Showing papers on "Hyperbolic partial differential equation published in 2006"

Exp-function method for nonlinear wave equations

Abstract: In this paper, a new method, called Exp-function method, is proposed to seek solitary solutions, periodic solutions and compacton-like solutions of nonlinear differential equations. The modified KdV equation and Dodd–Bullough–Mikhailov equation are chosen to illustrate the effectiveness and convenience of the suggested method.

Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks

Abstract: In this paper we consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and by using some observability inequalities. If one of the above assumptions is not satisfied, some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero.

Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach

Abstract: The world of quantitative finance (QF) is one of the fastest growing areas of research and its practical applications to derivatives pricing problem. Since the discovery of the famous Black-Scholes equation in the 1970's we have seen a surge in the number of models for a wide range of products such as plain and exotic options, interest rate derivatives, real options and many others. Gone are the days when it was possible to price these derivatives analytically. For most problems we must resort to some kind of approximate method.

Homotopy–perturbation method for pure nonlinear differential equation

Abstract: In this paper, the homotopy–perturbation method proposed by J.-H. He is adopted for solving pure strong nonlinear second-order differential equation. For the oscillatory differential equation the initial approximate solution is assumed in the form of Jacobi elliptic function and the forementioned method is used for obtaining of the approximate analytic solution. Two types of differential equations are considered: with strong cubic and strong quadratic nonlinearity. The obtained solution is compared with exact numerical one. The difference between these solutions is negligible for a long time period. The method is found to work extremely well in the examples, but the theoretical reasons are not yet clear.

The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations

Abstract: The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations. Unlike our approach in [5] for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in strong norms of our sequence of regularized problems.

Particle-size segregation and diffusive remixing in shallow granular avalanches

Abstract: Segregation and mixing of dissimilar grains is a problem in many industrial and pharmaceutical processes, as well as in hazardous geophysical flows, where the size-distribution can have a major impact on the local rheology and the overall run-out. In this paper, a simple binary mixture theory is used to formulate a model for particle-size segregation and diffusive remixing of large and small particles in shallow gravity-driven free-surface flows. This builds on a recent theory for the process of kinetic sieving, which is the dominant mechanism for segregation in granular avalanches provided the density-ratio and the size-ratio of the particles are not too large. The resulting nonlinear parabolic segregation–remixing equation reduces to a quasi-linear hyperbolic equation in the no-remixing limit. It assumes that the bulk velocity is incompressible and that the bulk pressure is lithostatic, making it compatible with most theories used to compute the motion of shallow granular free-surface flows. In steady-state, the segregation–remixing equation reduces to a logistic type equation and the ‘S’-shaped solutions are in very good agreement with existing particle dynamics simulations for both size and density segregation. Laterally uniform time-dependent solutions are constructed by mapping the segregation–remixing equation to Burgers equation and using the Cole–Hopf transformation to linearize the problem. It is then shown how solutions for arbitrary initial conditions can be constructed using standard methods. Three examples are investigated in which the initial concentration is (i) homogeneous, (ii) reverse graded with the coarse grains above the fines, and, (iii) normally graded with the fines above the coarse grains. Time-dependent two-dimensional solutions are also constructed for plug-flow in a semi-infinite chute.

Hyperbolic Partial Differential Equations

Abstract: Basic notions Finite speed of propagation of signals Hyperbolic equations with constant coefficients Hyperbolic equations with variable coefficients Pseudodifferential operators and energy inequalities Existence of solutions Waves and rays Finite difference approximation to hyperbolic equations Scattering theory Hyperbolic systems of conservation laws Huygens' principle for the wave equation on odd-dimensional spheres Hyperbolic polynomials The multiplicity of eigenvalues Mixed initial and boundary value problems Energy decay for star-shaped obstacles

On potential wells and applications to semilinear hyperbolic equations and parabolic equations

Abstract: In this paper we generalize the family of potential wells to the initial boundary value problem of semilinear hyperbolic equations and parabolic equations u tt - Δ u = f ( u ) , x ∈ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ Ω , u ( x , t ) = 0 , x ∈ ∂ Ω , t ⩾ 0 and u t - Δ u = f ( u ) , x ∈ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω , u ( x , t ) = 0 , x ∈ ∂ Ω , t ⩾ 0 , not only give a threshold result of global existence and nonexistence of solutions, but also obtain the vacuum isolating of solutions Finally we prove the global existence of solutions for above problem with critical initial conditions I ( u 0 ) ⩾ 0 , E ( 0 ) = d or I ( u 0 ) ⩾ 0 , J ( u 0 ) = d So Payne and Sattinger's results are generalized and improved in essential

On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, II: Universality of Critical Behaviour

Abstract: Hamiltonian perturbations of the simplest hyperbolic equation ut + a(u) ux = 0 are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution. Moreover, this behaviour is described by a special solution to an integrable fourth order ODE.

An Anisotropic Acoustic Wave Equation for VTI Media

Abstract: Here we propose a new anisotropic acoustic wave equation based on the same dispersion relation as Alkhalifah’s (2000), but introducing an auxiliary function which allows the original fourth-order differential equation to become a coupled system of lower-order differential equations. Of these two equations, one equation can be considered as a hyperbolic wave equation for elliptical anisotropy, but with a correction term that compensates for the loss of anisotropy for VTI media, while the other can be considered as the additional expansion or contraction of the wavefront in the lateral directions. This two-way anisotropic wave equation can be used for both modeling and reverse-time migration. The new anisotropic acoustic equation has the obvious physical meaning and is much easier to implement. Impulse responses for both modeling and migration have been shown to validate the proposed anisotropic acoustic equation.

A note on partial functional differential equations with state-dependent delay

Abstract: In this paper we study the existence of mild solutions for a class of abstract partial functional differential equation with state-dependent delay.

Metric Operator in Pseudo-Hermitian Quantum Mechanics and the Imaginary Cubic Potential

Abstract: We present a systematic perturbative construction of the most general metric operator (and positive-definite inner product) for quasi-Hermitian Hamiltonians of the standard form, H = 1 2 p 2 + v(x), in one dimension. We show that this problem is equivalent to solving an infinite system of iteratively decoupled hyperbolic partial differential equations in (1 + 1)-dimensions. For the case that v(x} is purely imaginary, the latter have the form of a nonhomogeneous wave equation which admits an exact solution. We apply our general method to obtain the most general metric operator for the imaginary cubic potential, v(x) = i∈x 3 . This reveals an infinite class of previously unknown CPT- as well as non-CPT-inner products. We compute the physical observables of the corresponding unitary quantum system and determine the underlying classical system. Our results for the imaginary cubic potential show that, unlike the quantum system, the corresponding classical system is not sensitive to the choice of the metric operator. As another application of our method we give a complete characterization of the pseudo-Hermitian canonical quantization of a free particle moving in R that is consistent with the usual choice for the quantum Hamiltonian. Finally, we discuss subtleties involved with higher dimensions and systems having a fixed length scale.

Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms

Abstract: In this paper we develop an intrinsic approach to derivation of energy decay rates for the semilinear wave equation with localized interior nonlinear monotone damping g ( u t ) and a source term f ( u ) . The proposed approach allows to consider, in an unified way, much more general classes of hyperbolic problems than addressed before in the literature. These generalizations refer to both geometric and topological aspects of the problem. The method leads to optimal decay rates for solutions of semilinear hyperbolic equations driven by a source of critical exponent and subjected to a nonlinear damping localized in a small region adjacent to a portion of the boundary. The distinct features of the model include: (i) Neumann boundary conditions are assumed and, (ii) no growth conditions are imposed on the damping g ( s ) . It is well known that Neumann boundary does not satisfy Lopatinski condition and, therefore, the analysis of propagation of energy in the absence of the damping on the Neumann part of the boundary requires special geometric considerations. In addition, the sole conditions assumed on g ( s ) are monotonicity, continuity and g ( 0 ) = 0 . In particular, no differentiability and no growth conditions are imposed on the damping both at the origin and at the infinity. The asymptotic decay rates for the energy function are obtained from an intrinsic algorithm driven by solutions of simple ODE. Several examples illustrate the theory by exhibiting various decay rates (exponential, algebraic, rational, logarithmic, etc.) for the energy functional. An important corollary of our energy decay theorem is a stability result which shows that, under certain conditions, when dissipation is sublinear at infinity, the solution of the system remains uniformly bounded for all time in the norms above the finite energy level, even in the presence of a nonlinear source term.

On the Cauchy Problem of the Camassa-Holm Equation

Abstract: The purpose of this paper is to investigate the Cauchy problem of the Camassa-Holm equation. By using the abstract method proposed and studied by T. Kato and priori estimates, the existence and uniqueness of the global solution to the Cauchy problem of the Camassa-Holm equation in Lp frame under certain conditions are obtained. In addition, the continuous dependence of the solution of this equation on the linear dispersive coefficient contained in the equation is obtained.

Lipschitz stability of an inverse problem for an acoustic equation

Abstract: An inverse problem of the determination of the coefficient p(x) in the equation is considered. The main difficulty here as compared with the previous results is that the function p(x) is involved together with its derivatives. A Lipschitz stability estimate is obtained using the method of Carleman estimates.

Cauchy problem of the generalized double dispersion equation

Abstract: In this paper, the existence and the uniqueness of the global solution for the Cauchy problem of the generalized double dispersion equation are proved. The blow-up of the solution for the Cauchy problem of the generalized double dispersion equation is discussed by the concavity method under some conditions.

The repeated homogeneous balance method and its applications to nonlinear partial differential equations

Abstract: In this letter, a new method, called the repeated homogeneous balance method, is proposed for seeking the traveling wave solutions of nonlinear partial differential equations. The Burgers–KdV equation is chosen to illustrate our method. It has been confirmed that more traveling wave solutions of nonlinear partial differential equations can be effectively obtained by using the repeated homogeneous balance method.

Existence results for an impulsive abstract partial differential equation with state-dependent delay

Abstract: In this paper, we establish the existence of mild solutions for a class of impulsive abstract partial functional differential equation with state-dependent delay

Instabilities in the Chapman-Enskog Expansion and Hyperbolic Burnett Equations

Abstract: It is well-known that the classical Chapman-Enskog procedure does not work at the level of Burnett equations (the next step after the Navier-Stokes equations) Roughly speaking, the reason is that the solutions of higher equations of hydrodynamics (Burnett's, etc) become unstable with respect to short-wave perturbations This problem was recently attacked by several authors who proposed different ways to deal with it We present in this paper one of possible alternatives First we deduce a criterion for hyperbolicity of Burnett equations for the general molecular model and show that this criterion is not fulfilled in most typical cases Then we discuss in more detail the problem of truncation of the Chapman-Enskog expansion and show that the way of truncation is not unique The general idea of changes of coordinates (based on analogy with the theory of dynamical systems) leads finally to nonlinear Hyperbolic Burnett Equations (HBEs) without using any information beyond the classical Burnett equations It is proved that HBEs satisfy the linearized H-theorem The linear version of the problem is studied in more detail, the complete Chapman-Enskog expansion is given for the linear case A simplified proof of the Slemrod identity for Burnett coefficients is also given

On the Controllability of a Fractional Order Parabolic Equation

Abstract: The null-controllability property of a $1-d$ parabolic equation involving a fractional power of the Laplace operator, $(-\Delta)^\alpha$, is studied. The control is a scalar time-dependent function $g=g(t)$ acting on the system through a given space-profile $f=f(x)$ on the interior of the domain. Thus, the control $g$ determines the intensity of the space control $f$ applied to the system, the latter being given a priori. We show that, if $\alpha\leq 1/2$ and the shape function $f$ is, say, in $L^2$, no initial datum belonging to any Sobolev space of negative order may be driven to zero in any time. This is in contrast with the existing positive results for the case $\alpha >1/2$ and, in particular, for the heat equation that corresponds to $\alpha=1$. This negative result exhibits a new phenomenon that does not arise either for finite-dimensional systems or in the context of the heat equation. On the contrary, if more regularity of the shape function $f$ is assumed, then we show that there are initial data in any Sobolev space $H^m$ that may be controlled. Once again this is precisely the opposite behavior with respect to the control properties of the heat equation in which, when increasing the regularity of the control profile, the space of controllable data decreases. These results show that, in order for the control properties of the heat equation to be true, the dynamical system under consideration has to have a sufficiently strong smoothing effect that is critical when $\alpha=1/2$ for the fractional powers of the Dirichlet Laplacian in $1-d$. The results we present here are, in nature and with respect to techniques of proof, similar to those on the control of the heat equation in unbounded domains in [S. Micu and E. Zuazua, Trans. Amer. Math. Soc., 353 (2000), pp. 1635-1659] and [S. Micu and E. Zuazua, Portugal. Math., 58 (2001), pp. 1-24]. We also discuss the hyperbolic counterpart of this problem considering a fractional order wave equation and some other models.