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 Methods for solving the initial value problem with a two-sided estimate of the local error

2021 ◽  
pp. 88-92
Author(s):  
Yaroslav Pelekh ◽  
Andrii Kunynets ◽  
Halyna Beregova ◽  
Tatiana Magerovska
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Local Error ◽  
Initial Value ◽  
Order Of Accuracy ◽  
Embedded Methods ◽  
The Right

Numerical methods for solving the initial value problem for ordinary differential equations are proposed. Embedded methods of order of accuracy 2(1), 3(2) and 4(3) are constructed. To estimate the local error, two-sided calculation formulas were used, which give estimates of the main terms of the error without additional calculations of the right-hand side of the differential equation, which favorably distinguishes them from traditional two-sided methods of the Runge- Kutta type.

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 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 nonstandard first order initial value problems

1992 ◽  
Vol 5 (1) ◽  
pp. 69-82 ◽  
Author(s):  
M. Venkatesulu ◽  
P. D. N. Srinivasu
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Unique Solution ◽  
Initial Value Problem ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Initial Value ◽  
First Order ◽  
Physical Phenomena

Differential equations of the form y′=f(t,y,y′), where f is not necessarily linear in its arguments, represent certain physical phenomena and are known for quite some time. The well known Clairut's and Chrystal's equations fall into this category. Earlier, we established the existence of a (unique) solution of the nonstandard initial value problem (NSTD IV P) y′=f(t,y,y′), y(t0)=y0 under certain natural hypotheses on f. In this paper we present some first order convergent numerical methods for finding the approximate solutions of the NST D I V Ps.

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 Numerical computation for solving fuzzy differential equations

2019 ◽  
Vol 16 (2) ◽  
pp. 1026
Author(s):  
Fuziyah Ishak ◽  
Najihah Chaini
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Efficient Method ◽  
Dynamic Systems ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Absolute Error ◽  
Initial Value ◽  
First Order

Fuzzy differential equations (FDEs) play important roles in modeling dynamic systems in science, economics and engineering. The modeling roles are important because most problems in nature are indistinct and uncertain. Numerical methods are needed to solve FDEs since it is difficult to obtain exact solutions. Many approaches have been studied and explored by previous researchers to solve FDEs numerically. Most FDEs are solved by adapting numerical solutions of ordinary differential equations. In this study, we propose the extended Trapezoidal method to solve first order initial value problems of FDEs. The computed results are compared to that of Euler and Trapezoidal methods in terms of errors in order to test the accuracy and validity of the proposed method. The results shown that the extended Trapezoidal method is more accurate in terms of absolute error. Since the extended Trapezoidal method has shown to be an efficient method to solve FDEs, this brings an idea for future researchers to explore and improve the existing numerical methods for solving more general FDEs.

On Nagumo's Condition

1972 ◽  
Vol 15 (4) ◽  
pp. 609-611 ◽  
Author(s):  
Thomas Rogers
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Uniqueness Theorem ◽  
Initial Value Problem ◽  
Initial Value ◽  
Image Position

The classical uniqueness theorem of Nagumo [1] for ordinary differential equations is as follows.Theorem. If f(t, y) is continuous on 0≤t≤1, -∞<y<∞ and ifthen there is at most one solution to the initial value problem y'=f(t, y), y(0)=0.

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