Second-order Linear Difference Research Articles

Numerical analysis of a linear second-order finite difference scheme for space-fractional Allen–Cahn equations

In this paper, we construct a new linear second-order finite difference scheme with two parameters for space-fractional Allen–Cahn equations. We first prove that the discrete maximum principle holds under reasonable constraints on time step size and coefficient of stabilized term. Secondly, we analyze the maximum-norm error. Thirdly, we can see that the proposed scheme is unconditionally energy-stable by defining the modified energy and selecting the appropriate parameters. Finally, two numerical examples are presented to verify the theoretical results.

Properties of critical and subcritical second order self-adjoint linear equations

Abstract We discuss critical and subcritical linear second-order difference equations, and we observe several identities and inequalities which such equations satisfy depending on their coefficients. Later, we investigate the limit behaviour depending on the coefficients of solutions and of the sequences which appear when finding said solutions. We will see that certain identity is preserved in limit under weaker assumptions. Finally, we investigate a class of fourth-order linear difference equations and show that they are always 1-critical.

Riccati technique and oscillation of linear second-order difference equations

Riccati technique and oscillation of linear second-order difference equations

Revisiting Samuelson\u2019s models, linear and nonlinear, stability conditions and oscillating dynamics

In this work, we reconsider the dynamics of a few versions of the classical Samuelson’s multiplier–accelerator model for national economy. First we recall that the classical one with constant governmental expenditure, represented by a linear second-order difference equation, is able to generate oscillations converging to the equilibrium for a wide range of values of the parameters, and give its analytic solution for all the possible cases. A delayed version proposed in the recent literature, represented by a linear third-order difference equation, is also considered. We show that also this model is able to produce converging oscillations, and give a complete analysis of the stability region of the equilibrium. A new simple nonlinear model is proposed, showing that it keeps oscillatory behavior, although coupled with other dynamics related to global effects. Our analysis confirms that the seminal work of Samuelson and simple modifications of it, may give powerful tools in the study of the business cycles.

A full probabilistic solution of a stochastic red blood cells model using RVT technique

Mathematical modeling is one of the most interesting ways to express the problem that describe the dynamics of the number of red blood cells count in the bloodstream of the human body. This problem has been deterministically solved based on continuous or discrete differential models. However, the stochastic models of this problem are rarely available and inadequate. This paper is organized to solve the random homogeneous linear second-order difference equation that describes a stochastic discrete red blood cells model. A complete probabilistic solution of this problem is conducted via applying the random variable transformation technique. This is achieved by deriving the first probability density function of the solution processes. The probabilistic behavior of the steady-state case (when time tends to infinity) is also studied. Moreover, the fundamental statistical measures, related to the stochastic solutions, such as the mean, the variance and the confidence intervals, are obtained. For the sake of clarity, numerical results (for pre-assigned distributions to the model parameters and initial conditions) with conclusions are presented.

A maximum-principle preserving and unconditionally energy-stable linear second-order finite difference scheme for Allen–Cahn equations

In this paper, we propose a new linear second-order finite difference scheme for Allen–Cahn equations. We use a modified Leap-Frog finite difference scheme with stabilized term and a central finite difference scheme for temporal and spatial discretization respectively. It is shown that the discrete maximum principle holds under reasonable constraints on time step size and coefficient of stabilized term. Based on the maximum stability, the maximum-norm error is analyzed. Moreover, we can see that the proposed scheme is unconditionally energy-stable. Finally, a numerical experiment is performed to verify the theoretical results.

On the evaluation of an integral involving the Whittaker [formula omitted] function

The topic of interest is the evaluation of the integral I(α;β)≔∫α+∞W1,β2(x)dxx2withα∈(0,∞)andβ∈ℂsuch thatW1,β(α)=0,where Wa,b(z) denotes the Whittaker W function. The question is brought about by a certain Sturm–Liouville problem: the (discrete) spectrum of the respective Sturm–Liouville operator is captured by β=βk, k∈N, and I(α;β) gives the (positive but finite) “length” of the kth non-normalized eigenfunction. The Sturm–Liouville problem is of importance in financial mathematics, mathematical physics, stochastic processes, and quickest change-point detection. We obtain a new analytic expression for I(α;β). The expression is in the form of a (convergent) series, which can be computed numerically via successive iteration of a homogeneous linear second-order difference equation. We also offer an exact closed-form solution to the difference equation. We conclude with a curious connection of our result to the latest developments in the theory of hypergeometric functions.

Chebyshev polynomials, Catalan numbers, and tridiagonal matrices

We establish a relation between linear second-order difference equations corresponding to Chebyshev polynomials and Catalan numbers. The latter are the limit coefficients of a converging series of rational functions corresponding to the Riccati equation. As the main application, we show a relation between the polynomials $$ \varphi _n(\mu)$$ that are solutions of the problem of commutation of a tridiagonal matrix with the simplest Vandermonde matrix and Chebyshev polynomials.

A discrete and q asymptotic iteration method

ABSTRACTWe introduce a finite difference and q-difference analogues of the Asymptotic Iteration Method of Ciftci, Hall, and Saad. We give necessary, and sufficient condition for the existence of a polynomial solution to a general linear second-order difference or q-difference equation subject to a ‘terminating condition’, which is precisely defined. When a difference or q-difference equation has a polynomial solution, we show how to find the second solution.

An indirect comparison quasi-static calibration method for piezoelectric pressure sensors based on an inverse model

This graph represents the design schematic of an inverse model, which is used to predict the reference pressure curves and reference peak pressures. The inverse model is established using a linear second-order difference equation. It is supposed that p represents the input pressure curve of the calibrated sensor and y represents the voltage response curve of the calibrated sensor. The input pressure curve of the calibrated sensor is the average response curve of the reference pressure sensors. Where y (1) represents the 1-derivative of y , and y (2) represents the 2-derivative of y . • An novel indirect comparison quasi-static calibration method for piezoelectric pressure sensors was proposed. • An inverse model was established to predict reference pressures. • Reference pressure sensors were used only once during the acquisition of modeling data of the inverse model. • The calibration accuracy of the proposed method was compared with the direct comparison quasi-static method. Quasi-static calibration is an effective method to ensure the measurement accuracy of piezoelectric pressure sensors. In this paper, a novel indirect comparison quasi-static calibration method is proposed. An inverse model is established by a linear second-order differential equation and used to predict reference pressure curves and reference peak pressures. The working principle of the drop-weight device and calibration steps are described. Calibration tests show that the inverse model has good prediction accuracy. Compared with the peak pressures measured by the reference pressure sensors, the maximum prediction error of the peak pressures calculated by the inverse model is 0.29%. Reference pressure sensors are used only once during the acquisition of modeling data of the inverse model, which can effectively extend the service life of the reference pressure sensors. Furthermore, the proposed method has good calibration accuracy as well as the direct comparison method.

Invariance of Deficiency Indices of Second-Order Symmetric Linear Difference Equations under Perturbations

This paper focuses on the invariance of deficiency indices of second-order symmetric linear difference equations under perturbations. By applying the perturbation theory of Hermitian linear relations, the invariance of deficiency indices of the corresponding minimal subspaces under bounded and relatively bounded perturbations is built. As a consequence, the invariance of limit types of second-order symmetric linear difference equations under bounded and relatively bounded perturbations is obtained.

Canonical forms and discrete Liouville\u2013Green asymptotics for second-order linear difference equations

Liouville–Green (WKB) asymptotic approximations are constructed for some classes of linear second-order difference equations. This is done starting from certain “canonical forms” for the three-term linear recurrence. Rigorous explicit bounds are established for the error terms in the asymptotic approximations of recessive as well as dominant solutions. The asymptotics with respect to parameters affecting the equation is also discussed. Several illustrative examples are given.

Modelling vibrating nano-strings by lattice, finite difference and Eringen's nonlocal models

This paper focuses on the modelling of a tightly stretched nano-string with elastically supported ends by the central finite difference model (FDM), the lattice model and the Eringen's nonlocal model (ENM). The equivalence between the FDM and the lattice model is confirmed provided that the nodal spacing of FDM is equal to the segmental length of lattice model. The analytical vibration solutions for lattice model or FDM can be obtained by solving the linear second-order finite difference equation derived herein. By matching the vibration solutions for lattice model (or FDM) and ENM, the Eringen's small length scale coefficient e0 is calibrated, which is a constant equal to 0.289, regardless of the vibration modes or boundary conditions. The influence of end lateral spring stiffnesses on the vibration solutions for a taut local string is also discussed. For a taut symmetrically restrained local string with sliding ends, the fundamental frequency always approaches to zero and the second frequency always approaches to π.

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 Solutions to Nonhomogeneous Linear Second-Order Difference Equations

By using some solvability methods and the contraction mapping principle are investigated bounded, as well as periodic solutions to some classes of nonhomogeneous linear second-order difference equations on domains N 0 , Z ∖ N 2 and Z . The case when the coefficients of the equation are constant and the zeros of the characteristic polynomial associated to the corresponding homogeneous equation do not belong to the unit circle is described in detail.

Liouville–Green (WKB) asymptotics for almost-diagonal linear second-order matrix difference equations

A Liouville–Green (or WKB) asymptotic approximation theory is developed for a class of almost-diagonal (‘asymptotically diagonal’) linear second-order matrix difference equations. Rigorous and explicitly computable bounds for the error terms are obtained, the asymptotics being made with respect to both, the index and some parameter affecting the equation. The case of the associated inhomogeneous equations is also considered in detail. Some examples and a number of applications are presented for the purpose of illustration.

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.

Approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations

This paper is concerned with approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations with at least one endpoint in the limit point case. A sufficient condition is firstly given for that the k-th eigenvalue of a self-adjoint subspace (relation) below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of self-adjoint subspaces. Then, by applying it to singular second-order symmetric linear difference equations, the approximation of eigenvalues below the essential spectra is obtained, i.e., for any given self-adjoint subspace extension of the corresponding minimal subspace, its k-th eigenvalue below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of constructed induced regular self-adjoint subspace extensions.

Equivalence of one-dimensional second-order linear finite difference operators

The direct, gauge and projective equivalence problems for one-dimensional second-order linear finite difference operators are solved using the method of equivariant moving frames.

Regular approximations of isolated eigenvalues of singular second-order symmetric linear difference equations

This paper is concerned with regular approximations of isolated eigenvalues of singular second-order symmetric linear difference equations. It is shown that the kth eigenvalue of any given self-adjoint subspace extension is exactly the limit of the kth eigenvalues of the induced regular self-adjoint subspace extensions in the case that each endpoint is either regular or in the limit circle case. Furthermore, error estimates for the approximations of eigenvalues are given in this case. In addition, it is shown that isolated eigenvalues in every gap of the essential spectrum of any self-adjoint subspace extension are exactly the limits of eigenvalues of suitably chosen induced regular self-adjoint subspace extensions in the case that at least one endpoint is in the limit point case.