scholarly journals Cubature method for the numerical solution of the characteristic initial value problem

1968 ◽  
Vol 8 (2) ◽  
pp. 355-368 ◽  
Author(s):  
M. K. Jain ◽  
K. D. Sharma
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Goursat Problem ◽  
Euler Method ◽  
Hyperbolic Partial Differential Equations ◽  
Initial Value ◽  
Runge Kutta Method ◽  
Quadrature Methods ◽  
Adams Methods ◽  
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The resemblance of the Goursat problem for the hyperbolic partial differential equations to the initial value problem for the ordinary differential equations has suggested the extension of many well known numerical methods existing for (1.2) to the numerical treatment of (1.1). Day [2] discusses the quadrature methods while Diaz [3] generalizes the simple Euler-method. Moore [6] gives an analogue to the fourth order Runge-Kutta-method and Tornig [7] generalizes the explicit and implicit Adams-methods.

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 ◽  
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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.

Asymptotic behaviour of functional-differential equations with proportional time delays

1996 ◽  
Vol 7 (1) ◽  
pp. 11-30 ◽  
Author(s):  
Yunkang Liu
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Initial Value Problem ◽  
Time Delays ◽  
Functional Differential Equations ◽  
Analytic Solutions ◽  
Comprehensive Analysis ◽  
Initial Value ◽  
Functional Differential ◽  
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This paper discusses the initial value problemwhereA, BiandCiared × dcomplex matrices,pi,qi∈ (0, 1),i= 1, 2, …, andy0is a column vector in ℂd. By using ideas from the theory of ordinary differential equations and the theory of functional equations, we give a comprehensive analysis of the asymptotic behaviour of analytic solutions of this initial value problem.

Solvability of Ordinary Differential Equations Near Singular Points: An Analytic Case

1967 ◽  
Vol 19 ◽  
pp. 1303-1313
Author(s):  
Homer G. Ellis
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Singular Point ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Singular Points ◽  
Initial Value ◽  
Analytic Case ◽  
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The question of solvability of the differential equation1with x ranging over an interval (0, a], and with the boundary condition ƒ(0+) = 0, can be investigated as an initial-value problem at 0, which may be a singular point for the equation.

On the asymptotics of solutions of a class of linear functional-differential equations

1996 ◽  
Vol 7 (5) ◽  
pp. 511-518 ◽  
Author(s):  
G. Derfel ◽  
F. Vogl
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Linear Functional ◽  
Sharp Estimate ◽  
Functional Differential Equations ◽  
Initial Value ◽  
Asymptotics Of Solutions ◽  
Functional Differential ◽  
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Growth Of Solutions

A sharp estimate of the growth of solutions of the initial value problem for systems of the formwhere Cj(t) are matrices with elements of power growth, is found. As a corollary of this result, it follows, for instance, that each solution of the initial value problem satisfies the estimate ‖u(t)‖ ≤ Cexp{γln2(1+|t|)} for some C > 0 and γ > 0.

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.

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