scholarly journals On Generating Functions

1930 ◽  
Vol 2 (2) ◽  
pp. 71-82 ◽  
Author(s):  
W. L. Ferrar
Keyword(s):  
Differential Equation ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Order Differential Equation ◽  
Second Order ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Three Term Recurrence Relation ◽  
Sequence Of Functions

It is well known that the polynomial in x,has the following properties:—(A) it is the coefficient of tn in the expansion of (1–2xt+t2)–½;(B) it satisfies the three-term recurrence relation(C) it is the solution of the second order differential equation(D) the sequence Pn(x) is orthogonal for the interval (— 1, 1),i.e. whenSeveral other familiar polynomials, e.g., those of Laguerre Hermite, Tschebyscheff, have properties similar to some or all of the above. The aim of the present paper is to examine whether, given a sequence of functions (polynomials or not) which has one of these properties, the others follow from it : in other words we propose to examine the inter-relation of the four properties. Actually we relate each property to the generating function.

On the Forced Lienard Equation

1969 ◽  
Vol 12 (1) ◽  
pp. 79-84 ◽  
Author(s):  
R.R. Stevens
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Second Order ◽  
Liénard Equation ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Lienard Equation

We consider the second order differential equation(1)with the assumptions that(2) f(x) is continuous (- ∞ < x < ∞) and p(t) is continuous and bounded: |p(t)| ≤ E, - ∞ < t < ∞.Also, throughout this paper, F(x) denotes an antiderivative of f(x).

Periodic solutions for a second-order differential equation with indefinite weak singularity

2019 ◽  
Vol 149 (5) ◽  
pp. 1135-1152 ◽  
Author(s):  
José Godoy ◽  
Manuel Zamora
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Second Order ◽  
Constant Sign ◽  
Piecewise Constant ◽  
Weak Singularity ◽  
Second Order Differential Equation ◽  
Image Position

AbstractAs a consequence of the main result of this paper efficient conditions guaranteeing the existence of a T −periodic solution to the second-order differential equation $${u}^{\prime \prime} = \displaystyle{{h(t)} \over {u^\lambda }}$$are established. Here, h ∈ L(ℝ/Tℤ) is a piecewise-constant sign-changing function and the non-linear term presents a weak singularity at 0 (i.e. λ ∈ (0, 1)).

On the Uniqueness of the Green's Function Associated with a Second-order Differential Equation

1949 ◽  
Vol 1 (2) ◽  
pp. 191-198 ◽  
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Arbitrary Function ◽  
Order Differential Equation ◽  
Second Order ◽  
Partial Differential ◽  
Second Order Differential Equation ◽  
Image Position

The Green's function G(x, ξ, λ) associated with the differential equation is of importance in the theory of the expansion of an arbitrary function in terms of the solutions of the differential equation. It is proved that this function is unique if q(x) ≧ — Ax2— B, where A and B are positive constants or zero. A similar theorem is proved for the Green's function G(x, y, ξ, η, λ) associated with the partial differential equation

On the stability of a second-order differential equation

1965 ◽  
Vol 5 (1) ◽  
pp. 8-16 ◽  
Author(s):  
D. E. Daykin ◽  
K. W. Chang
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Second Order ◽  
Second Order Differential Equation ◽  
The Stability ◽  
Image Position ◽  
Familiar Form

In this note we discuss the stability at the origin of the solutions of the differential equation where a dot indicates a differentiation with respect to time, and α, β are real-valued functions of any arguments. We tacitly assume that α, β are such that solutions to (1) do in fact exist. Under the transformation equation (1) takes the equivalent familiar form .

The Existence of Continuable Solutions of a Second Order Differential Equation

1977 ◽  
Vol 29 (3) ◽  
pp. 472-479 ◽  
Author(s):  
G. J. Butler
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Real Line ◽  
Order Differential Equation ◽  
Nonlinear Ordinary Differential Equation ◽  
Second Order ◽  
The Real ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Sign Restriction

A much-studied equation in recent years has been the second order nonlinear ordinary differential equationwhere q and f are continuous on the real line and, in addition, f is monotone increasing with yf(y) > 0 for y ≠ 0. Although the original interest in (1) lay largely with the case that q﹛t) ≧ 0 for all large values of t, a number of papers have recently appeared in which this sign restriction is removed.

A mechanical method for the solution of second-order linear differential equations

1931 ◽  
Vol 27 (4) ◽  
pp. 546-552 ◽  
Author(s):  
E. C. Bullard ◽  
P. B. Moon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Order Differential Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mechanical Method ◽  
Order Linear ◽  
Second Order Differential Equation ◽  
Linear Differential

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.

scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

Numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument

2017 ◽  
Vol 23 (2) ◽  
Author(s):  
Muhad H. Abregov ◽  
Vladimir Z. Kanchukoev ◽  
Maryana A. Shardanova
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Second Order ◽  
Deviating Argument ◽  
Second Order Differential Equation

AbstractThis work is devoted to the numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument in minor terms. The sufficient conditions of the one-valued solvability are established, and the a priori estimate of the solution is obtained. For the numerical solution, the problem studied is reduced to the equivalent boundary value problem for an ordinary linear differential equation of fourth order, for which the finite-difference scheme of second-order approximation was built. The convergence of this scheme to the exact solution is shown under certain conditions of the solvability of the initial problem. To solve the finite-difference problem, the method of five-point marching of schemes is used.

Sign in / Sign up

Export Citation Format

Share Document