Problem For Wave Equation Research Articles

Seismic modeling with radial basis function-generated finite differences (RBF-FD) – a simplified treatment of interfaces

In a previous study of seismic modeling with radial basis function-generated finite differences (RBF-FD), we outlined a numerical method for solving 2-D wave equations in domains with material interfaces between different regions. The method was applicable on a mesh-free set of data nodes. It included all information about interfaces within the weights of the stencils (allowing the use of traditional time integrators), and was shown to solve problems of the 2-D elastic wave equation to 3rd-order accuracy. In the present paper, we discuss a refinement of that method that makes it simpler to implement. It can also improve accuracy for the case of smoothly-variable model parameter values near interfaces. We give several test cases that demonstrate the method solving 2-D elastic wave equation problems to 4th-order accuracy, even in the presence of smoothly-curved interfaces with jump discontinuities in the model parameters.

Determining distributed parameters in a neuronal cable model on a tree graph

For graph domains without cycles, we show how unknown coefficients and source terms for a parabolic equation can be recovered from the dynamical Neumann‐to‐Dirichlet map associated with the boundary vertices. Through use of a companion wave equation problem, the topology of the tree graph, degree of the vertices, and edge lengths can also be recovered. The motivation for this work comes from a neuronal cable equation defined on the neuron's dendritic tree, and the inverse problem concerns parameter identification of k unknown distributed conductance parameters. Copyright © 2017 John Wiley & Sons, Ltd.

The renormalization method based on the Taylor expansion and applications for asymptotic analysis

Based on the Taylor expansion, we propose a renormalization method (TR method, for simplicity) for asymptotic analysis. The standard renormalization group (RG) method for asymptotic analysis can be derived out from this new method, and hence, the mathematical essence of the RG method is also recovered. The biggest advantage of the proposed method is that the secular terms in perturbation series are automatically eliminated, but in usual perturbation theory, we need more efforts and tricks to eliminate these terms. At the same time, the mathematical foundation of the method is simple, and the logic of the method is very clear. Therefore, it is very easy to use in practice. As application, we obtain the uniform valid asymptotic solutions to some typical problems including vector field, boundary layer, boundary value problems of nonlinear wave equations, the normal form theory and reduction equations of dynamical systems. Further, by combining the topological deformation with the TR method, a modified method, namely the homotopy renormalization method (for simplicity, HTR), is proposed to overcome some weaknesses of the standard RG method and TR method. In this HTR method, since there is a freedom to choose the first-order approximate solution in perturbation expansion, we can improve the global solution. In particular, for those equations not including a small parameter, the HTR method can also be applied. Some concrete applications including multi-solution problems, the forced Duffing equation and the Blasius equation are given.

On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation

In this paper, we study on the initial-boundary value problem for nonlinear wave equations of higher-order Kirchhoff type with Strong Dissipation: . At first, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. Then, by “Concavity” method we establish three blow-up results for certain solutions in the case 1): , in the case 2): and in the case 3): . At last, we consider that the estimation of the upper bounds of the blow-up time is given for deferent initial energy.

Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping

In this paper, we consider the initial boundary problem for the Kirchhoff type wave equation. We prove that the Kirchhoff wave model is globally well-posed in the sufficiently regular space \begin{document}$ (H^2(\Omega)\cap H^1_0(\Omega))\times H^1_0(\Omega) $\end{document} , then, we also obtain that the semigroup generated by the equation has a global attractor in the corresponding phase space, in the presence of a quite general nonlinearity of supercritical growth.

Damped wave equation with a critical nonlinearity in higher space dimensions

We study the Cauchy problem for nonlinear damped wave equations with a critical defocusing power nonlinearity |u|2nu, where n denotes the space dimension. For n=1,2,3, global in time existence of small solutions was shown in [4]. In this paper, we generalize the results to any spatial dimension via the method of decomposition of the equation into the high and low frequency components under the assumption that the initial data are small and decay rapidly at infinity. Furthermore we present a sharp time decay estimate of solutions with a logarithmic correction.

Global existence and explosion of the stochastic viscoelastic wave equation driven by multiplicative noises

In this paper, we discuss an initial boundary value problem of stochastic viscoelastic wave equation driven by multiplicative noise involving the nonlinear damping term utq−2ut and a source term of the type up−2u. We first establish the local existence and uniqueness of solution by the iterative technique truncation function method. Moreover, we also show that the solution is global for q ≥ p. Lastly, by modifying the energy functional, we give sufficient conditions such that the local solution of the stochastic equations will blow up with positive probability or explode in energy sense for p > q.

Null controllability of some degenerate wave equations

This paper is devoted to a study of the null controllability problems for one-dimensional linear degenerate wave equations through a boundary controller. First, the well-posedness of linear degenerate wave equations is discussed. Then the null controllability of some degenerate wave equations is established, when a control acts on the non-degenerate boundary. Different from the known controllability results in the case that a control acts on the degenerate boundary, any initial value in state space is controllable in this case. Also, an explicit expression for the controllability time is given. Furthermore, a counterexample on the controllability is given for some other degenerate wave equations.

Finite time blow‐up for a kind of initial‐boundary value problem of semilinear damped wave equation

In this paper, we consider the initial‐boundary value problem of semilinear damped wave equation utt−Δu + ut=|u|p with power in an exterior domain. Blow‐up result in a finite time will be established in higher dimensions (n≥3), no matter how small the initial data are. A special test function will be constructed, and then, we obtain the blow‐up result by a contradiction argument. Copyright © 2016 John Wiley & Sons, Ltd.

Observer design for wave equation with a forcing term in the boundary

This paper investigates the observer design problem for 1D wave equation with nonlinear boundary condition containing a forcing term, whose dynamics presents spatiotemporal chaotic behaviors. By introducing a linear error input on the left-end boundary, we construct an observer via the method of characteristics. Moreover, we present a sufficient and necessary condition for the stability of the error dynamics system. Numerical simulations are presented to illustrate the theoretical outcomes.

Nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions

The aim of this paper is to study the long time behavior of solutions to the initial boundary value problem of critical semilinear wave equations with small data in exterior domain in high dimensions (n≥5). We prove that solutions cannot exist globally in time. The novelty is that we first use a cutoff function to transform the exterior problem to a Cauchy problem. Then we divide the time into two intervals, in one time interval we establish an energy estimate to control the error terms while in the other interval the asymptotic behavior of two test functions is used. Furthermore, we obtain the upper bound of the lifespan by following the idea of Zhou and Han (2014).

Theory of damped wave models with integrable and decaying in time speed of propagation

We study the Cauchy problem for damped wave equations with a time-dependent propagation speed and dissipation. The model of interest is [Formula: see text] We assume [Formula: see text]. Then we propose a classification of dissipation terms in non-effective and effective. In each case we derive estimates for kinetic and elastic type energies by developing a suitable WKB analysis. Moreover, we show optimality of results by the aid of scale-invariant models. Finally, we explain by an example that in some estimates a loss of regularity appears.

On the solvability of a boundary value problem for nonlinear wave equations in angular domains

For a one-dimensional wave equation with a weak nonlinearity, we study the Darboux boundary value problem in angular domains, for which we analyze the existence and uniqueness of a global solution and the existence of local solutions as well as the absence of global solutions.

Some remarks on quasilinear wave equations with null condition in 3‐D

In this paper, we give an alternative proof to the global existence result, which is originally owing to the pioneering work of Klainerman and Christodoulou, for the Cauchy problem of quasilinear wave equations with null condition in three space dimensions. The proof can display the following three features simultaneously: the Lorentz boost operator is not employed in the generalized energy estimates; the generalized energy of the solution will be always small, which was first observed by Alinhac; and the initial data are not assumed to have a compact support. Copyright © 2016 John Wiley & Sons, Ltd.

A note on quasilinear wave equations in two space dimensions II: Almost global existence of classical solutions

We give an alternative proof of Alinhac's almost global existence result for the Cauchy problem of quasilinear wave equations with quadratic nonlinearity satisfying the null condition in 2-D (S. Alinhac, 2001, [3]). The main innovation of our proof is that the Lorentz boost operator is not employed in the generalized energy estimates.

Behavior of the formal Fourier solution of the wave equation with a summable potential

The convergence of the formal Fourier solution to a mixed problem for the wave equation with a summable potential is analyzed under weaker assumptions imposed on the initial position u(x, 0) = φ(x) than those required for a classical solution up to the case φ(x)∈ L p [0,1] for p > 1. It is shown that the formal solution series always converges and represents a weak solution of the mixed problem.

РЕЗОЛЬВЕНТНЫЙ ПОДХОД К МЕТОДУ ФУРЬЕ В СМЕШАННОЙ ЗАДАЧЕ ДЛЯ НЕОДНОРОДНОГО ВОЛНОВОГО УРАВНЕНИЯ

РЕЗОЛЬВЕНТНЫЙ ПОДХОД К МЕТОДУ ФУРЬЕ В СМЕШАННОЙ ЗАДАЧЕ ДЛЯ НЕОДНОРОДНОГО ВОЛНОВОГО УРАВНЕНИЯ

ОБОСНОВАНИЕ МЕТОДА ФУРЬЕ В СМЕШАННОЙ ЗАДАЧЕ ДЛЯ ВОЛНОВОГО УРАВНЕНИЯ С НЕНУЛЕВОЙ НАЧАЛЬНОЙ СКОРОСТЬЮ

ОБОСНОВАНИЕ МЕТОДА ФУРЬЕ В СМЕШАННОЙ ЗАДАЧЕ ДЛЯ ВОЛНОВОГО УРАВНЕНИЯ С НЕНУЛЕВОЙ НАЧАЛЬНОЙ СКОРОСТЬЮ

A note on quasilinear wave equations in two space dimensions

In this paper, we give an alternative proof of Alinhac's global existence result for the Cauchy problem of quasilinear wave equations with both null conditions in two space dimensions[S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001) 597--618]. The innovation in our proof is that when applying the vector fields method to do the generalized energy estimates, we don't employ the Lorentz boost operator and only use the general space-time derivatives, spatial rotation and scaling operator.

The g-analytic Function Theory and Wave Equation

In this paper we develop a $g$-analytic function and a $g$-harmonic function theory for one-dimensional wave equation in the Minkowski space. In terms of the Minkowskian polar coordinates we can derive a set of complete hyperbolic type Trefftz bases, which can be transformed to polynomials as the bases for a trial solution of wave equation. The Cauchy-Riemann equations and the Cauchy theoremfor $g$-analytic functions are proved, and meanwhilethe existence of Cauchy integral formula is disproved from thenon-uniqueness of the Dirichlet problem for wave equation under the boundary conditions on whole boundary, which isalso known as the backward wave problem (BWP).Examples are used to demonstrate these results.