Second-order Linear Equations Research Articles

The posteriori analysis of the spectral element discretization of the wave equation

This work focuses on discretizing a second-order linear wave equation using the implicit Euler scheme for time discretization and the spectral element method for spatial discretization. We prove that optimal adaptivity can be achieved by combining adaptive time steps with adaptive spectral mesh. We introduce two sets of error indicators for time and space, respectively, and derive optimal estimates.

On the Boyarsky-Meyers estimate for the solution of the Dirichlet problem for a second-order linear elliptic equation with drift

We establish the increased integrability of the gradient of the solution to the Dirichlet problem for the Laplace operator with lower terms and prove the unique solvability of this problem.

The Lattice Boltzmann Method Using Parallel Computation: A Great Potential Solution for Various Complicated Acoustic Problems

This paper proposes the D2Q5 Lattice Boltzmann method (LBM) method, in two dimensions with five discrete lattice velocities, for simulating linear sound wave propagation in closed rooms. A second-order linear acoustic equation obtained from the LBM method was used as the model equation. Boundary conditions at the domain boundary use the bounce-back scheme. The LBM numerical calculation algorithm in this paper is relatively simpler and easy to implement. Parallelization with the GPU CUDA was implemented to speed up the execution time. The calculation results show that the use of parallel GPU CUDA programming can accelerate the proposed simulation 27.47 times faster than serial CPU programming. The simulation results are validated with analytical solutions for acoustic pulse reflected by the flat and oblique walls, the comparisons show very good concordance, and the D2Q5 LBM has second-order accuracy. In addition, the simulation results in the form of wavefront propagation images in complicated shaped rooms are also compared with experimental photographs, and the comparison also shows excellent concordance. The numerical results of the D2Q5 LBM are promising and also demonstrate the great capability of the D2Q5 LBM for investigating room acoustics in various complexities.

Integral Transforms and the Hyers–Ulam Stability of Linear Differential Equations with Constant Coefficients

Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential equations with constant coefficients, utilizing the Kamal transform, where we focus on first- and second-order linear equations. In particular, in this work, we employ the Kamal transform to determine the Hyers–Ulam stability and Hyers–Ulam stability constants for first-order complex constant coefficient differential equations and, for second-order real constant coefficient differential equations, improving previous results obtained by using the Kamal transform. In a section of examples, we compare and contrast our results favorably with those established in the literature using means other than the Kamal transform.

ОБ АСИМПТОТИЧЕСКОЙ УСТОЙЧИВОСТИ РЕШЕНИЙ ЛИНЕЙНОГО ВОЛЬТЕРРОВА ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ ВТОРОГО ПОРЯДКА В СЛУЧАЕ НЕДИФФЕРЕНЦИРУЕМОСТИ СРЕЗАННЫХ ФУНКЦИЙ

Sufficient conditions are established for the asymptotic stability of any solution of a linear second-order Volterra integro-differential equation in the case when the truncated functions may be non-differentiable at some points of the semiaxis. To do this, first, using a non-standard method of reduction to a system, the equation under consideration is reduced to an equivalent system, and the method of transforming equations and the method of cutting functions are applied to this system. The truncated kernel and free term are associated with the coefficient of the new unknown function using the inequality between the geometric and arithmetic means of two functions. Using the method of integral inequalities, sufficient conditions are established for the boundedness on the semiaxis of all solutions of the system. Then the method of squaring equations is used to the first equation of the system and the Lyusternik-Sobolev lemma is applied. Finally, based on a non-standard substitution, we complete the proof of the asymptotic stability of solutions to a given second-order linear integro-differential equation. An illustrative example is given.

L2 estimates for weak Galerkin finite element methods for second-order wave equations with polygonal meshes

In this paper, we describe weak Galerkin finite element methods for solving hyperbolic problems on polygonal meshes. We propose both semidiscrete and fully discrete schemes to numerically solve the second-order linear wave equation. For the time discretization, we have used implicit second order Newmark scheme. For sufficiently smooth solutions, optimal order error estimate in the L2 norm is shown to hold as O(hk+1+τ2), where h is the mesh size and τ the time step. An extensive set of numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the proposed method.

Operator estimates for non-periodically perforated domains: disappearance of cavities

ABSTRACT We consider a boundary value problem for a general second-order linear equation in a perforated domain. The perforation is made by small cavities, a minimal distance between the cavities is also small. We impose minimal natural geometric conditions on the shapes of the cavities and no conditions on their distribution in the domain. On the boundaries of the cavities, a nonlinear Robin condition is imposed. The sizes of the cavities and the minimal distance between them are supposed to satisfy a certain simple condition ensuring that under the homogenization the cavities disappear and we obtain a similar problem in a non-perforated domain. Our main results state the convergence of the solution of the perturbed problem to that of the homogenized one in W 2 1 - and L 2 -norms uniformly in L 2 -norm of the right-hand side in the equation and provide the estimates for the convergence rates.

Mechanical Analysis and Optimal Design of Wave Energy Device

At present, the global energy problem is serious. Wave energy is the kinetic energy and potential energy of waves on the ocean surface. Because of its high energy density, wide distribution, and long continuous operation of the device, it has become the focus of academic research. The key problem to be solved urgently for the large-scale application of wave energy is how to optimize the energy conversion efficiency of wave energy devices. This paper starts from the Newtonian dynamic equation, analyzes the motion state of the wave energy device in different wave environments, and studies the optimization and control of its power output. This paper first adopts the analysis method of first whole and then isolation, and establishes a one-dimensional coordinate system with the zero point of sea level. Then, the force analysis of the system is carried out, the dynamic differential equation is established, and the initial draft of the system is solved through the initial balance equation. Since the dynamic equation is a second-order linear ordinary differential inhomogeneous linear equation, the fourth-order Runge is used. The Kutta method is used to solve the problem, and the numerical solution for the position and velocity of the float is obtained. Secondly, the movable coordinate system is established with the float as the reference system, and the force analysis of the oscillator is carried out to obtain its relative dynamic equation. Then, the initial relative position of the oscillator is solved through the initial balance equation, and the relative dynamic differential equation is obtained by using the fourth-order Runge-Kutta method to obtain the numerical solution of the relative position and relative velocity of the oscillator with respect to time. Finally, the numerical solution of the absolute position and absolute velocity of the oscillator with respect to time is obtained through coordinate transformation.

The method of stochastic characteristics for linear second-order hypoelliptic equations

We study hypoelliptic stochastic differential equations (SDEs) and their connection to degenerate-elliptic boundary value problems on bounded or unbounded domains. In particular, we provide probabilistic conditions that guarantee that the formal stochastic representation of a solution is smooth on the interior of the domain and continuously approaches the prescribed boundary data at a given boundary point. The main general results are proved using fine properties of the process stopped at the boundary of the domain combined with hypoellipticity of the operators associated to the SDE. The main general results are then applied to deduce properties of the associated Green’s functions and to obtain a generalization of Bony’s Harnack inequality. We moreover revisit the transience and recurrence dichotomy for hypoelliptic diffusions and its relationship to invariant measures.

Explicit Richardson extrapolation methods and their analyses for solving two-dimensional nonlinear wave equation with delays

<abstract><p>In this study, we construct two explicit finite difference methods (EFDMs) for nonlinear wave equation with delay. The first EFDM is developed by modifying the standard second-order EFDM used to solve linear second-order wave equations, of which stable requirement is accepted. The second EFDM is devised for nonlinear wave equation with delay by extending the famous Du Fort-Frankel scheme initially applied to solve linear parabolic equation. The error estimations of these two EFDMs are given by applying the discrete energy methods. Besides, Richardson extrapolation methods (REMs), which are used along with them, are established to improve the convergent rates of the numerical solutions. Finally, numerical results confirm the accuracies of the algorithms and the correctness of theoretical findings. There are few studies on numerical solutions of wave equations with delay by Du Fort-Frankel-type scheme. Therefore, a main contribution of this study is that Du Fort-Frankel scheme and a corresponding new REM are constructed to solve nonlinear wave equation with delay, efficiently.</p></abstract>

Numerical wave propagation aided by deep learning

We propose a deep learning approach for wave propagation in media with multiscale wave speed, using a second-order linear wave equation model. We use neural networks to enhance the accuracy of a given inaccurate coarse solver, which under-resolves a class of multiscale wave media and wave fields of interest. Our approach involves generating training data by the given computationally efficient coarse solver and another sufficiently accurate solver, applied to a class of wave media (described by their wave speed profiles) and initial wave fields. We find that the trained neural networks can approximate the nonlinear dependence of the propagation on the wave speed as long as the causality is appropriately sampled in training data. We combine the neural-network-enhanced coarse solver with the parareal algorithm and demonstrate that the coupled approach improves the stability of parareal algorithms for wave propagation and improves the accuracy of the enhanced coarse solvers.

A Solution to Linear Black-Scholes Second-order Parabolic equation in sobolev space

In this paper, a set of functions were constructed that transforms Black-Scholes partial differential equation into weak formulations. The analytical solutions: existence, uniqueness and other estimates were also obtained in weak form with the use of boundary conditions to establish the effects of its financial implications in Sobolev spaces. The regularity conditions of the problem were considered which the coefficients, the boundary of the domain are all smooth functions. To this end, the definitions, assumptions which paved way to useful assertions are illustrated in this paper.

On criticality theory for elliptic mixed boundary value problems in divergence form

The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain [Formula: see text] that satisfy an oblique boundary condition on a portion of [Formula: see text]. First, we study weak solutions for the degenerate mixed boundary value problem [Formula: see text] where [Formula: see text] is a bounded Lipschitz domain, [Formula: see text] is a relatively open portion of [Formula: see text], and [Formula: see text] is an oblique (Robin) boundary operator defined on [Formula: see text] in a weak sense. In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a minimal positive Green function. Then we establish a criticality theory for positive weak solutions of the operator [Formula: see text] in a general domain [Formula: see text] with no boundary condition on [Formula: see text] and no growth condition at infinity. The paper extends results obtained by Pinchover and Saadon for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of [Formula: see text], and the boundary [Formula: see text] are assumed.

Solving the nonlinear Riccati equation for the 1D plane-wave reflection response from a velocity gradient interface by a Heun to hypergeometric reduction

For 1D acoustic wave propagation in the frequency domain, the complex nonlinear Riccati differential equation describes the dependence of the plane-wave reflection response in depth, defined as the upgoing part of the wavefield divided by the downgoing part of the wavefield at that depth. In the case that the model is a velocity gradient interface expressed in terms of a smooth Heaviside function defined by the Fermi-Dirac function, the Riccati equation is shown to have an analytical solution for the reflection response. The solution accounts for all wave phenomena in the gradient zone. The solution of the Riccati equation for the reflection response is obtained by introducing the Riccati equation for the reciprocal of the reflection response which can be related to a second-order self-adjoint linear differential equation. The two latter equations obey the same radiation condition. Furthermore, a connection between the second-order linear equation and the Heun equation is obtained, both being characterized by having four regular singular points. By a change of variables, the Heun equation can be transformed into Gauss’ hypergeometric equation, whereby the solution is expressed as the sum of two linearly independent functions containing hypergeometric functions. The advantage of applying the Heun-to-hypergeometric reduction to the nonlinear Riccati equation is that solutions to the hypergeometric equation are well known. The two functions allow us to analytically determine the Riccati solution because the ratio of the linearly independent functions at infinity is constant. In contrast to the classic frequency-independent reflection coefficient from two layers in welded contact, the reflection coefficient at the surface associated with the gradient interface depends on frequency. In the limit of zero frequency, the reflection coefficient approaches the classic one. We have implemented the iterative solution of the integral form of the Riccati equation and found that the solution builds up nonlinearly to the correct solution. The model is verified by a numerical example valid for 1D wave propagation.

On meromorphic solutions of the equations related to the first Painlevé equation

In this paper, we consider the generalised hierarchy of the first Painlevé equation which is a sequence of polynomial ordinary differential equations of even order that have a uniform differential-algebraic structure determined by the operator L~n. The first member of this hierarchy for n = 2 is the first Painlevé equation, and the subsequent equations of order 2n – 2 contain arbitrary parameters. They are named as higher analogues of the first Painlevé equation of 2n – 2 order. The article considers the analytical properties of solutions to the equations of the generalised hierarchy of the first Painlevé equation and the related linear equations. It is established that each hierarchy equation has one dominant term, and an arbitrary meromorphic solution of any hierarchy equation cannot have a finite number of poles. The character of the mobile poles of meromorphic solutions is determined. Using the Frobenius method, sufficient conditions are obtained for the meromorphicity of the general solution of the second-order linear equations with a linear potential defined by meromorphic solutions of the first three equations of the hierarchy.

On the numerical solution of second-order stiff linear differential-algebraic equations

This article addresses systems of linear ordinary differential equations with an identically degenerate matrix in the main part. Such formulations of problems in literature are usually called differential-algebraic equations. In this work, attention is paid to the problems of the second order. Basing on the theory of matrix pencils and polynomials, sufficient conditions for existence and uniqueness of the equations’ solution are given. To solve them numerically, authors investigate a multistep method and its version based on a reformulated notation of the original problem. This representation makes it possible to construct methods whose coefficient matrices can be calculated at previous points. This approach has delivered good results in numerical solution of first-order differential-algebraic equations that contain stiff and rapidly oscillating components and have singular matrix pencil. The stability of proposed numerical algorithm is investigated for the well-known test equation. It is shown that this difference scheme has the first order of convergence. Numerical calculations of the model problem are presented.

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.