Ordinary Differential Equations: Initial Value Problem

2017 ◽  
pp. 69-117
Author(s):  
H. Aref ◽  
S. Balachandar
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Initial Value

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.

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.

The asymptotic discretization error of a class of methods for solving ordinary differential equations

1967 ◽  
Vol 63 (2) ◽  
pp. 461-472 ◽  
Author(s):  
J. M. Watt
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Initial Value Problem ◽  
General Class ◽  
Asymptotic Form ◽  
Multistep Methods ◽  
Discretization Error ◽  
Linear Multistep Methods ◽  
Initial Value

AbstractThe order and asymptotic form of the error of a general class of numerical method for solving the initial value problem for systems of ordinary differential equations is considered. Previously only the convergence of the methods, which include Runge-Kutta and linear multistep methods, has been discussed.

scholarly journals Singular Initial Value Problem for Certain Classes of Systems of Ordinary Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Josef Diblík ◽  
Josef Rebenda ◽  
Zdeněk Šmarda
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
Singular Initial Value Problem ◽  
The Right

The paper is devoted to the study of the solvability of a singular initial value problem for systems of ordinary differential equations. The main results give sufficient conditions for the existence of solutions in the right-hand neighbourhood of a singular point. In addition, the dimension of the set of initial data generating such solutions is estimated. An asymptotic behavior of solutions is determined as well and relevant asymptotic formulas are derived. The method of functions defined implicitly and the topological method (Ważewski's method) are used in the proofs. The results generalize some previous ones on singular initial value problems for differential equations.

SIMPLE ALGORITHM'S FOR THE CALCULATION OF HEAT TRANSPORT PROBLEM IN PLATE

2001 ◽  
Vol 6 (1) ◽  
pp. 85-96
Author(s):  
H. Kalis ◽  
I. Kangro
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Heat Transport ◽  
Initial Value Problem ◽  
Transport Problem ◽  
Two Dimensional ◽  
Initial Value ◽  
Equation Solution ◽  
Averaging Methods ◽  
First Order

The approximations of some heat transport problem in a thin plate are based on the finite volume and conservative averaging methods [1,2]. These procedures allow one to reduce the two dimensional heat transport problem described by a partial differential equation to an initial‐value problem for a system of two ordinary differential equations (ODEs) of the first order or to an initial‐value problem for one ordinary differential equations of the first order with one algebraic equation. Solution of the corresponding problems is obtained by using MAPLE routines “gear”, “mgear” and “lsode”.

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