Second-order Linear Equations Research Articles

The numerical solution of semidiscrete linear evolution problems on the finite interval using the Unified Transform Method

We study a semidiscrete analogue of the Unified Transform Method introduced by A. S. Fokas, to solve initial-boundary-value problems for linear evolution partial differential equations with constant coefficients on the finite interval x ∈ ( 0 , L ) x \in (0,L) . The semidiscrete method is applied to various spatial discretizations of several first and second-order linear equations, producing the exact solution for the semidiscrete problem, given appropriate initial and boundary data. From these solutions, we derive alternative series representations that are better suited for numerical computations. In addition, we show how the Unified Transform Method treats derivative boundary conditions and ghost points introduced by the choice of discretization stencil and we propose the notion of “natural” discretizations. We consider the continuum limit of the semidiscrete solutions and compare with standard finite-difference schemes.

An Algebraic Elimination by Substitution Algorithm for Vehicle–Bridge Interaction Problems

This paper proposes an algorithm to solve vehicle–bridge interaction (VBI) problems. It describes the VBI system in a linear second-order differential-algebraic equation (DAE) manner, incorporating the motions of both the vehicle and the bridge subsystems, together with the contact forces. On account of the conjugation between the VBI system’s motions and contact forces, the key feature of the proposed algorithm is to solve the DAE with the the algebraic method of elimination by substitution, avoiding the singularity of the differential operator. It enables a versatile and consensus solution scheme for different types of contact mechanisms, including the rigid contact model with constraints, the elastic contact model with the compliance method, and the creep tangential contact model, as well as different wheel–rail profiles. Combined with an appropriate numerical integration scheme, it allows an explicit expression of the contact forces, subsequently leading to the partitioned non-iterative analysis of the motions of both the vehicle and the bridge subsystems. In addition, by incorporating the dynamic condensation of wheels’ degrees of freedom into those of upper components in a matrix form, i.e. a reduced system approach, the proposed algorithm can avoid the numerical drift associated with the kinematic constraints on the acceleration level. To demonstrate the accuracy of the proposed approach, this study first examines a benchmark example of a single degree of freedom sprung mass model traversing a simply supported bridge. Further, it considers a more realistic problem of one and eight train vehicles crossing three three-span continuous bridges. The comparisons and the agreements with existing algorithms prove the accuracy and computational efficiency of the proposed analysis scheme.

Half-Sweep Refinement of SOR Iterative Method via Linear Rational Finite Difference Approximation for Second-Order Linear Fredholm Integro-Differential Equations

The numerical solutions of the second-order linear Fredholm integro-differential equations have been considered and discussed based on several discretization schemes. In this paper, the new schemes are developed derived on the hybrid of the three-point half-sweep linear rational finite difference (3HSLRFD) approaches with the half-sweep composite trapezoidal (HSCT) approach. The main advantage of the established schemes is that they discretize the differential terms and integral term of second-order linear Fredholm integro-differential equations into the algebraic equations and generate the corresponding linear system. Furthermore, the half-sweep (HS) concept is combined with the refinement of the successive over-relaxation (RSOR) iterative method to create the new half-sweep successive over-relaxation (HSRSOR) iterative method, which is implemented to get the numerical solution of a system of linear algebraic equations. Apart from that, the classical or full-sweep Gauss-Seidel (FSGS) and full-sweep successive over-relaxation iterative (FSSOR) methods are presented, which serve as the control method in this paper. In the end, we employed FSGS, FSRSOR and HSRSOR methods to obtain numerical solutions of three examples and make a detailed comparison from three aspects of the number of iterations, elapsed time and maximum absolute error. Numerical results demonstrate that FSRSOR and HSRSOR methods have lesser iterations, faster elapsed time, and are more accurate than FSGS. In addition, HSRSOR is the most effective of the three methods. To sum up, this paper has successfully proposed the applicability and superiority of the new HSRSOR method based on 3HSLRFD-HSCT schemes.

Reconstruction method of equilibrated flux for a posteriori error estimate of elliptic problems

In this paper, a method which reconstructs an H(div)-conforming local equilibrated flux is presented for equilibrated flux-based a posteriori error estimate for the finite element method of the second-order elliptic problem. The flux is reconstructed in the lowest-order Raviart–Thomas spaces for finite element approximation. For a simplicial mesh, the reconstruction which performed on every element rather than on the patch of the elements of the mesh or on the dual mesh is achieved by solving a third (or fourth)-order linear equations on every element and a second-order linear equations on every edge or face. So, the amount of computational work is small. Numerical examples demonstratex the effectiveness and improvements of our method.

Initial-Boundary Value Problems for Anisotropic Parabolic Equations with Variable Exponents of the Nonlinearity in Unbounded Domains with Conditions at Infinity

We deal with the initial-boundary value problems with some restrictions at infinity for linear and nonlinear anisotropic parabolic second-order equations in unbounded domains with respect to the spatial variables. The weak solutions of our problem in Lebesgue and Sobolev spaces with variable exponents is considered. We prove theorems on the existence and uniqueness of the weak solutions using the method based on Saint- Venant principle, and the monotonicity method. Moreover, we obtain estimate of the weak solutions.

Refinement of SOR Iterative Method for the Linear Rational Finite Difference Solution of Second-Order Fredholm Integro-Differential Equations

The primary objective of this paper is to develop the Refinement of Successive Over-Relaxation (RSOR) method based on a three-point linear rational finite difference-quadrature discretization scheme for the numerical solution of second-order linear Fredholm integro-differential equation (FIDE). Besides, to illuminate the superior performance of the proposed method, some numerical examples are presented and solved by implementing three approaches which are the Gauss-Seidel (GS), the Successive Over-Relaxation (SOR) and the RSOR methods. Lastly, through the comparison of the results, it is verified that the RSOR method is more effective than the other two methods, especially when considering the aspects of the number of iterations and running time.

Estimates of the Fundamental Solution for Elliptic Equation in Divergence Form with Drift

For a linear divergent second-order elliptic equation with uniformly elliptic measurable matrix of the principal coefficients and drift, a condition on the low order coefficients is found, which guarantees the existence of classical two-sided estimates for the fundamental solution.

Solving random fractional second-order linear equations via the mean square Laplace transform: Theory and statistical computing

This paper deals with random fractional differential equations of the form, CD0+αX(t)+AX˙(t)+BX(t)=0, t>0, with initial conditions, X(0)=C0 and X˙(0)=C1, where CD0+αX(t) stands for the Caputo fractional derivative of X(t). We consider the case that the fractional differentiation order is 1<α<2. For the sake of generality, we further assume that C0, C1, A and B are random variables satisfying certain mild hypotheses. Then, we first construct a solution stochastic process, via a generalized power series, which is mean square convergent for all t>0. Secondly, we provide explicit approximations of the expectation and variance functions of the solution. To complete the random analysis and from this latter key information, we take advantage of the Principle of Maximum Entropy to calculate approximations of the first probability density function of the solution. All the theoretical findings are illustrated via numerical experiments.

Inverse, ill-posed problems, essentially different solutions and explicit formulas for solutions

A request for an inverse problem, as well as for an incorrect problem produces tens of millions of answers in the Internet. In the past few decades, hundreds of international conferences on these topics have been held annually. Problems of this kind are quite involved, and their numerical analysis requires the development of special methods and numerical algorithms. Explicit formulas provide the main tool for testing these methods and numerical algorithms. The Cauchy problem for an elliptic equation is a classical ill-posed problem, which serves as a model for many inverse and incorrect problems. In the present paper we give a numerically realizable explicit formula for solving the Cauchy problem in a two-dimensional domain for a general second-order linear elliptic equation with analytic coefficients and the Cauchy analytic data on the analytic boundary.

Non-polynomial Cubic Spline Method for the Solution of Second-order Linear Hyperbolic Equation

Second-order linear hyperbolic equations are solved by using a new three level method based on nonpolynomial spline in the space direction and Taylor expansion in the time direction. Numerical results reveal that three level method based on non-polynomial spline is implemented and effective

New efficient time-stepping schemes for the Navier–Stokes–Cahn–Hilliard equations

In this paper, we construct and analyze a class of efficient time-stepping schemes for the Navier–Stokes–Cahn–Hilliard equation system stemming from phase field modeling of two-phase incompressible flows. The proposed schemes are based on an auxiliary variable approach for the Cahn–Hilliard equation and a delicate treatment of the terms coupling the Navier–Stokes equation and the Cahn–Hilliard equation. The former has been found to be an efficient tool for numerical integration of the Cahn–Hilliard equation in recent years. A detailed comparison with existing schemes is given, and the advantage of the new schemes are emphasized. In the theoretical aspect we rigorously prove that the designed schemes are unconditionally stable in the sense that some kind of energy remains bounded during the time stepping. In the implementation, we show how the schemes can be reformulated into six linear second-order elliptic equations with constant coefficients. The efficiency of the proposed algorithm is verified through several numerical examples.

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.

Computing differential Galois groups of second-order linear q-difference equations

We apply the differential Galois theory for difference equations developed by Hardouin and Singer to compute the differential Galois group for a second-order linear q-difference equation with rational function coefficients. This Galois group encodes the possible polynomial differential relations among the solutions of the equation. We apply our results to compute the differential Galois groups of several concrete q-difference equations, including for the colored Jones polynomial of a certain knot.

A performance assessment of an HDG method for second-order Fredholm integro-differential equation: existence-uniqueness and approximation

In this paper, we obtain the existence--uniqueness of solution to the second-order linear Fredholm integro-differential equation (FIDE) with Dirichlet boundary condition by hybridizable discontinuous Galerkin (HDG) method. A key property of these methods is to eliminate all internal degrees of freedom and to construct a linear system that only includes globally coupled unknowns at the element interfaces. After designing and implementing HDG algorithm, we provide some necessary and sufficient conditions based on the stabilization parameter and kernel function to guarantee the existence-uniqueness of the approximate solution. Then, some numerical examples are carried out to assess the performance of the present method. When comparing with existing some methods in literature, the experimental studies verify the reliability and feasibility of the HDG method for the problem under consideration.

Riccati technique and oscillation of linear second-order difference equations

Riccati technique and oscillation of linear second-order difference equations

Asymptotic iteration method for solving Hahn difference equations

Hahn’s difference operator D_{q;w}f(x) =({f(qx+w)-f(x)})/({(q-1)x+w}), qin (0,1), w>0, xneq w/(1-q) is used to unify the recently established difference and q-asymptotic iteration methods (DAIM, qAIM). The technique is applied to solve the second-order linear Hahn difference equations. The necessary and sufficient conditions for polynomial solutions are derived and examined for the (q;w)-hypergeometric equation.

Hyers—Ulam Stability of Second-Order Linear Dynamic Equations on Time Scales

We investigate the Hyers—Ulam stability (HUS) of certain second-order linear constant coefficient dynamic equations on time scales, building on recent results for first-order constant coefficient time-scale equations. In particular, for the case where the roots of the characteristic equation are non-zero real numbers that are positively regressive on the time scale, we establish that the best HUS constant in this case is the reciprocal of the absolute product of these two roots. Conditions for instability are also given.

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.

Review and Recent Developments on the Perfectly Matched Layer (PML) Method for the Numerical Modeling and Simulation of Elastic Wave Propagation in Unbounded Domains

This review article revisits and outlines the perfectly matched layer (PML) method and its various formulations developed over the past 25 years for the numerical modeling and simulation of wave propagation in unbounded media. Based on the concept of complex coordinate stretching, an efficient mixed displacement-strain unsplit-field PML formulation for second-order (displacementbased) linear elastodynamic equations is then proposed for simulating the propagation and absorption of elastic waves in unbounded (infinite or semi-infinite) domains. Both time-harmonic (frequency-domain) and time-dependent (timedomain) PML formulations are derived for two-and three-dimensional linear elastodynamic problems. Through the introduction of only a few additional variables governed by low-order auxiliary differential equations, the resulting mixed timedomain PML formulation is second-order in time, thereby allowing the use of standard time integration schemes commonly employed in computational structural dynamics and thus facilitating the incorporation into existing displacementbased finite element codes. For computational efficiency, the proposed timedomain PML formulation is implemented using a hybrid approach that couples a mixed (displacement-strain) formulation for the PML region with a classical (displacement-based) formulation for the physical domain of interest, using a standard Galerkin finite element method (FEM) for spatial discretization and a Newmark time scheme coupled with a finite difference (Crank-Nicolson) time scheme for time sampling. Numerical experiments show the performances of the PML method in terms of accuracy, efficiency and stability for two-dimensional linear elastodynamic problems in single-and multi-layer isotropic homogeneous elastic media.

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.