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.

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.

Limit theorems for sequences of jump Markov processes approximating ordinary differential processes

1971 ◽  
Vol 8 (02) ◽  
pp. 344-356 ◽  
Author(s):  
T. G. Kurtz
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Markov Processes ◽  
Limit Theorems ◽  
First Order ◽  
Image Position ◽  
Jump Markov Processes

In [3] this author gave conditions under which a sequence of jump Markov processes Xn (t) will converge to the solution X(t) of a system of first order ordinary differential equations, in the sense that for every δ > 0.

Limit theorems for sequences of jump Markov processes approximating ordinary differential processes

1971 ◽  
Vol 8 (2) ◽  
pp. 344-356 ◽  
Author(s):  
T. G. Kurtz
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Markov Processes ◽  
Limit Theorems ◽  
First Order ◽  
Image Position ◽  
Jump Markov Processes

In [3] this author gave conditions under which a sequence of jump Markov processes Xn(t) will converge to the solution X(t) of a system of first order ordinary differential equations, in the sense that for every δ > 0.

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 ◽  
Image Position

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.

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 ◽  
Image Position

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 ◽  
Image Position

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.

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.

Oscillation Criteria For Second Order Superlinear Differential Equations

1989 ◽  
Vol 41 (2) ◽  
pp. 321-340 ◽  
Author(s):  
CH. G. Philos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Continuous Function ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Real Line ◽  
Nonlinear Ordinary Differential Equation ◽  
Second Order ◽  
Oscillation Criteria ◽  
Image Position

This paper is concerned with the question of oscillation of the solutions of second order superlinear ordinary differential equations with alternating coefficients.Consider the second order nonlinear ordinary differential equationwhere a is a continuous function on the interval [t0, ∞), t0 > 0, and / is a continuous function on the real line R, which is continuously differentia t e , except possibly at 0, and satisfies.

Oscillation Theorems for Nonlinear Ordinary Differential Equations of Even Order

1981 ◽  
Vol 24 (4) ◽  
pp. 409-413
Author(s):  
Kurt Kreith ◽  
Takaŝi Kusano
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Ordinary Differential Equations ◽  
Even Order ◽  
Image Position ◽  
Oscillation Theorems

Consider the differential equation1where n is even and f(t, y) is subject to the following conditions:(a) f(t, y) is continuous on [0, ∞)× R;(2) (b) f(t, y) is nondecreasing in y for each fixed t∈[0,∞);(c) yf(t, y ) > 0 for y ≠ 0 and t∈[0,∞).

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