◽  
Runge Kutta

We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedded integration formulas which appear in pairs. Typically adaptive methods monitor the truncation error at each integration step and automatically adjust the step size to keep the error within prescribed limit. Numerical solutions to different nonlinear initial value problems (IVPs) attained by RKF method are compared with corresponding classical Runge-Kutta (RK4) approximations in order to investigate the computational superiority of the former. The resulting gain in efficiency is compatible with the theoretical prediction. Moreover, with the aid of a suitable time-stepping scheme, we show that the RKF method invariably requires less number of steps to arrive at the right endpoint of the finite interval where the IVP is being considered.

scholarly journals Formulative Visualization of Numerical Methods for Solving Non-Linear Ordinary Differential Equations

2021 ◽  
Vol 2 (2) ◽  
pp. 79-88
Author(s):  
Jeevan Kafle ◽  
Bhogendra Kumar Thakur ◽  
Grishma Acharya
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Real Life ◽  
Euler Method ◽  
Linear Ordinary Differential Equations ◽  
Life Problems ◽  
Backward Euler ◽  
Non Linear ◽  
Runge Kutta

Many physical problems in the real world are frequently modeled by ordinary differential equations (ODEs). Real-life problems are usually non-linear, numerical methods are therefore needed to approximate their solution. We consider different numerical methods viz., Explicit (Forward) and Implicit (Backward) Euler method, Classical second-order Runge-Kutta (RK2) method (Heun’s method or Improved Euler method), Third-order Runge-Kutta (RK3) method, Fourth-order Runge-Kutta (RK4) method, and Butcher fifth-order Runge-Kutta (BRK5) method which are popular classical iteration methods of approximating solutions of ODEs. Moreover, an intuitive explanation of those methods is also be presented, comparing among them and also with exact solutions with necessary visualizations. Finally, we analyze the error and accuracy of these methods with the help of suitable mathematical programming software.

scholarly journals Accuracy Analysis on Solution of Initial Value Problems of Ordinary Differential Equations for Some Numerical Methods with Different Step Sizes

2021 ◽  
Vol 10 (1) ◽  
pp. 118-133
Author(s):  
Mohammad Asif Arefin ◽  
Biswajit Gain ◽  
Rezaul Karim
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Numerical Solutions ◽  
Initial Value ◽  
Step Size ◽  
Runge Kutta Method ◽  
Efficiency Comparison

In this article, three numerical methods namely Euler’s, Modified Euler, and Runge-Kutta method have been discussed, to solve the initial value problem of ordinary differential equations. The main goal of this research paper is to find out the accurate results of the initial value problem (IVP) of ordinary differential equations (ODE) by applying the proposed methods. To achieve this goal, solutions of some IVPs of ODEs have been done with the different step sizes by using the proposed three methods, and solutions for each step size are analyzed very sharply. To ensure the accuracy of the proposed methods and to determine the accurate results, numerical solutions are compared with the exact solutions. It is observed that numerical solutions are best fitted with exact solutions when the taken step size is very much small. Consequently, all the proposed three methods are quite efficient and accurate for solving the IVPs of ODEs. Error estimation plays a significant role in the establishment of a comparison among the proposed three methods. On the subject of accuracy and efficiency, comparison is successfully implemented among the proposed three methods.

scholarly journals Numerical Solutions of Stochastic Differential Equations Driven by Poisson Random Measure with Non-Lipschitz Coefficients

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Hui Yu ◽  
Minghui Song
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Linear Growth ◽  
Numerical Solutions ◽  
Random Measure ◽  
Euler Method ◽  
Poisson Random Measure ◽  
Euler’S Method ◽  
Linear Growth Condition

The numerical methods in the current known literature require the stochastic differential equations (SDEs) driven by Poisson random measure satisfying the global Lipschitz condition and the linear growth condition. In this paper, Euler's method is introduced for SDEs driven by Poisson random measure with non-Lipschitz coefficients which cover more classes of such equations than before. The main aim is to investigate the convergence of the Euler method in probability to such equations with non-Lipschitz coefficients. Numerical example is given to demonstrate our results.

scholarly journals Generalized Runge-Kutta Method with respect to the Non-Newtonian Calculus

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Uğur Kadak ◽  
Muharrem Özlük
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Generating Functions ◽  
Kutta Method ◽  
Science And Engineering ◽  
Runge Kutta Method ◽  
Wide Range ◽  
Runge Kutta

Theory and applications of non-Newtonian calculus have been evolving rapidly over the recent years. As numerical methods have a wide range of applications in science and engineering, the idea of the design of such numerical methods based on non-Newtonian calculus is self-evident. In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functions. The efficiency of the proposed non-Newtonian Euler and Runge-Kutta methods is exposed by examples, and the results are compared with the exact solutions.

scholarly journals DEGENERATE MATRIX METHOD FOR SOLVING NONLINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

1998 ◽  
Vol 3 (1) ◽  
pp. 45-56
Author(s):  
T. Cîrulis ◽  
O. Lietuvietis
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Matrix Method ◽  
Iteration Procedure ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Systems Of Differential Equations ◽  
Runge Kutta ◽  
Degenerate Matrix

Degenerate matrix method for numerical solving nonlinear systems of ordinary differential equations is considered. The method is based on an application of special degenerate matrix and usual iteration procedure. The method, which is connected with an implicit Runge‐Kutta method, can be simply realized on computers. An estimation for the error of the method is given.

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