Abstract Hyperbolic Equations Research Articles

A parallel type decomposition scheme for quasi-linear abstract hyperbolic equation

Abstract The Cauchy problem for an abstract hyperbolic equation with the Lipschitz continuous operator is considered in the Hilbert space. The operator corresponding to the elliptic part of the equation is the sum of operators A 1 , A 2 , … , A m {A_{1},A_{2},\ldots,A_{m}} . Each summand is a self-adjoint and positive definite operator. A parallel type decomposition scheme for an approximate solution of the stated problem is constructed. The main idea of the scheme is that on each local interval the classical difference problems are solved in parallel (independently from each other) with the operators A 1 , A 2 , … , A m {A_{1},A_{2},\ldots,A_{m}} . The weighted average of the obtained solutions is announced as an approximate solution at the right end of the local interval. The convergence of the proposed scheme is proved and the approximate solution error is estimated, as well as the error of the difference analogue for the first-order derivative for the case when the initial problem data satisfy the natural sufficient conditions for solution existence.

Damped quantum wave equation from non-standard Lagrangians and damping terms removal

Hyperbolic equations are used in several physical phenomena to describe dynamical processes where information propagates with a finite speed. Recently, the wave equations with damping terms turned out to be fundamental hyperbolic equations in certain branches of physics mainly scattering processes and fractal medium. The aim of the present study is double shooting, first to prove that damped quantum wave equations may be obtained using the notion of non-standard Lagrangians and second to show that linear and nonlinear damping terms may be obtained if the concept of ‘two-occurrences of integrals’ is used, hence reducing the damped quantum wave equation to the conventional quantum wave equation known as the Klein-Gordon equation. This study supports the idea of non-standard Lagrangians and its usefulness in the theory of partial differential equations.

Operator Splitting for Quasi-Linear Abstract Hyperbolic Equation

We consider an abstract hyperbolic equation with a Lipschitz continuous operator, where the main operator is self-adjoint and positive definite and represents a sum of two similar operators. For this equation, we construct a decomposition scheme of high order of accuracy. This scheme is based on rational splitting of cosine-operator function.

The Finite Element Approximation in Hyperbolic Equation and Its Application—The Pollution of the Water in the West of Algeria as an Example

Hyperbolic variational equations are discussed and their existence and uniqueness of weak solution is established over in the last six decades. In this paper the hyperbolic equations (strong formula) can be transformed into a Hyperbolic variational equations. In this research, we propose a time-space discretization to show the existence and uniqueness of the discrete solution and how we apply it in the transport problem. The proposed approach stands on a discrete L∞-stability property with respect to the right-hand side and the boundary conditions of our problem which has been proposed. Furthermore the numerical example is given for the pollution in the smooth fluid as water and we have taken the pollution of the water in the west of Algeria as an example.

Monotone bicompact scheme for quasilinear hyperbolic equations

Hyperbolic equations and systems of equationscomprise a considerable part of mathematical modelsused in various applications [1]. Among them, someproblems are related to finding weak solutions, whichare described by piecewise smooth functions. Highorder accurate shockcapturing schemes are widelyused for the numerical solution of nonlinear problems.Important properties to be possessed by such schemesare monotonicity [1] and conservation [2]. For thisreason, the design of monotone and conservative difference schemes has been addressed in numerous publications [1, 2].For hyperbolic quasilinear equations and systemsof equations, bicompact difference schemes on a twopoint stencil were proposed in [3]. They have fourthorder accuracy in space and are absolutely stable, conservative, and monotone for local Courant numbersκ≥ [4]. Moreover, they are efficient and can besolved using by the running calculation method.In this paper, a new highorder accurate monotonebicompact scheme is proposed for quasilinear hyperbolic equations. Like the schemes in [3], it is absolutely stable, conservative, and efficient but, in contrast to the former, has the following additional advantages. First, the new scheme remains monotone forsmaller Courant numbers than that in [3]. Second,when applied to stiff problems, the scheme is morerobust than that in [3]. An

Well-defined solvability and spectral analysis of abstract hyperbolic integrodifferential equations

The authors study integrodifferential equations in Hilbert space. The coefficients of the equations are unbounded and the principal part is an abstract hyperbolic equation perturbed by terms with Volterra integral operators. Such equations can be regarded as an abstract generalization of the Gurtin–Pipkin integrodifferential equation that describes heat transfer in materials with memory and has a number of other applications. Well-defined solvability of initial boundary value problems for such equations is established in weighted Sobolev spaces on the positive semi-axis. The authors examine spectral problems for operator-valued functions representing the symbols of the said equations and study the spectrum of the abstract Gurtin–Pipkin integrodifferential equation.

Well-defined solvability and spectral properties of abstract hyperbolic equations with aftereffect

We study functional differential equations with unbounded operator coefficients in Hilbert spaces such that the principal part of the equation is an abstract hyperbolic equation perturbed by terms with delay and terms containing Volterra integral operators. The well-posed solvability of initial boundary-value problems for the specified problems in weighted Sobolev spaces on the positive semi-axis is established.

On the Existence of Almost Automorphic Solutions to Some Abstract Hyperbolic Differential Equations

In this paper we give some sufficient conditions, which do ensure the existence and uniqueness of a compact almost automorphic solution to some hyperbolic differential equations. As an illustration, we study the existence and uniqueness of almost automorphic solutions to a one-dimensional heat equation with small delays.

Spectral analysis and solvability of abstract hyperbolic equations with aftereffect

We analyze functional-differential equations with unbounded operator coefficients in a Hilbert space whose leading part is an abstract hyperbolic equation perturbed by terms with a retarded argument and by terms with Volterra integral operators.

Error estimate in the projection-difference method for an abstract quasilinear hyperbolic equation with a nonsmooth right-hand side

A three-level scheme of the projection-difference method for an abstract quasilinear hyperbolic equation is analyzed for convergency in a Hilbert space with variable operator coefficients and a non-smooth (only Bochner integrable) right-hand side. An energy error estimate is derived without any special conditions imposed on projection subspaces.

The Fourth Order of Accuracy Decomposition Scheme for Abstract Hyperbolic Equation

Abstract Using the rational splitting of a cosine operator-function, the fourth order accuracy decomposition scheme is constructed for hyperbolic equation when the principal operator is self-adjoint positively defined and is represented as a sum of two summands. Stability of the constructed scheme is shown and the error of an approximate solution is estimated.

Error estimates for schemes of the projection-difference method for abstract quasilinear hyperbolic equations

We study the convergence of the three-layer scheme of the projection-difference method for abstract quasilinear hyperbolic equations in Hilbert space. We establish asymptotic energy error estimates for an arbitrary choice of finite-dimensional subspaces in which the approximation problems are solved.

Estimates for the Rate of Convergence of the Galerkin Method for Abstract Hyperbolic Equations

We study the rate of convergence of the semidiscrete Galerkin method for linear hyperbolic equations in a Hilbert space. We establish asymptotic estimates for the error arising as a result of the arbitrariness in the choice of subspaces in which the approximation problems are solved.

Solutions of abstract hyperbolic equations by Rothe method.

Solutions of abstract hyperbolic equations by Rothe method.

Equipartition of energy for higher order abstract hyperbolic equations

(1982). Equipartition of energy for higher order abstract hyperbolic equations. Communications in Partial Differential Equations: Vol. 7, No. 10, pp. 1217-1251.

Uniqueness theorem for abstract hyperbolic equations with application to the uniqueness of the harmonic coordinate system in general relativity

Uniqueness theorem for abstract hyperbolic equations with application to the uniqueness of the harmonic coordinate system in general relativity

Comments on the derivation of finite-element approximations from conservation principles

Abstract Hyperbolic equations are often derived from conservation principles. We will show in this paper that in those cases, expressing this principles in a natural discrete form provides a procedure for the derivation of finite-element approximations. Likewise, the solution of parabolic equations also satisfy natural conservation principles. The derivation of finite-element approximations from their use will also shown to be feasible.

On optimal control of processes governed by abstract functional, integral and hyperbolic differential equations1

This paper presents a functional analytic model for a class of problems of optimal control, derives necessary optimality criterions and applies them to a feneralized version of Butkovski's problem ...

On optimal control of processes governed by abstract functional, integral and hyperbolic differential equations

On optimal control of processes governed by abstract functional, integral and hyperbolic differential equations