scholarly journals SOLVING DIFFERENCE EQUATIONS IN SEQUENCES: UNIVERSALITY AND UNDECIDABILITY

2020 ◽  
Vol 8 ◽  
Author(s):  
GLEB POGUDIN ◽  
THOMAS SCANLON ◽  
MICHAEL WIBMER
Keyword(s):  
Difference Equations ◽  
Difference Schemes ◽  
Ground Field ◽  
Standard Difference ◽  
Solutions Of Equations ◽  
Image Position ◽  
Monoid Action

We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (for example, standard difference schemes) and difference equations in functions on words. On the universality side, we prove a version of strong Nullstellensatz for such difference equations under the assumption that the cardinality of the ground field is greater than the cardinality of the monoid and construct an example showing that this assumption cannot be omitted. On the undecidability side, we show that the following problems are undecidable:

Stability of Solutions of Differential-operator and Operator-difference Equations in the Sense of Perturbation of Operators

2006 ◽  
Vol 6 (3) ◽  
pp. 269-290 ◽  
Author(s):  
B. S. Jovanović ◽  
S. V. Lemeshevsky ◽  
P. P. Matus ◽  
P. N. Vabishchevich
Keyword(s):  
Differential Operator ◽  
Difference Equations ◽  
Hyperbolic Equations ◽  
Second Order ◽  
Difference Schemes ◽  
Order Differential Operator ◽  
Stability Of Solutions ◽  
Operator Equations ◽  
The Difference ◽  
Coefficient Stability

Abstract Estimates of stability in the sense perturbation of the operator for solving first- and second-order differential-operator equations have been obtained. For two- and three-level operator-difference schemes with weights similar estimates hold. Using the results obtained, we construct estimates of the coefficient stability for onedimensional parabolic and hyperbolic equations as well as for the difference schemes approximating the corresponding differential problems.

scholarly journals The evaluation of certain continued fractions

1937 ◽  
Vol 30 ◽  
pp. vi-x
Author(s):  
C. G. Darwin
Keyword(s):  
Difference Equation ◽  
Hypergeometric Function ◽  
Difference Equations ◽  
Continued Fractions ◽  
Continued Fraction ◽  
The Difference ◽  
Image Position

1. If the approximate numerical value of e is expressed as a continued fraction the result isand it was in finding the proof that the sequence extends correctly to infinity that the following work was done. First the continued fraction may be simplified by setting down the difference equations for numerator and denominator as usual, and eliminating two out of every successive three equations. A difference equation is thus formed between the first, fourth, seventh, tenth … convergents , and this equation will generate another continued fraction. After a little rearrangement of the first two members it appears that (1) implies2. We therefore consider the continued fractionwhich includes (2), and also certain continued fractions which were discussed by Prof. Turnbull. He evaluated them without solving the difference equations, and it is the purpose here to show how the difference equations may be solved completely both in his cases and in the different problem of (2). It will appear that the work is connected with certain types of hypergeometric function, but I shall not go into this deeply.

On the spectral analysis of self adjoint operators generated by second order difference equations

1991 ◽  
Vol 118 (1-2) ◽  
pp. 139-151 ◽  
Author(s):  
Dale T. Smith
Keyword(s):  
Essential Spectrum ◽  
Difference Equations ◽  
Difference Operator ◽  
Simple Criterion ◽  
Order Difference ◽  
Forward Difference ◽  
Other Information ◽  
Image Position ◽  
Adjoint Operators ◽  
Oscillation Constant

SynopsisIn this paper, I shall consider operators generated by difference equations of the formwhere Δ is the forward difference operator, and a, p, and r are sequences of real numbers. The connection between the oscillation constant of this equation and the bottom of the essential spectrum of self-adjoint extensions of the operator generated by the equation is given, as well as various other information about the spectrum of such extensions. In particular, I derive conditions for the spectrum to have only countably many eigenvalues below zero, and a simple criterion for the invariance of the essential spectrum.

Finite Difference Schemes for Diffusion Problems Based on a Hybrid Perturbation Galerkin Method

2006 ◽  
Author(s):  
James Geer ◽  
John Fillo
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Galerkin Method ◽  
Difference Equations ◽  
Space Variable ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Hybrid Technique ◽  
Finite Difference Equations ◽  
Hybrid Equations

A new technique for the development of finite difference schemes for diffusion equations is presented. The model equations are the one space variable advection diffusion equation and the two space variable diffusion equation, each with Dirichlet boundary conditions. A two-step hybrid technique, which combines perturbation methods, based on the parameter ρ = Δt / (Δx)2, with the Galerkin method, provides a systematic way to develop new finite difference methods, referred to as hybrid equations. The main contributions of this paper include: 1) recovery of classical explicit or implicit finite difference schemes using only the perturbation terms; 2) development of new finite difference schemes, referred to as hybrid equations, which have better stability properties than the classical finite difference equations, permitting the use of larger values of the parameter ρ; and 3) higher order accurate methods, with either O((Δx)4) or O((Δx)6) truncation error, formed by convex linear combinations of the classical and hybrid equations. The solution of the hybrid finite difference equations requires only a tridiagonal equation solver and, hence, does not lead to excessive computational effort.

The Stability of Solutions of Generalized Emden-Fowler Equations

1974 ◽  
Vol 17 (3) ◽  
pp. 397-401 ◽  
Author(s):  
Hugo Teufel
Keyword(s):  
Oscillatory Solution ◽  
Stability Of Solutions ◽  
Oscillatory Solutions ◽  
Monotonicity Properties ◽  
Solutions Of Equations ◽  
The Stability ◽  
Image Position

AbstractThis paper gives several monotonicity properties of all oscillatory solutions of equations with separable and nonseparable nonlinearities which are more general than the Emden- Fowler equations*Principally, if x(t) is an oscillatory solution, conditions are given such that; if a(t)↑ ∞ as t → ∞, then x(t) → 0; and, if a(t) ↓ 0 as t → ∞, then lim sup | x(t) | = ∞.

scholarly journals A stability condition for nth order difference equations

1971 ◽  
Vol 12 (1) ◽  
pp. 24-30
Author(s):  
Russell A. Smith
Keyword(s):  
Difference Equations ◽  
Hermitian Form ◽  
Real Variable ◽  
Necessary Condition ◽  
Quadratic Lyapunov Function ◽  
Globally Asymptotically Stable ◽  
Order Difference ◽  
Constant Coefficients ◽  
Image Position ◽  
Special Case

Consider the system of difference equationsin which the unknown x(t) is a complex m-vector, t is a real variable and a1, …, an are complex m × m matrices whose elements are functions of t, x(t), x(t+1), …, x(t+n – 1). A positive definite hermitian form V(x1x2, …, xn), with constant coefficients, is called a strong autonomous quadratic Lyapunov function (written strong AQLF) of (1) if there exists a constant K > 1 such that K2v(t+1) < v(t) for all non-zero solutions x(t)of (1), where v(t) = V(x(t), x(t+ 1), …, x(t+n —1)). The existence of a strong AQLF is a sufficient condition for the trivial solution x =0 of (1) to be globally asymptotically stable. It is a necessary condition only in the special case of an equation

Continuants and precontinuants

1939 ◽  
Vol 35 (4) ◽  
pp. 548-561 ◽  
Author(s):  
G. T. Bennett
Keyword(s):  
Difference Equations ◽  
Continued Fractions ◽  
Initial Values ◽  
Image Position

1. It is here expedient to ignore the continued fractions from which they are usually derived and to define the simple continuants determined by the parameters a1, a2, a3, …, an, … as given by the chain of difference equationstogether with the initial values u0 = 0, u1 = 1: and these data lead successively to the functionsand

XXIII.—On a Class of Partial Differential Equations

1925 ◽  
Vol 44 ◽  
pp. 242-247 ◽  
Author(s):  
E. L. Ince
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Wave Motion ◽  
Partial Differential ◽  
Symbolic Form ◽  
Independent Variable ◽  
Solutions Of Equations ◽  
The One ◽  
Image Position

In but a few cases are solutions of equations of the typeknown, and even in the simplest cases the most general of the known solutions suffers from the disadvantage of being in a symbolic form. The equations to be studied in the present paper are those which can be derived fromby a change in the independent variable. This equation is the one which most naturally claims the attention of the investigator, after the equation of wave-motionhas been disposed of.

scholarly journals Second Order Equations in Functional Spaces: Qualitative and Discrete Well-Posedness

2015 ◽  
Vol 2015 ◽  
pp. 1-63 ◽  
Author(s):  
A. Ashyralyev ◽  
J. Pastor ◽  
S. Piskarev ◽  
H. A. Yurtsever
Keyword(s):  
Approximate Solution ◽  
Difference Equations ◽  
Second Order ◽  
Difference Schemes ◽  
Well Posedness ◽  
Present Survey ◽  
The Stability ◽  
Well Posed ◽  
Second Order Equations

The present survey contains the recent results on the local and nonlocal well-posed problems for second order differential and difference equations. Results on the stability of differential problems for second order equations and of difference schemes for approximate solution of the second order problems are presented.

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