scholarly journals Small Modification on Modified Euler Method for Solving Initial Value Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Abushet Hayalu Workie
Keyword(s):  
Initial Value Problems ◽  
Euler Method ◽  
Numerical Results ◽  
Initial Value ◽  
Cpu Time ◽  
Stability And Consistency

In this article, small modification to the Modified Euler Method is proposed. Stability and consistency were tested to determine the end result, and some numerical results were presented, and the CPU time was compared again, and it is recognized that the proposed method is more reliable and compatible with higher efficiency.

A P-STABLE EIGHTEENTH-ORDER SIX-STEP METHOD FOR PERIODIC INITIAL VALUE PROBLEMS

2007 ◽  
Vol 18 (03) ◽  
pp. 419-431 ◽  
Author(s):  
CHUNFENG WANG ◽  
ZHONGCHENG WANG
Keyword(s):  
Initial Value Problems ◽  
Numerical Results ◽  
New Method ◽  
Initial Value ◽  
Step Method ◽  
Periodic Initial Value Problems

In this paper we present a new kind of P-stable eighteenth-order six-step method for periodic initial-value problems. We add the fourth derivatives to our previous P-stable six-step method to increase the accuracy. We apply two classes of well-known problems to our new method and compare it with the previous methods. The numerical results show that the new method is much more stable, accurate and efficient than the previous methods.

A P-stable complete in phase Obrechkoff trigonometric fitted method for periodic initial-value problems

1993 ◽  
Vol 441 (1912) ◽  
pp. 283-289 ◽  
Keyword(s):  
Initial Value Problems ◽  
Numerical Results ◽  
New Method ◽  
Phase Lag ◽  
Initial Value ◽  
Periodic Initial Value Problems

In this paper we derive a P-stable trigonometric fitted Obrechkoff method with phase-lag (frequency distortion) infinity. It is easy to see, from numerical results presented, that the new method is much more accurate than previous methods.

scholarly journals Direct integrator of block type methods with additional derivative for general third order initial value problems

2020 ◽  
Vol 12 (10) ◽  
pp. 168781402096618
Author(s):  
Mohammed Yousif Turki ◽  
Fudziah Ismail ◽  
Norazak Senu ◽  
Zarina Bibi Ibrahim
Keyword(s):  
Initial Value Problems ◽  
Numerical Solutions ◽  
Numerical Results ◽  
Order System ◽  
Block Type ◽  
Initial Value ◽  
Third Order ◽  
First Order ◽  
Block Methods ◽  
Hermite Interpolating Polynomial

This paper presents the construction of the two-point and three-point block methods with additional derivatives for directly solving [Formula: see text]. The proposed block methods are formulated using Hermite Interpolating Polynomial and approximate the solution of the problem at two or three-point concurrently. The block methods obtain the numerical solutions directly without reducing the equation into the first order system of initial value problems (IVPs). The order and zero-stability of the proposed methods are also investigated. Numerical results are presented and comparisons with other existing block methods are made. The performance shows that the proposed methods are very efficient in solving the general third order IVPs.

scholarly journals On Solving Systems of ODEs Numerically

2005 ◽  
Vol 3 (4) ◽  
Author(s):  
Temple Fay ◽  
Stephan Joubert ◽  
Andrew Mkolesia
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Simple Procedure ◽  
Numerical Results ◽  
Initial Value ◽  
Computer Laboratory ◽  
Laboratory Component ◽  
Systems Of Odes

Many beginning courses on ordinary differential equations have a computer laboratory component in which the students are asked to solve initial value problems numerically. But little attention in texts is given to the question of how accurate such solutions are. In this article we offer a simple procedure that not only can provide a measure of accuracy, but also often produces superior numerical results.

scholarly journals Optimized Hybrid Methods for Solving Oscillatory Second Order Initial Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
N. Senu ◽  
F. Ismail ◽  
S. Z. Ahmad ◽  
M. Suleiman
Keyword(s):  
Hybrid Method ◽  
Initial Value Problems ◽  
Scientific Literature ◽  
Hybrid Methods ◽  
Second Order ◽  
Numerical Results ◽  
Phase Lag ◽  
Initial Value ◽  
Oscillatory Initial Value Problems

Two-step optimized hybrid methods of order five and order six are developed for the integration of second order oscillatory initial value problems. The optimized hybrid method (OHMs) are based on the existing nonzero dissipative hybrid methods. Phase-lag, dissipation or amplification error, and the differentiation of the phase-lag relations are required to obtain the methods. Phase-fitted methods based on the same nonzero dissipative hybrid methods are also constructed. Numerical results show that OHMs are more accurate compared to the phase-fitted methods and some well-known methods appeared in the scientific literature in solving oscillating second order initial value problems. It is also found that the nonzero dissipative hybrid methods are more suitable to be optimized than phase-fitted methods.

Fuzzy form of Euler Method to Solve Fuzzy Differential Equations

2021 ◽  
Vol 01 ◽  
Author(s):  
Umme Salma Pirzada ◽  
S. Rama Mohan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Riccati Equation ◽  
Initial Value Problems ◽  
Euler Method ◽  
Fuzzy Differential Equation ◽  
Fuzzy Arithmetic ◽  
Local Error ◽  
Initial Value

: This paper proposes fuzzy form of Euler method to solve fuzzy initial value problems. By this method, fuzzy differential equations can be solved directly using fuzzy arithmetic. The solution by this method is readily available in a form of fuzzy-valued function. The method does not require to re-write fuzzy differential equation into system of two crisp ordinary differential equations. Algorithm of the method and local error expression are discussed. An illustration and solution of fuzzy Riccati equation are provided for the applicability of the method.

scholarly journals New Analytic Solution to the Lane-Emden Equation of Index 2

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
S. S. Motsa ◽  
S. Shateyi
Keyword(s):  
Analytical Methods ◽  
Initial Value Problems ◽  
Analytic Solution ◽  
Mathematical Physics ◽  
Numerical Results ◽  
Initial Value ◽  
Emden Equation ◽  
New Methods ◽  
Index 2 ◽  
Good Agreement

We present two new analytic methods that are used for solving initial value problems that model polytropic and stellar structures in astrophysics and mathematical physics. The applicability, effectiveness, and reliability of the methods are assessed on the Lane-Emden equation which is described by a second-order nonlinear differential equation. The results obtained in this work are also compared with numerical results of Horedt (1986) which are widely used as a benchmark for testing new methods of solution. Good agreement is observed between the present results and the numerical results. Comparison is also made between the proposed new methods and existing analytical methods and it is found that the new methods are more efficient and have several advantages over some of the existing analytical methods.

scholarly journals Chebyshev neural network model with linear and nonlinear active functions

2016 ◽  
Vol 5 (3) ◽  
pp. 182
Author(s):  
Sarkhosh Seddighi Chaharborj ◽  
Yaghoub Mahmoudi
Keyword(s):  
Neural Network ◽  
Neural Network Model ◽  
Initial Value Problems ◽  
High Accuracy ◽  
Numerical Results ◽  
Initial Value ◽  
Linear Ordinary Differential Equations ◽  
Chebyshev Neural Network ◽  
Active Function ◽  
Linear And Nonlinear

In this paper the second order non-linear ordinary differential equations of Lane-Emden type as singular initial value problems using Chebyshev Neural Network (ChNN) with linear and nonlinear active functions has been studied. Active functions as, \(\texttt{F(z)=z}, \texttt{sinh(x)}, \texttt{tanh(z)}\) are considered to find the numerical results with high accuracy. Numerical results from Chebyshev Neural Network shows that linear active function has more accuracy and is more convenient compare to other functions.

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