Showing papers on "Hyperbolic partial differential equation published in 1992"
TL;DR: In this article, a method for obtaining traveling-wave solutions of nonlinear wave equations that are essentially of a localized nature is proposed based on the fact that most solutions are functions of a hyperbolic tangent.
Abstract: A method is proposed for obtaining traveling‐wave solutions of nonlinear wave equations that are essentially of a localized nature. It is based on the fact that most solutions are functions of a hyperbolic tangent. This technique is straightforward to use and only minimal algebra is needed to find these solutions. The method is applied to selected cases.
1,394 citations
406 citations
TL;DR: In this article, the authors consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time-asymptotically.
Abstract: We consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time-asymptotically. The hyperbolic model may be viewed as isentropic Euler equations with friction term added to the momentum equation to model gas flow through a porous media. In this case our result justifies Darcy's law time-asymptotically. Our model may also be viewed as an elastic model with damping.
359 citations
TL;DR: In this paper, the authors interpret the following fully nonlinear second-order partial differential equation as the value function of a certain optimal controlled diffusion problem, where is a second order elliptic partial differential operator parametrized by the control variable αϵA: with Here σ,b and c are functions defined on with values respectively in and is a real function defined on.
Abstract: We interpret the following fully nonlinear second-order partial differential equation as the value function of a certain optimal controlled diffusion problem, where is a second order elliptic partial differential operator parametrized by the control variable αϵA: with Here σ,b, and c are functions defined on with values respectively in and is a real function defined on . A particular case of this equation is when . In this case, the equation is the well-known Hamilton-Jacobi-Bellman equation. The problem is formulated as follows: The state equation of the control problem is a classical one. The cost function is described by an adapted solution of a certain backward stochastic differential equation. The paper discusses Bellman's dynamic programming principle for this problem The value function is proved to be a viscosity solution of the above possibly degenerate fully nonlinear equation
328 citations
Journal Article•
250 citations
TL;DR: A damped hyerbolic equation with critical exponent was proposed in this paper, where the critical exponent is defined as the number of elements in a damped hyperbolic equation. Communications in Partial Differential Equations: Vol 17, No. 5-6, pp. 841-866.
Abstract: (1992). A damped hyerbolic equation with critical exponent. Communications in Partial Differential Equations: Vol. 17, No. 5-6, pp. 841-866.
192 citations
TL;DR: In this article, it is shown that if the solution of the zero dissipation problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding system with visco-ity that converge to the solutions of the system without visco -coverage away from shock discontinuities at a rate of order e as the viscoity coefficient e goes to zero.
Abstract: In this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity. It is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero. The proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles.
191 citations
TL;DR: The Riemann problem for a general inhomogeneous system of conservation laws is solved in a neighborhood of a state at which one of the nonlinear waves in the problem takes on a zero speed as mentioned in this paper.
Abstract: The Riemann problem for a general inhomogeneous system of conservation laws is solved in a neighborhood of a state at which one of the nonlinear waves in the problem takes on a zero speed. The inhomogeneity is modeled by a linearly degenerate field. The solution of the Riemann problem determines the nature of wave interactions, and thus the Riemann problem serves as a canonical form for nonlinear systems of conservation laws. Generic conditions on the fluxes are stated and it is proved that under these conditions, the solution of the Riemann problem exists, is unique, and has a fixed structure; this demonstrates that, in the above sense, resonant inhomogeneous systems generically have the same canonical form. The wave curves for these systems are only Lipschitz continuous in a neighborhood of the states where the wave speeds coincide, and so, in contrast to strictly hyperbolic systems, the implicit function theorem cannot be applied directly to obtain existence and uniqueness. Here we show that existence ...
181 citations
TL;DR: In this paper, the limit set of an orbit belongs to the set of equilibrium points, and easily applied conditions to determine when this limit set is a single point are given to parabolic equations, linearly damped hyperbolic equations as well as their discretizations.
Abstract: In gradient-like systems, the limit set of an orbit belongs to the set of equilibrium points We give easily applied conditions to determine when this limit set is a single point Applications are given to parabolic equations, linearly damped hyperbolic equations as well as their discretizations
143 citations
TL;DR: In this paper, a triangle-based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures.
140 citations
TL;DR: In this article, a hyperbolic grid generation scheme formulated from grid orthogonality and cell volume specification has been significantly enhanced so that high quality three-dimensional grids can be obtained for a wide variety of geometries.
TL;DR: For a damped hyperbolic equation in R3 over a bounded smooth domain in R2, it is proved that the global attractors are upper semicontinuous as mentioned in this paper.
Abstract: For a damped hyperbolic equation in a thin domain in R3 over a bounded smooth domain in R2 , it is proved that the global attractors are upper semicontinuous. It is shown also that a global attractor exists in the case of the critical Sobolev exponent.
TL;DR: In this paper, the global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain ofn-dimensional Euclidean space with smooth boundary.
Abstract: The global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain ofn-dimensional Euclidean space with smooth boundary. Dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions ast tending to ∞ is described. Time periodic solutions are also studied. As an application of our main theorem, nonlinear viscoelasticity, strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated.
TL;DR: In this paper, it was shown that the eigenvalues at corresponding periodic orbits form a complete set of invariants for the smooth conjugacy of low dimensional Anosov systems.
Abstract: We give a new proof of the fact that the eigenvalues at corresponding periodic orbits forms a complete set of invariants for the smooth conjugacy of low dimensional Anosov systems. We also show that, if a homeomorphism conjugating two smooth low dimensional Anosov systems is absolutely continuous, then it is as smooth as the maps. We furthermore prove generalizations of these facts for non-uniformly hyperbolic systems as well as extensions and counterexamples in higher dimensions.
TL;DR: In this article, the mathematical structure of a continuum reactive mixture model of the combustion of granular energetic materials was studied and the wave fields associated with this description were derived and classified.
Abstract: In this paper, we study the mathematical structure of a continuum reactive mixture model of the combustion of granular energetic materials. We obtain and classify the wave fields associated with this description. This analysis shows that this system of hyperbolic equations becomes degenerate when the relative flow is locally sonic. We derive the corresponding Riemann invariants and construct simple wave solutions. We also discuss special discontinuous solutions of the system of equations. For fixed upstream conditions, different downstream states are possible when the relative velocities exceed the speed of the sound gas.
TL;DR: In this paper, the authors define an algorithm to characterize all weak symmetry groups of a partial differential equation and find new invariant solutions to the Fokker-Planck equation.
TL;DR: In this article, an analysis by energy methods is given for fully discrete numerical methods for time-dependent partial integro-differential equations, where stability and error estimates are derived in H1 and L2.
Abstract: An analysis by energy methods is given for fully discrete numerical methods for time-dependent partial integro-differential equations. Stability and error estimates are derived in H1 and L2. The methods considered pay attention to the storage needs during time-stepping.
TL;DR: This paper provides a brief summary of the current state of the theory of 2-D implicit systems and suggests that the implicit models are more suited to the description of naturally occurring two-dimensional systems, such as are described by the hyperbolic equation and the heat equation.
TL;DR: For second-order partial differential equations, the question of whether they can have solutions decaying superexponentially at infinity is studied in this paper, and a negative answer to the question is given.
Abstract: For second-order partial differential equations the question of whether they can have solutions decaying superexponentially at infinity is studied. An example is constructed of an equation Δu = q(x)u on the plane with bounded coefficients q having a nonzero solution decaying superexponentially. This example provides a negative answer to a familiar question of E. M. Landis. These questions are also studied for hyperbolic and parabolic equations on manifolds. An example is constructed of a parabolic equation having a nonzero solution u(x, t) decaying superexponentially as t→∞.
Book•
01 Jan 1992
TL;DR: In this paper, partial differential equation applications, including heat conduction, wave propagation, vibrations, traffic flow shocks, evolution of population densities, fluid flow, electrostatics, minimal surfaces, gravitation, and quantum mechanics, are discussed.
Abstract: This text documents partial differential equation applications, including: heat conduction; wave propagation; vibrations; traffic flow shocks; evolution of population densities; fluid flow; electrostatics; minimal surfaces; gravitation; and quantum mechanics.
TL;DR: In this article, the authors obtained stability estimates of recovery of two coefficients of a hyperbolic partial differential equation from all possible measurements implemented at a part of the lateral boundary in the plane case and of Holder type in the three-dimensional case.
Abstract: The authors obtain stability estimates of recovery of two coefficients of a hyperbolic partial differential equation from all possible measurements implemented at a part of the lateral boundary. These estimates are of logarithmic type in the plane case and of Holder type in the three-dimensional case. As an important auxiliary result they have proved stability estimates in (attenuated) integral geometry with incomplete data.
TL;DR: In this paper, hyperbolic heat conduction is studied by considering all the thermophysical properties, except the thermal diffusivity, to be temperature dependent, and the resulting nonlinear hyper-bolic equations are linearized by using Kirchhoff transformation.
Abstract: With the advent of lasers with very short pulse durations and their use in materials processing, the effect of thermal wave propagation velocity becomes important. Also, localized heating in laser-aided materials processing causes significant variations in the material properties. To account for these two effects, hyperbolic heat conduction is studied in this paper by considering all the thermophysical properties, except the thermal diffusivity, to be temperature dependent. The resulting nonlinear hyperbolic equations are linearized by using Kirchhoff transformation. Both analytical and numerical solutions are obtained for finite domains. Results are presented and compared with parabolic conduction results.
TL;DR: In this article, the existence of absorbing balls in every Sobolev norm is proved and an upper estimate for the Hausdorff dimension of the attractor is given, and the transition of energy from low modes to high ones is observed.
Abstract: The global behavior of the Kuramoto-Sivashinsky equation is studied. The existence of an absorbing ball in every Sobolev norm is proved. The transition of energy from low modes to high ones is observed. An upper estimate for the Hausdorff dimension of the attractor is given. The main tool is to use the methods of the theory of ordinary differential equations in the investigation of partial differential equations.
TL;DR: In this article, the authors introduce a method for constructing solutions of homogeneous partial differential equations based on constructing the space-time Fourier transform of a function so that it satisfies the transformed partial differential equation.
Abstract: We introduce a method for constructing solutions of homogeneous partial differential equations. This method can be used to construct the usual, well-known, separable solutions of the wave equation, but it also easily gives the non-separable localized wave solutions. These solutions exhibit a degree of focusing about the propagation axis that is dependent on a free parameter, and have many important potential applications. The method is based on constructing the space-time Fourier transform of a function so that it satisfies the transformed partial differential equation. We also apply the method to construct localized wave solutions of the wave equation in a lossy infinite medium, and of the Klein-Gordon equation. The localized wave solutions of these three equations differ somewhat, and we discuss these differences. A discussion of the properties of the localized waves, and of experiments to launch them, is included in the Appendix.
TL;DR: This work shows how to construct an adaptively implicit scheme which is nearly fully implicit in regions of fast flow, but which may be explicit at sharp fronts which are moving more slowly, and demonstrates that these methods are more accurate than conventional implicit algorithms and more efficient than fully explicit methods, for which smaller time steps must be used.
TL;DR: A survey of the literature and general comments are presented in Section 4.1 as mentioned in this paper, where the connections between superdiffusion processes and one class of nonlinear parabolic differential equations are established.
Abstract: We establish connections between superdiffusion processes and one class of nonlinear parabolic differential equations. Analytic results due to Brezis and Friedman, Baras and Pierre and others are used to investigate the graphs of superdiffusions. A survey of the literature and general comments are presented in Section 4.
Book•
01 Aug 1992
TL;DR: Papers presented at the May 1991 symposium reflect continuing interest in the role of domain decomposition in the effective utilization of parallel systems; applications in fluid mechanics, structures, biology, and design optimization; and maturation of analysis of elliptic equations.
Abstract: Papers presented at the May 1991 symposium reflect continuing interest in the role of domain decomposition in the effective utilization of parallel systems; applications in fluid mechanics, structures, biology, and design optimization; and maturation of analysis of elliptic equations, with theoretic
TL;DR: In this article, the hyperbolic heat conduction equation is used to predict the temperature distributions in both semi-infinite and finite isotropic media due to a train of temporally rectangular pulses which approximate the Gaussian temporal profile of mode-locked laser pulses.
TL;DR: A uniqueness theorem for second order hyperbolic differential equations is given in this paper, where it is shown that the uniqueness theorem holds for all second order HDEs in the context of partial differential equations.
Abstract: (1992). A uniqueness theorem for second order hyperbolic differential equations. Communications in Partial Differential Equations: Vol. 17, No. 5-6, pp. 699-714.
TL;DR: In this paper, the authors describe the partial differential equations (PDE) on thin domains and present recent results on the dynamics of a dissipative parabolic equation and a damped hyperbolic equation.
Abstract: Publisher Summary This chapter describes the partial differential equations (PDE) on thin domains. In many applications, the chapter encounters PDE defined on domains for which the size in some directions is much larger than the size in others. In physics and engineering, the chapter often encounters the expressions thin rods, thin plates, thin shells, etc. The chapter also interested in the dynamics of fluids which lie between two rotating cylinders or spheres which are very close together or the conduction of heat through thin materials. For conservative PDE on thin domains, it is more difficult to make a comparison with the PDE on the reduced domain. In general, the chapter presents recent results on the dynamics of a dissipative parabolic equation and a damped hyperbolic equation on thin domains.