Linear Second-order Hyperbolic Equation Research Articles

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

On a Method for Constructing the Riemann Function for Partial Differential Equations with a Singular Bessel Operator

A linear second-order hyperbolic equation of two independent variables with a singular Bessel operator is considered. For particular types of such equations, a detailed literature review of known methods for constructing Riemann functions is given. It is shown that to construct the Riemann function for equations with a singular Bessel operator, we can use the Erdelyi–Kober transmutation operator. The Riemann function for the Euler–Poisson–Darboux differential equations is found in explicit form. In this paper, we give examples and an algorithm for constructing the Riemann function for second-order hyperbolic equations with the Bessel operator.

A collocation approach for solving two-dimensional second-order linear hyperbolic equations

In this study, a collocation approach is introduced to solve second-order two-dimensional hyperbolic telegraph equation under the initial and boundary conditions. The method is based on the Bessel functions of the first kind, matrix operations and collocation points. The method is constructed in four steps for the considered problem. In first step we construct the fundamental relations for the solution method. By using the collocation points and matrix operations, second step gives the constructing of the main matrix equation. In third step, matrix forms are created for the initial and boundary conditions. We compute the approximate solutions by combining second and third steps. Algorithm of the proposed method is given. Later, error estimation technique is presented and the approximate solutions are improved. Numerical applications are included to demonstrate the validity and applicability of the presented method.

Symmetries of Fundamental Solutions and Their Application in Continuum Mechanics

An application of the symmetries of fundamental solutions in continuum mechanics is presented. It is shown that the Riemann function of a second-order linear hyperbolic equation in two independent variables is invariant with respect to the symmetries of fundamental solutions, and a method is proposed for constructing such a function. A fourth-order linear elliptic partial differential equation is considered that describes the displacements of a transversely isotropic linear elastic medium. The symmetries of this equation and the symmetries of the fundamental solutions are found. The symmetries of the fundamental solutions are used to construct an invariant fundamental solution in terms of elementary functions.

Error Estimates for Semi-Discrete and Fully Discrete Galerkin Finite Element Approximations of the General Linear Second-Order Hyperbolic Equation

ABSTRACTIn this article, we derive error estimates for the semi-discrete and fully discrete Galerkin approximations of a general linear second-order hyperbolic partial differential equation with general damping (which includes boundary damping). The results can be applied to a variety of cases (e.g. vibrating systems of linked elastic bodies). The results generalize pioneering work of Dupont and complement a recent article by Basson and Van Rensburg.

An unconditionally stable spatial sixth-order CCD-ADI method for the two-dimensional linear telegraph equation

The telegraph equation is one of the important models in many physics and engineering. In this work, we discuss the high-order compact finite difference method for solving the two-dimensional second-order linear hyperbolic equation. By using a combined compact finite difference method for the spatial discretization, a high-order alternating direction implicit method (ADI) is proposed. The method is O(ź2 + h6) accurate, where ź, h are the temporal step size and spatial size, respectively. Von Neumann linear stability analysis shows that the method is unconditionally stable. Finally, numerical examples are used to illustrate the high accuracy of the new difference scheme.

Boundary value problems for a weakly loaded operator of a second-order hyperbolic equation in a cylindrical domain

By using methods of functional analysis, we prove the existence and uniqueness, in appropriate function spaces, of solutions of boundary value problems for a second-order linear hyperbolic equation weakly loaded by a differential operator. The order of the operator in the loaded term with respect to its derivatives is equal to 2.

High-accuracy finite-difference schemes for solving elastodynamic problems in curvilinear coordinates within multiblock approach

We propose highly accurate finite-difference schemes for simulating wave propagation problems described by linear second-order hyperbolic equations. The schemes are based on the summation by parts (SBP) approach modified for applications with violation of input data smoothness. In particular, we derive and implement stable schemes for solving elastodynamic anisotropic problems described by the Navier wave equation in complex geometry. To enhance potential of the method, we use a general type of coordinate transformation and multiblock grids. We also show that the conventional spectral element method (SEM) can be treated as the multiblock finite-difference method whose blocks are the SEM cells with SBP operators on GLL grid.

Some properties and applications of the Riemann and Green-Hadamard functions for linear second-order hyperbolic equations

We study some properties of the Riemann and Green-Hadamard functions for linear second-order hyperbolic equations of general form. We consider their sign definiteness and symmetry in a certain sense and prove a comparison type theorem. Some applications are presented.

Solvability of some spatially nonlocal boundary value problems for second-order linear hyperbolic equations

Solvability of some spatially nonlocal boundary value problems for second-order linear hyperbolic equations

An unconditionally stable spline difference scheme of [formula omitted] for solving the second-order 1D linear hyperbolic equation

In this paper, the second-order linear hyperbolic equation is solved by using a new three-level difference scheme based on quartic spline interpolation in space direction and finite difference discretization in time direction. Stability analysis of the scheme is carried out. The proposed scheme is second-order accurate in time direction and fourth-order accurate in space direction. Finally, numerical examples are tested and results are compared with other published numerical solutions.

Contact-equivalence problem for linear hyperbolic equations

We consider the local equivalence problem for the class of linear second-order hyperbolic equations in two independent variables under an action of the pseudo-group of contact transformations. É. Cartan’s method is used for finding the Maurer-Cartan forms for symmetry groups of equations from the class and computing structure equations and complete sets of differential invariants for these groups. The solution of the equivalence problem is formulated in terms of these differential invariants.

Carleman Estimates for Second-Order Hyperbolic Equations

In the space of variables (x, t) ∈ ℝ n+1, we consider a linear second-order hyperbolic equation with coefficients depending only on x. Given a domain D ⊂ ℝ n+1 whose projection to the x-space is a compact domain Ω, we consider the question of construction of a stability estimate for a solution to the Cauchy problem with data on the lateral boundary S of D. The well-known method for obtaining such estimates bases on the Carleman estimates with an exponential-type weight function exp(2τϕ(x, t)) whose construction faces certain difficulties in case of hyperbolic equations with variable coefficients. We demonstrate that if D is symmetric with respect to the plane t = 0 then we can take ϕ(x, t) to be the function ϕ(x, t) = s 2(x, x 0) − pt 2, where s(x, x 0) is the distance between points x and x 0 in the Riemannian metric induced by the differential equation, p is some positive number less than 1, and the fixed point x 0 can either belong to the domain Ω or lie beyond it. As for the metric, we suppose that the sectional curvature of the corresponding Riemannian space is bounded above by some number k 0 ≥ 0. In case of space of nonpositive curvature the parameter p can be taken arbitrarily close to 1; in this case as p → 1 the stability estimates lead to a uniqueness theorem which describes exactly the domain of the solution continuation through S. It turns out that, in case of space of bounded positive curvature, construction of a Carleman estimate is possible only if the product of k 0 and sup x∈Ω s 2(x, x 0) satisfies some smallness condition.

TIME AND SPACE ADAPTIVITY FOR THE SECOND-ORDER WAVE EQUATION

The aim of this paper is to show that, for a linear second-order hyperbolic equation discretized by the backward Euler scheme in time and continuous piecewise affine finite elements in space, the adaptation of the time steps can be combined with spatial mesh adaptivity in an optimal way. We derive a priori and a posteriori error estimates which admit, as much as it is possible, the decoupling of the errors committed in the temporal and spatial discretizations.

Energy estimates for solutions of the mixed problem for linear second-order hyperbolic equations

A mixed problem for a linear second-order hyperbolic equation with antidissipation inside the domain and dissipation on a part of the boundary is considered. It is proved that for certain relations between the antidissipation inside the domain and the dissipation on the part of the boundary, the energy of the system exponentially decreases, whereas for sufficiently large antidissipation inside the domain the boundary dissipation has no effect on the energy of the system; in this case the energy remains unbounded.

Exceptional hyperbolic systems of Hamiltonian form

Some years ago, Bluman [1] gave the ‘integrability conditions’ for a linear second-order hyperbolic Equation, i.e. the conditions under which it can be mapped invertibly to a constant coefficient wave equation. Considering that a second-order hyperbolic equation in two dimensions can be regarded as the hodograph representation of a 2x2 quasi-linear homogeneous system, one may wonder how the fulfilment of the above-mentioned integrability conditions is reflected in the structure of the quasi-linear system. The question is especially interesting if the quasi-linear system derives from a Hamiltonian density.

The existence of a classical solution to the mixed problem for a linear second-order hyperbolic equation

The paper deals with the problem of solvability of the mixed problem for a linear second-order hyperbolic partial differential equation. The minimal necessary and sufficient conditions for the existence of a unique classical solution to this problem are established.

Norm estimates in besov and Lizorkin-Triebel spaces for the solutions of second-order linear hyperbolic equations

In the paper one considers the nonhomogeneous hyperbolic equation $$\partial _t^2 u + iB\left( t \right)\partial _t u + A\left( t \right)u = h$$ on\(\left[ {0, T} \right] \times \mathfrak{M}\), where\(\mathfrak{M}\)=Rn or\(\mathfrak{M}\) is a smooth closed manifold, A(t) and B(t) are pseudodifferential operators on\(\mathfrak{M}\), depending on t e [0, T], of orders 2 and 1, respectively. For the solutions of equation (1) for small t one establishes estimates of the form With arbitrary r eR and integer l≥0, where for G:,. and E:,. one can take the Besov space B:,.\(\left( \mathfrak{M} \right)\) or the Lizorkin-Triebet space F:,.\(\left( \mathfrak{M} \right)\), depending on the values n, v, p, q1 q2, and the “Brenner number” m, which are determined from the principal symbols of the operators A(0) and B(0); also the actual form of the scalar function σv,p, n(t) depends on n, v, p, q1 q2, and m: it may be power-like ∣t∣v−n+2n/p, or logarithmic ¦log¦t¦∥, or a constant. In addition, one obtains estimates of the form characterizing the integrability properties over timespace and the smoothing (for t>0) of the solutions of equation (1).

Some boundary-value problems for linear multidimensional second-order hyperbolic equations

For the linear hyperbolic equations $$\sum\limits_{i,j = 1}^{m + 1} {a_{ij} \left( {x,x_{m + 1} } \right)u_{x_i x_j } + \sum\limits_{i = 1}^{m + 1} {a_i \left( {x,x_{m + 1} } \right)u_{x_i } + c\left( {x,x_{m + 1} } \right)u = 0,x = \left( {x_1 ,...,x_m } \right)} ,} m \geqslant 2,$$ the correctness of multidimensional analogues of the problems of Darboux and Goursat is established and a theorem on the uniqueness of a solution of the Cauchy characteristic problem is proved.

Absorbing boundary conditions for second-order hyperbolic equations

A uniform approach to construct absorbing artificial boundary conditons for second-order linear hyperbolic equations is proposed. The nonlocal boundary condition is given by a pseudodifferential operator that annihilates travelling waves. It is obtained through the dispersion relation of the differential equation by requiring that the initial-boundary value problem admits the wave solutions travelling in one direction only. Local approximations of this global boundary condition yield an nth-order differential operator. It is shown that the best approximations must be in the canonical forms which can be factorized into first-order operators. These boundary conditions are perfectly absorbing for wave packets propagating at certain group velocities. A hierarchy of absorbing boundary conditions is derived for transonic small perturbation equations of unsteady flows. These examples illustrate that the absorbing boundary conditions are easy to derive, and the effectiveness is demonstrated by the numerical experiments.