scholarly journals Given a one-step numerical scheme, on which ordinary differential equations is it exact?

2009 ◽  
Vol 223 (2) ◽  
pp. 1058-1065 ◽  
Author(s):  
Francisco R. Villatoro
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Scheme ◽  
One Step

scholarly journals The Treatment of Second Order Ordinary Differential Equations Using Equidistant One-step Block Hybrid

2019 ◽  
pp. 1-9
Author(s):  
Y. Skwame ◽  
J. Sabo ◽  
M. Mathew
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Second Order ◽  
Direct Solution ◽  
Block Method ◽  
Initial Value ◽  
One Step ◽  
Better Than

A general one-step hybrid block method with equidistant of order 6 has been successfully developed for the direct solution of second order IVPs in this article. Numerical analysis shows that the developed method is consistent and zero-stable which implies its convergence. The analysis of the new method is examined on two highly and mildly stiff second-order initial value problems to illustrate the efficiency of the method. It is obvious that our method performs better than the existing method within which we compare our result with. Hence, the approach is an adequate one for solving special second order IVPs.

scholarly journals Numerical Solution of Stieltjes Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1571
Author(s):  
Francisco J. Fernández ◽  
F. Adrián F. Tojo
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Explicit Solution ◽  
Numerical Scheme ◽  
Linear Ordinary Differential Equation ◽  
Stieltjes Integral ◽  
Novel Method ◽  
Theoretical Results

This work is devoted to the obtaining of a new numerical scheme based on quadrature formulae for the Lebesgue–Stieltjes integral for the approximation of Stieltjes ordinary differential equations. This novel method allows us to numerically approximate models based on Stieltjes ordinary differential equations for which no explicit solution is known. We prove several theoretical results related to the consistency, convergence, and stability of the numerical method. We also obtain the explicit solution of the Stieltjes linear ordinary differential equation and use it to validate the numerical method. Finally, we present some numerical results that we have obtained for a realistic population model based on a Stieltjes differential equation and a system of Stieltjes differential equations with several derivators.

A NEW CLASS OF L-STABLE HYBRID ONE-STEP METHODS FOR THE NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS

2015 ◽  
Vol 1 (2) ◽  
pp. 39-44
Author(s):  
Mohammad Mehdizadeh ◽  
Maryam Molayi
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Hybrid Methods ◽  
Akzo Nobel ◽  
New Class ◽  
Numerical Studies ◽  
One Step ◽  
Judicious Choice ◽  
Stable Hybrid

In this paper, a class of one-step hybrid methods for the numerical solution of ordinary differential equations (ODEs) are considered. The accuracy and stability properties of these methods are investigated. By judicious choice of the coefficients in these formulae a class of method is derived which is shown to be L-stable and so is appropriate for the solution of certain ordinary differential and stiff differential equations. We apply the new method for numerical integration of some famous stiff chemical problems such chemical Akzo-Nobel problem, ROBER problem (suggested by Robertson) and some others which are very popular in numerical studies.

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