Boundary Value Problems For Hyperbolic Equations Research Articles

Collocation and Least Residuals Method and Its Applications

The collocation and least residuals (CLR) method combines the methods of collocations (CM) and least residuals. Unlike the CM, in the CLR method an approximate solution of the problem is found from an overdetermined system of linear algebraic equations (SLAE). The solution of this system is sought under the requirement of minimizing a functional involving the residuals of all its equations. On the one hand, this added complication of the numerical algorithm expands the capabilities of the CM for solving boundary value problems with singularities. On the other hand, the CLR method inherits to a considerable extent some convenient features of the CM. In the present paper, the CLR capabilities are illustrated on benchmark problems for 2D and 3D Navier–Stokes equations, the modeling of the laser welding of metal plates of similar and different metals, problems investigating strength of loaded parts made of composite materials, boundary-value problems for hyperbolic equations.

Modeling of Wave Propagation in Degenerate Hyperbolic Systems

Degeneration in a small spatial coordinate of the original initial---boundary-value problem for hyperbolic equations is considered in order to construct simpler models. The problem is investigated in a domain whose one spatial scale is much less than the other scales. This makes it possible to expand the desired functions in power series and hence to reduce the dimension of the problem. However, this comes at the cost of degeneration of the spectrum of the original three-dimensional problem.

Asymptotic integration of degenerate singularly perturbed systems of parabolic partial differential equations

Eidel’man, Butuzov, and Ivasishen. In particular, Ladyzhenskaya and Ural’tseva established sufficient conditions for the existence and uniqueness of classical and generalized solutions of main boundary-value problems in the linear case and substantiated the applicability of the Fourier method to the solution of these problems [1, pp. 295‐299]. Analogous results on the existence and uniqueness of solutions of the corresponding problems for singularly perturbed equations were obtained by Oleinik in [2]. In [3], Butuzov proposed an original method for the solution of boundary-value problems for singularly perturbed partial differential equations. According to the method of angular boundary functions developed by Butuzov, asymptotic solutions of boundary-value problems are constructed in the form of regular and boundary-layer parts. In this case, functions of the boundary-layer part are solutions of certain differential equations with constant coefficients that satisfy certain boundary conditions. In the present paper, we consider the first boundary-value problem for a degenerate, linear, singularly perturbed parabolic system. The statement of the problem is close to problems investigated by Mitropol’skii and Khoma for regularly perturbed quasilinear and nonlinear equations of the hyperbolic type [4, pp. 137‐ 225]. In the solution of the first boundary-value problem for hyperbolic equations with slowly varying coefficients, Feshchenko and Shkil’ used the Fourier method [5, pp. 226‐245]. However, the question of the convergence of the corresponding series and the possibility of their term-by-term differentiation remains open. Note that equations with slowly varying coefficients can be reduced to singularly perturbed equations by a change of the independent variable. Consider the problem

Regularization of boundary-value problems for hyperbolic equations

The problem of the stability of wave propagation in anisotropic inhomogeneous media is considered. The class of approximate solutions possessing the stability property with respect to the small deviations of the input data in the form regularizing the operators R(φ, ψ, x, t, α) is constructed. Here an important role is played by the choice of the smoothing function and by the conditions for matching the regularization parameter with the error.

Boundary-Value Problems for Hyperbolic Equations Related to Steady Granular Flow

Boundary value problems for steady-state flow in elastoplasticity are a topic of mathematical and physical interest. In particular, the underlying PDE may be hyperbolic, and uncertainties surround the choice of physically appropriate stress and velocity boundary conditions. The analysis and numerical simulations of this paper address this issue for a model problem, a system of equations describing antiplane shearing of an elastoplastic material. This system retains the relevant mathematical structure of elastoplastic planar flow. Even if the flow rule is associative, two significant phenomena appear: (i) For boundary conditions suggestive of granular flow in a hopper, in which it seems physically natural to specify the velocity everywhere along a portion of the boundary, no such solutions of the equations exist; rather, we construct a solution with a shear band (velocity jump) along part of the boundary, and an appropriate relaxed boundary condition is satisfied there. (ii) Rigid zones appear inside deforming regions of the flow, and the stress field in such a zone is not uniquely determined. For a nonassociative flow rule, an extreme form of nonuniqueness—both velocity and stress—is encountered.

An initial-boundary-value problem for hyperbolic differential-operator equations on a finite interval

In this paper we give, for the first time, an abstract interpretation of initial--boundary-value problems for hyperbolic equations such that a part of the boundary-value conditions contains also a differentiation of the time $t$ of the same order as the equations. Initial--boundary-value problems for hyperbolic equations are reduced to the Cauchy problem for a system of hyperbolic differential-operator equations. A solution of this system is not a vector function but one function. At the same time, the system is not overdetermined. We prove the well-posedness of the Cauchy problem, and for some special cases we give an expansion of a solution to the series of eigenvectors. As application we show, in particular, a generalization of the classical Fourier method of separation of variables.

A Note on the Difference Schemes of the Nonlocal Boundary Value Problems for Hyperbolic Equations

The nonlocal boundary-value problem for hyperbolic equations in a Hilbert space H with the self-adjoint positive definite operator A is considered. Applying the operator approach, we establish the stability estimates for solution of this nonlocal boundary-value problem. In applications, the stability estimates for the solution of the nonlocal boundary value problems for hyperbolic equations are obtained. The first and second order of accuracy difference schemes generated by the integer power of A for approximately solving this abstract nonlocal boundary-value problem are presented. The stability estimates for the solution of these difference schemes are obtained. The theoretical statements for the solution of this difference schemes are supported by the results of numerical experiments.

Conditions of solvability of quasilinear periodic boundary-value problems for hyperbolic equations of the second order

On the basis of properties of the Vejvoda-Shtedry operator, we obtain solvability conditions for the 2π-periodic problem $$u_{tt} - u_{xx} = F\left[ {u,u_t } \right], u\left( {0,t} \right) = u\left( {\pi ,t} \right) = 0, u\left( {x,t + 2\pi } \right) = u\left( {x,t} \right)$$ .

A boundary-value problem for hyperbolic equations

The purpose of this paper is to study the following boundary-value problem for a hyperbolic equation with two independent variables: find the solution of this equation satisfying the boundary conditions given on two smooth curves. The curves emanate from a common point and lie inside the characteristic angle with its vertex at this point.

Boundary-value problems for hyperbolic equations with constant coefficients

By using a metric approach, we study the problem of well-posedness of boundary-value problems for hyperbolic equations of ordern (n≥2) with constant coefficients in a cylindrical domain. Conditions of existence and uniqueness of solutions are formulated in number-theoretic terms. We prove a metric theorem on lower estimates of small denominators that appear when constructing solutions.

The stability of the solutions of some boundary-value problems for hyperbolic equations

The behaviour of the solutions of linear hyperbolic equations is investigated as t→ ∞ in the half-space x > 0, - ∞ < y α < ∞, α = 1,2, …, r, with boundary conditions defined on the boundary x = 0. The equations and the boundary condition are assumed to be homogeneous with respect to the order of differentiation and all coefficients are assumed to be constant. A problem of this type has been previously studied in detail in connection with the stability of shock waves in gas dynamics [1–4] and some particular results have also been obtained for magnetohydrodynamic shocks [5–7]. In general, as will be shown below, the disturbances may have the same types of behaviour as t→∞ as in [2,4]: the disturbances increase exponentially (instability), decay as a power function (stability), or remain bounded (neutral stability). The transitions of the system to an unstable, stable, and neutrally stable state are investigated and the criteria for these transitions are derived. These criteria are used to establish the existence of neutrally stable magnetohydrodynamic shocks even in the case of an ideal gas, a phenomenon that has not been previously documented [5–7]. The existence of an a priori bound on the solution has been proved for these systems in cases of stability and neutral stability [8,9]. The interaction of disturbances with the boundary in the case of neutral stability produces a non-smooth solution, so that the a priori bound of [9] is unimprovable. An elementary explanation of this effect is proposed. It is shown that the addition of small non-differential terms to the equations and the boundary conditions does not cause the problems to become ill-posed if the parameters of the original problem ensure neutral stability. The behaviour of disturbances on the boundary of the half-space is described by the solution of the Cauchy problem for some systems of partial differential equations of a high order with special conditions on the external forces and the initial values. This result is similar to that observed in gas dynamics [10]. The stability of solutions with boundary conditions at x = 0 for x>0 and x<0 is analysed similarly and does not require a separate treatment.

A Test of Moving Mesh Refinement for 2-D Scalar Hyperbolic Problems

We present an algorithm for numerically solving hyperbolic equations with moving shock fronts. The central idea is a moving, finer mesh which follows the shock. Computational results are presented.

Hybrid Difference Methods for the Initial Boundary-Value Problem for Hyperbolic Equations

The use of lower order approximations in the neighborhood of boundaries coupled with higher order interior approximations is examined for the mixed initial boundary-value problem for hyperbolic partial differential equations. Uniform error can be maintained using smaller grid intervals with the lower order approximations near the boundaries. Stability results are presented for approximations to the initial boundary-value problem for the model equation ${u_t} + c{u_x} = 0$ which are fourth order in space and second order in time in the interior and second order in both space and time near the boundaries. These results are generalized to a class of methods of this type for hyperbolic systems. Computational results are presented and comparisons are made with other methods.

Hybrid difference methods for the initial boundary-value problem for hyperbolic equations

The use of lower order approximations in the neighborhood of boundaries coupled with higher order interior approximations is examined for the mixed initial boundary-value problem for hyperbolic partial differential equations. Uniform error can be maintained using smaller grid intervals with the lower order approximations near the boundaries. Stability results are presented for approximations to the initial boundary-value problem for the model equation u t + c u x = 0 {u_t} + c{u_x} = 0 which are fourth order in space and second order in time in the interior and second order in both space and time near the boundaries. These results are generalized to a class of methods of this type for hyperbolic systems. Computational results are presented and comparisons are made with other methods.

Dispersion and Gibbs phenomenon associated with difference approximations to initial boundary-value problems for hyperbolic equations

In this paper, we analyze the problem of the semidiscretized approximation for the initial boundary-value problem of the wave equation. Point-wise convergence properties for the propagation of discontinuities are investigated via a uniformly valid asymptotic expansion. An approximate error analysis using matched asymptotic expansions is constructed and compared with the asymptotic expansion of the exact solution.

Fourth Order Difference Methods for the Initial Boundary-Value Problem for Hyperbolic Equations

Centered difference approximations of fourth order in space and second order in time are applied to the mixed initial boundary-value problem for the hyperbolic equation ${u_t} = - c{u_x}$. A method utilizing third order uncentered differences at the boundaries is shown to be stable and to retain an overall fourth order convergence estimate. Several computational examples illustrate the success of these methods for problems with one and two spacial dimensions. Further examples illustrate the effects of approximations of various orders of accuracy used at the boundaries.

Fourth order difference methods for the initial boundary-value problem for hyperbolic equations

Centered difference approximations of fourth order in space and second order in time are applied to the mixed initial boundary-value problem for the hyperbolic equation u t = − c u x {u_t} = - c{u_x} . A method utilizing third order uncentered differences at the boundaries is shown to be stable and to retain an overall fourth order convergence estimate. Several computational examples illustrate the success of these methods for problems with one and two spacial dimensions. Further examples illustrate the effects of approximations of various orders of accuracy used at the boundaries.