Linear First-order Difference Equations Research Articles

Analytical Calculation of Static Deflection of Biperiodic Stepped Euler–Bernoulli Beam

In this paper, we investigate the lateral deflection of a simply supported periodic stepped beam under uniform load by using an analytical method. This study considers each element of the biperiodic stepped beam as a Euler–Bernoulli beam. By using the local coordinates alongside with the boundary and continuity conditions, the different coefficients for each element caused by the jump of the bending rigidity are calculated. The continuous deflection problem of the multi-stepped repetitive beam is formulated as a linear first-order difference equation with second member. With these coefficients, the deflection at mid-span of the biperiodic beam is analytically found in exact form. This deflection is satisfactory compared to the results of a finite element model based on beam discretization techniques using Hermitian cubic shape functions. The normalized deflection at mid span converges non-monotonically towards the homogenization beam model based on equivalent homogenized stiffness.

Modeling by Using Linear First-Order Difference Equations and its Application in Economics

This research discusses the modeling by using linear difference equations that conclude to understand and explain different economic phenomena, we have divided this research into two parts, the first one focused in the Linear First-Order Difference Equations with Constant Coefficients with example like explaining the “Cobweb” model to study the relationship between supply & demand and Harrod model to study the relationship between national income and economic growth. As for the second one was about case study: entitled modeling the impact of government support on the budget balance in Algeria.
 We have founded that using modeling by using linear first-order difference equations with constant coefficients offers opportunities for the researchers in economics to model Economic phenomena in equations, and by trying to solve those equations to reach logic results, to eventually analyzingand explaining those results based on mathematical model.

Solvability of a one-parameter class of nonlinear second-order difference equations by invariants

By using an invariant we show in an original and quite unexpected way that a one-parameter class of nonlinear second-order difference equations is solvable in closed form, improving and theoretically explaining a recent result in the literature. As a motivation for this and also for general use of invariants for difference equations in solvability, we also demonstrate an application of the method of invariants on a very basic example of linear first-order difference equation with constant coefficients explaining a frequently confused detail. We also give a general hint how the method can be applied for the case of difference equations of any order.

Testing Rational Addiction: When Lifetime is Uncertain, One Lag is Enough

The rational addiction model is usually tested by estimating a linear second-order difference Euler equation, which may produce unreliable estimates. We show that a linear first-order difference equation is a better alternative. This empirical specification is appropriate under the reasonable assumption that people are uncertain about the time of their death, it is based on the same structural assumptions used in the literature, and it retains all policy implications of the deterministic rational addiction model. It is also empirically convenient because it is simple, it allows using efficient estimation strategies that do not require instrumental variables, and it is robust to the possible non-stationarity of the data. As an application, we estimate the demand for smoking in the US from 1970 to 2016, and we show that it is consistent with the rational addiction model.

Bounded and periodic solutions to the linear first-order difference equation on the integer domain

The existence of bounded solutions to the linear first-order difference equation on the set of all integers is studied. Some sufficient conditions for the existence of solutions converging to zero when nto -infty, as well as when nto+infty, are also given. For the case when the coefficients of the equation are periodic, the long-term behavior of non-periodic solutions is studied.

Existence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equation

We present some interesting facts connected with the following second-order difference equation: xn+2−qnxn=fn,n∈N0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$x_{n+2}-q_{n}x_{n}=f_{n},\\quad n\\in \\mathbb{N}_{0}, $$\\end{document} where (q_{n})_{ninmathbb{N}_{0}} and (f_{n})_{ninmathbb {N}_{0}} are given sequences of numbers. We give some sufficient conditions for the existence of a unique bounded solution to the difference equation and present an elegant proof based on a combination of theory of linear difference equations and the Banach fixed point theorem. We also deal with the equation by using theory of solvability of difference equations. A global convergence result of solutions to a linear first-order difference equation is given. Some comments on an abstract version of the linear first-order difference equation are also given.

Solution of Stochastic Non-Homogeneous Linear First-Order Difference Equations

In this paper, the closed form solution of the non-homogeneous linear first-order difference equation is given. The studied equation is in the form: xn = x0 + bn, where the initial value x0 and b, are random variables.

Dynamics of partisan representation the American south, 1898–2010

This study addresses institutional representation in legislative delegations through the decomposition of the southern U.S. House delegation over time. Linear first-order difference equations are calculated to show the shift from the Solid South and the disintegration of Democratic dominance. These calculations also show that the qualitative behavior of partisan control varies over time given a series of critical events, including the Dixiecrat experience, the Congressional reforms of the 1970s, and the Republican Revolution of 1994. However, I also argue that the Republican Revolution was actually predictable, given the twentieth-century experience of the southern delegation.

Dynamics of Southern Partisan Representation: 1898-2005

This study addresses institutional representation in legislative delegations through the decomposition of the southern U.S. House delegation over time. Linear first-order difference equations are calculated to show the shift from the Solid South and the disintegration of Democratic dominance. These calculations also show that the qualitative behavior of partisan control varies over time given a series of critical events, including the Dixiecrat experience, the Congressional reforms of the 1970s, and the Republican Revolution of 1994. However, I also argue that the Republican Revolution was actually predictable, given the twentieth-century experience of the southern delegation.

Long-Wave Extensional Buckling of Large Regular Frames

In very high and slender frames with very stiff members, the safety against long-wave buckling modes involving axial extensions requires investigation. The problem is solved exactly for a planar regular rectangular frame of rectangular boundary, compressed between two sliding rigid half spaces. The problem is reduced to a system of six linear first-order difference equations which are solved in terms of discrete complex exponentials. Because the coefficients of the determinant to varnish depend on axial loads nonlinearly, by means of explicitly inexpressible complex roots and eigenvectors of another 6 times 6 matrix and the well-known nonlinear stability functions, the critical load is determined numerically by a procedure analogous to the method regula falsi. The ratio of column critical force to column Euler load depends eessentially on the ratio of column and beam bending stiffnesses, the ratio of building slenderness to column slenderness, and weakly on column slenderness ratio.

Recursive computation of certain derivatives&a study of error propagation

A brief study is made of the propagation of errors in linear first-order difference equations. The recursive computation of successive derivatives of e x / x and (cos x )/ x is considered as an llustration.