Order Hyperbolic Equations Research Articles

Asymptotic property of solutions of some higher order hyperbolic equations, I

Asymptotic property of solutions of some higher order hyperbolic equations, I

Задача Коши и смешанная задача для вырождающхся гиперболических уравнений второго порядка

Задача Коши и смешанная задача для вырождающхся гиперболических уравнений второго порядка

Решение задачи Коши и смешанной задачи для вырождающихся гиперболических уравнений второго порядка методом конечных разностей

Решение задачи Коши и смешанной задачи для вырождающихся гиперболических уравнений второго порядка методом конечных разностей

The Krein-Krasnoselskii-Kooi Condition for a Second Order Hyperbolic Partial Differential Equation with Complete Second Member

The Krein-Krasnoselskii-Kooi Condition for a Second Order Hyperbolic Partial Differential Equation with Complete Second Member

On some mixed problems for fourth order hyperbolic equations

On some mixed problems for fourth order hyperbolic equations

On some mixed problems for fourth order hyperbolic equations

On some mixed problems for fourth order hyperbolic equations

On some generalised solution of a nonlinear first order hyperbolic partial differential equation

On some generalised solution of a nonlinear first order hyperbolic partial differential equation

A Runge-Kutta Method for the Numerical Solution of the Goursat Problem in Hyperbolic Partial Differential Equations

A Runge-Kutta type method is developed for the numerical solution of second order hyperbolic partial differential equations. Numerical examples of the method are given.

On a mixed boundary value problem for linear third order hyperbolic partial differential equations

On a mixed boundary value problem for linear third order hyperbolic partial differential equations

Explicit difference scheme for the solution of a mixed problem for a general second order hyperbolic equation

In [1] an inexplicit finite difference scheine for the solution of a mixed problem for a general second order hyperbolic equation was put forward, by means of which the existence of a generalized solution of this problem was proved and its differential properties were investigated. In [2]–[4] an explicit difference scheme was used for the solution of the Cauchy problem for hyperbolic (linear and quasilinear) equations and linear hyperbolic systems. This scheme is used below to solve a mixed problem for a second order hyperbolic equation. The stability of this scheme (energy inequality) is proved and the convergence of the solutions of the corresponding difference equations, extended in a definite way, to the generalized solution of a mixed problem. The regularity of this solution may be Investigated by means of the given scheme as was done in [1].

Asymptotic properties of the dynamic variables in the theory of spinor and scalar meson fields

THE present paper will discuss some asymptotic properties of the dynamic variables and of the conservation laws in field theory, that correspond to fields with progressive singularities. The method which we shall adopt throughout is based on interpretation of all the equations in the sense of the theory of Schwartz distributions. In our case it is completely equivalent to a description of the field with the aid of weak solutions, though it possesses considerable technical advantages. As regards the equations discussed, we shall adopt the point of view of [1], namely, we shall consider the equations whose characteristic manifolds are described by the eikonal equation∗∗ ▪ Here A i are the electromagnetic potentials, c is the velocity of light, s the action, and e and m respectively the charge and mass of the particle described by (1). We shall consider below the second order hyperbolic equation in the scalar function W, describing certain types of meson, G σμ δ 2W gdx σδx μ = 0 (2) and the first order hyperbolic system, describing the spinor particles, γ σ δU δx σ = 0 , (3) where U is the four-component spinor and γ σ are the familiar Dirac matrices, satisfying the anticommutational relations γ i γ j + γ j γ i + γ j γ i = − 2 δ ij , γ 4 = I − g i γ i .

Energy inequalities for the solution of differential equations

where the barred subscripts denote backward difference quotients and the unbarred subscripts denote forward difference quotients. The fundamental problem concerning such finite difference approximations to (1.1) is to show that their solutions tend with diminishing mesh size to the solution of (1.1). Actually, it is sufficient to prove that the difference equations are stable since the convergence of a finite difference scheme can be derived from its stability(2) in a way which, by now, has become standard (e.g. see Douglas [2], F. John [6] and Lax and Richtmyer [7]). The stability of the difference schemes considered in this paper are deduced from an energy inequality satisfied by the difference operator L. This energy inequality is a discrete analogue of the well-known energy inequality of Friedrichs and Lewy [5 ] for second order hyperbolic equations. The energy inequality for L states that any function qk together with its first order difference quotients can be estimated in the mean square along time lines in terms of Lo. It is the fact that the first difference quotients of 4 can be estimated in terms of Lo that enables us to treat difference approximations to differential equations which have nonconstant coefficients. It also enables us to treat certain quasi-linear equations.

Uniqueness In Boundary Value Problems For The Second Order Hyperbolic Equation

Introduction. We study linear normal hyperbolic partial differential equations of the second order, with one dependent variable u, and N independent variables xi (i = 1, … , N). The uniqueness theorem connected with the Cauchy problem for this type of equation is well known and in effect states that if u and its first normal derivatives vanish on a spacelike initial surface S then u vanishes in a certain conical region which contains S (1, p. 379).

Solutions of second order hyperbolic differential equations with constant coefficients in a domain with a plane boundary

It is known that solutions of partial differential equations often have special regularity properties that set them apart from the “arbitrary” functions of any class C m or even C ∞. For example solutions of analytic elliptic equations are themselves analytic. Similarly, solutions of the heat equation are analytic in the space variables and of class C ∞ in the time. Even solutions of hyperbolic or ultra-hyperbolic equations have their special regularity properties, where however “regularity” does not always consist in possessing a large number of derivatives. This may find its expression in the fact that local integral transforms of the solution have many derivatives or even are analytic. Often the solutions form a family of functions, which share with the analytic functions the property of unique continuation, at least within certain limits.

The solution of Cauchy's Problem for linear partial differential equations, with constant coefficients, by means of integrals involving complex variables

The object of this paper is to show how the theory of integrals involving complex variables may be applied to the integration of linear partial differential equations, possessing real, distinct characteristics and constant coefficients. The problem considered is a Cauchy problem (with analytic data)—typical of the equation of real characteristics and the method taken is that of Riemann. For simplicity of exposition, the second order hyperbolic equation is considered, but the results are given in such a form as to indicate an obvious generalisation to equations of higher order.

An equation-free approach for second order multiscale hyperbolic problems in non-divergence form

The present study concerns the numerical homogenization of second order hyperbolic equations in non-divergence form, where the model problem includes a rapidly oscillating coefficient function. The ...