scholarly journals On the Convergence of Successive Approximations in the Theory of Ordinary Differential Equations

1958 ◽  
Vol 1 (1) ◽  
pp. 9-20 ◽  
Author(s):  
W. A. J. Luxemburg
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Existence And Uniqueness ◽  
Successive Approximations ◽  
Initial Condition ◽  
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Let R denote the rectangle: |t-t0| ≤ a, | x-x0| ≤ b (a,b > 0) in the (t,x) plane and let f(t, x) be a function of two real variables t and x, defined and continuous on R. If I is the interval |t—t0| ≤ d with d = min(a,b/M), where M = max(|f(t, x)|, (t, x) ϵ R), then every solution x = x (t) of the differential equation x' = f(t, x) defined on I and which satisfies the initial condition x(t0) = x0, satisfies the integral equation1.1and conversely. In some cases, in order to prove the existence and uniqueness of the solutions of (1.1) on I, one forms the successive approximations1.2

scholarly journals On a Differential Equation and the Construction of Milner's Lamp

1886 ◽  
Vol 5 ◽  
pp. 99-101 ◽  
Author(s):  
Professor Cayley
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
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What sort of an equation isWriteand start with the equationsThis last givesand the system thus isviz., this is a system of ordinary differential equations between the five variables θ, r, X, Y, Z: the system can therefore be integrated with 4 arbitrary constants, and these may be so determined that for the value β of θ, X, Y, Z shall be each = 0; and r shall have the value r0.

Solutions of ordinary differential equations as limits of pure jump markov processes

1970 ◽  
Vol 7 (1) ◽  
pp. 49-58 ◽  
Author(s):  
Thomas G. Kurtz
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Population Growth ◽  
Ordinary Differential Equations ◽  
Markov Processes ◽  
Radioactive Decay ◽  
Dynamic Processes ◽  
Deterministic Models ◽  
Image Position ◽  
Jump Markov Processes

In a great variety of fields, e.g., biology, epidemic theory, physics, and chemistry, ordinary differential equations are used to give continuous deterministic models for dynamic processes which are actually discrete and random in their development. Perhaps the simplest example is the differential equation used to describe a number of processes including radioactive decay and population growth.

Application of Ambarzumian’s Method to Radiant Interchange in a Rectangular Cavity

1974 ◽  
Vol 96 (2) ◽  
pp. 191-196 ◽  
Author(s):  
A. L. Crosbie ◽  
T. R. Sawheny
Keyword(s):  
Differential Equation ◽  
Heat Flux ◽  
Integral Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Integral Equation ◽  
Initial Conditions ◽  
Rectangular Cavity ◽  
Wide Range ◽  
First Time

Ambarzumian’s method had been used for the first time to solve a radiant interchange problem. A rectangular cavity is defined by two semi-infinite parallel gray surfaces which are subject to an exponentially varying heat flux, i.e., q = q0 exp(−mx). Instead of solving the integral equation for the radiosity for each value of m, solutions for all values of m are obtained simultaneously. Using Ambarzumian’s method, the integral equation for the radiosity is first transformed into an integro-differential equation and then into a system of ordinary differential equations. Initial conditions required to solve the differential equations are the H functions which represent the radiosity at the edge of the cavity for various values of m. This H function is shown to satisfy a nonlinear integral equation which is easily solved by iteration. Numerical results for the H function and radiosity distribution within the cavity are presented for a wide range of m values.

scholarly journals Successive Approximation Method of Integro–Differential Equation With Applications

2019 ◽  
Vol 8 (3) ◽  
pp. 96
Author(s):  
Samir H. Abbas ◽  
Younis M. Younis
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Differential Equations ◽  
Successive Approximation ◽  
Approximation Method ◽  
Existence And Uniqueness ◽  
Successive Approximations ◽  
Successive Approximation Method ◽  
Integro Differential Equation

The aim of this paper is studying the existence and uniqueness solution of integro- differential equations by using Successive approximations method of picard. The results of written program in Mat-Lab show that the method is very interested and efficient with comparison the exact solution for solving of integro-differential equation.

Qualitative properties of nonlinear ordinary differential equations

1977 ◽  
Vol 79 (1-2) ◽  
pp. 79-85 ◽  
Author(s):  
Nelson Onuchic ◽  
Plácido Z. Táboas
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Ordinary Differential Equation ◽  
Point Of View ◽  
Nonlinear Ordinary Differential Equations ◽  
Qualitative Properties ◽  
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SynopsisThe perturbed linear ordinary differential equationis considered. Adopting the same approach of Massera and Schäffer [6], Corduneanu states in [2] the existence of a set of solutions of (1) contained in a given Banach space. In this paper we investigate some topological aspects of the set and analyze some of the implications from a point of view ofstability theory.

X.—The Two Parameter Sturm-Liouville Problem for Ordinary Differential Equations

1971 ◽  
Vol 69 (2) ◽  
pp. 139-148
Author(s):  
B. D. Sleeman
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Liouville Problem ◽  
Sturm Liouville ◽  
Image Position ◽  
Two Parameter ◽  
General Conditions

SynopsisThis paper discusses the existence, under fairly general conditions, of solutions of the two-parameter eigenvalue problem denned by the differential equation,and three point Sturm-Liouville boundary conditions.

The numerical integration of ordinary differential equations possessing exponential type solutions

1960 ◽  
Vol 56 (3) ◽  
pp. 240-246 ◽  
Author(s):  
S. C. R. Dennis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Exponential Type ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Standard Problem ◽  
Difference Series ◽  
The Common ◽  
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Although there are many variations of finite-difference methods of obtaining approximate numerical solutions to ordinary differential equations they share the common feature that they tend to treat an equation of a given type as a standard problem and take no account of any special characteristics the wanted solution may have. We here suggest an alternative procedure when the wanted solution exhibits exponential characteristics. In essence the idea is that if a differential equation has an exponential type solution y(x) it is useful to solve numerically, instead of the equation for y, the equation for u = logey. The error-building and stability characteristics are then those of u rather than y and consequently the accuracy of the solutions may be improved. Although there is nothing basically new in this, of course, the point that we demonstrate is that the differential equation in y can be solved numerically in such a manner that the transformation from y to u is not actually carried out, i.e. we retain the original dependent variable but take account of the exponential variation by modifying the integration formula. Consider for example, in the usual notation, the first-order equationwith a given initial condition y(x0) = y0. If x0, x1, …, xr, xn is a set of pivotal values of x;, usually assumed equally spaced so that xr+1 − xr = h, the usual approach replaces (1) by the formulawhich, once the integral is expressed in terms of pivotal values of f using a difference series, represents a step-by-step formula for constructing successive values of y.

A Minimum-Maximum Problem for Differential Expressions

1957 ◽  
Vol 9 ◽  
pp. 132-140 ◽  
Author(s):  
D. S. Carter
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Condition ◽  
Approximate Methods ◽  
First Order ◽  
Maximum Problem ◽  
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In the study of approximate methods for solving ordinary differential equations, an interesting question arises. To state it roughly for a single first order expression, let y0(t) be the solution of the equation(1.1)which satisfies the initial condition y(a) = na. Let nb be an approximation to the value of y0 at a later time, t = b.

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 ◽  
Image Position

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.

Solutions of ordinary differential equations as limits of pure jump markov processes

1970 ◽  
Vol 7 (01) ◽  
pp. 49-58 ◽  
Author(s):  
Thomas G. Kurtz
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Population Growth ◽  
Ordinary Differential Equations ◽  
Markov Processes ◽  
Radioactive Decay ◽  
Dynamic Processes ◽  
Deterministic Models ◽  
Image Position ◽  
Jump Markov Processes

In a great variety of fields, e.g., biology, epidemic theory, physics, and chemistry, ordinary differential equations are used to give continuous deterministic models for dynamic processes which are actually discrete and random in their development. Perhaps the simplest example is the differential equation used to describe a number of processes including radioactive decay and population growth.

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