DEVELOPMENT OF NEW HARMONIC EULER USING NONSTANDARD FINITE DIFFERENCE TECHNIQUE FOR SOLVING STIFF PROBLEMS

2015 ◽  
Vol 77 (20) ◽  
Author(s):  
Nurhafizah Moziyana Mohd Yusop ◽  
Mohammad Khatim Hasan
Keyword(s):  
Finite Difference ◽  
Numerical Algorithm ◽  
Mesh Size ◽  
High Accuracy ◽  
Euler Method ◽  
Stiff Problems ◽  
Harmonic Mean ◽  
Stiff Problem ◽  
Finite Difference Technique ◽  
Nonstandard Finite Difference

Solving stiff problem always required very tiny size of meshes if it is solved via traditional numerical algorithm. Using insufficient of mesh size, will triggered instabilities. In this paper, we develop an algorithm applying Harmonic Mean on Euler method to solve the stiff problems. The main purpose of this paper is to discuss the improvement of Harmonic Euler using Nonstandard Finite Difference (NSFD). The combination of these methods can provide new advantages that Euler method could offer. Four set of stiff problems are solved via three schemes, i.e. Harmonic Euler, Nonstandard Harmonic Euler and Nonstandard EO with Harmonic Euler. Findings show that both nonstandard schemes produce high accuracy results.

scholarly journals Dynamics of a Nonstandard Finite-Difference Scheme for a Limit Cycle Oscillator with Delayed Feedback

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Yuanyuan Wang ◽  
Xiaohua Ding
Keyword(s):  
Finite Difference ◽  
Limit Cycle ◽  
Euler Method ◽  
Delayed Feedback ◽  
Dimensional System ◽  
Two Dimensional ◽  
Limit Cycle Oscillator ◽  
The Stability ◽  
Nsfd Scheme ◽  
Nonstandard Finite Difference

We consider a complex autonomously driven single limit cycle oscillator with delayed feedback. The original model is translated to a two-dimensional system. Through a nonstandard finite-difference (NSFD) scheme we study the dynamics of this resulting system. The stability of the equilibrium of the model is investigated by analyzing the characteristic equation. In the two-dimensional discrete model, we find that there are stability switches on the time delay and Hopf bifurcation when the delay passes a sequence of critical values. Finally, computer simulations are performed to illustrate the theoretical results. And the results show that NSFD scheme is better than the Euler method.

scholarly journals An Accurate Predictor-Corrector-Type Nonstandard Finite Difference Scheme for an SEIR Epidemic Model

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Asma Farooqi ◽  
Riaz Ahmad ◽  
Rashada Farooqi ◽  
Sayer O. Alharbi ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Continuous System ◽  
Euler Method ◽  
State Variables ◽  
Nonstandard Finite Difference Scheme ◽  
Long Time ◽  
Predictor Corrector ◽  
Nonstandard Finite Difference

The present work deals with the construction, development, and analysis of a viable normalized predictor-corrector-type nonstandard finite difference scheme for the SEIR model concerning the transmission dynamics of measles. The proposed numerical scheme double refines the solution and gives realistic results even for large step sizes, thus making it economical when integrating over long time periods. Moreover, it is dynamically consistent with a continuous system and unconditionally convergent and preserves the positive behavior of the state variables involved in the system. Simulations are performed to guarantee the results, and its effectiveness is compared with well-known numerical methods such as Runge–Kutta (RK) and Euler method of a predictor-corrector type.

scholarly journals A Nonstandard Finite Difference Method for a Generalized Black–Scholes Equation

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 141
Author(s):  
Mohammad Mehdizadeh Khalsaraei ◽  
Mohammad Mehdi Rashidi ◽  
Ali Shokri ◽  
Higinio Ramos ◽  
Pari Khakzad
Keyword(s):  
Finite Difference ◽  
Order Of Convergence ◽  
Space Variable ◽  
Finite Difference Technique ◽  
Black Scholes Model ◽  
Implicit Finite Difference Scheme ◽  
Black Scholes ◽  
Implicit Finite Difference ◽  
Nonstandard Finite Difference ◽  
Good Agreement

An implicit finite difference scheme for the numerical solution of a generalized Black–Scholes equation is presented. The method is based on the nonstandard finite difference technique. The positivity property is discussed and it is shown that the proposed method is consistent, stable and also the order of the scheme respect to the space variable is two. As the Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset, the proposed method will be more appropriate for solving such symmetric models. In order to illustrate the efficiency of the new method, we applied it on some test examples. The obtained results confirm the theoretical behavior regarding the order of convergence. Furthermore, the numerical results are in good agreement with the exact solution and are more accurate than other existing results in the literature.

scholarly journals Nonstandard Finite Difference Schemes for an SIR Epidemic Model

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3082
Author(s):  
Mohammad Mehdizadeh Khalsaraei ◽  
Ali Shokri ◽  
Samad Noeiaghdam ◽  
Maryam Molayi
Keyword(s):  
Finite Difference ◽  
Epidemic Model ◽  
Euler Method ◽  
Finite Difference Schemes ◽  
Sir Epidemic Model ◽  
Standard Methods ◽  
Sir Epidemic ◽  
Nonstandard Finite Difference ◽  
Better Than ◽  
Positivity And Boundedness

This paper aims to present two nonstandard finite difference (NFSD) methods to solve an SIR epidemic model. The proposed methods have important properties such as positivity and boundedness and they also preserve conservation law. Numerical comparisons confirm that the accuracy of our method is better than that of other existing standard methods such as the second-order Runge–Kutta (RK2) method, the Euler method and some ready-made MATLAB codes.

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