Problem For Wave Equation Research Articles

Blow Up of Solutions to Wave Equations with Combined Logarithmic and Power-Type Nonlinearities

In this paper, we study the initial boundary value problem for wave equations with combined logarithmic and power-type nonlinearities. For arbitrary initial energy, we prove a necessary and sufficient condition for blow up at infinity of the global weak solutions. In addition, we derive a growth estimate for the blowing up global solutions.

Inverse problem for semilinear wave equation with strong damping

The initial-boundary and the inverse coefficient problems for the semilinear hyperbolic equation with strong damping are considered in this study. The conditions for the existence and uniqueness of solutions in Sobolev spaces to these problems have been established. The inverse problem involves determining the unknown time-dependent parameter in the right-hand side function of the equation using an additional integral type overdetermination condition.

Inverse problem for wave equation of memory type with acoustic boundary conditions

In this article, for the direct problem, which consists of finding the velocity potential and the displacement of boundary points in the wave equation of memory type with initial and boundary acoustic conditions, the inverse problem of determining the memory kernel by the integral overdetermination condition is studied. By introducing a new function, the problem is reduced to a problem with homogeneous boundary conditions. Using the technique of estimating integral equations and the contraction mappings principle in Sobolev spaces, the existence and uniqueness theorem for the inverse problem is proved.

Divergent series and generalized mixed problem for wave equation

Divergent series and generalized mixed problem for wave equation

Well-posedness and inverse problems for semilinear nonlocal wave equations

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form f(x,u) under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension n∈N.

Global existence of nonlinear elastic waves: Non-hyperelastic case

For the Cauchy problem of nonlinear elastic wave equations for three-dimensional isotropic, homogeneous and hyperelastic materials with null condition, global existence of classical solutions with small initial data was proved in Agemi (2000) [1] and Sideris (2000) [17] independently. How to treat the more general non-hyperelastic case is suggested as an open problem in Agemi (2005) [2]. The aim of this paper is to resolve this problem.

Blow-up of solutions to the wave equations with memory terms in Schwarzschild spacetime

The main goal of this work is to discuss formation of singularities of solutions to the Cauchy problems for single wave equation and coupled system of wave equations with nonlinear memory terms in Schwarzschild spacetime. For the initial value problem of nonlinear wave equations, blow-up results of sign changing solutions are derived by employing the suitable test function technique which is related to the Riemann-Liouville fractional derivative and contradiction argument. Additionally, upper bound lifespan estimates of solutions to the small initial value problem of wave equations are established by imposing certain conditions on the initial values. It is worth to mention that crucial difficulty is the setting of test function framework, which has not been applied to the Schwarzschild spacetime in the previous literatures. To the best of our knowledge, the results in Theorems 1.1-1.4 are new.

Blow-Up of Solutions to Boundary Value Problems of Coupled Wave Equations with Damping Terms on Exterior Domain

Blow-Up of Solutions to Boundary Value Problems of Coupled Wave Equations with Damping Terms on Exterior Domain

Fast energy decay for wave equation with a monotone potential and an effective damping

We consider the total energy decay of the Cauchy problem for wave equations with a potential and an effective damping. We treat it in the whole one-dimensional Euclidean space [Formula: see text]. Fast energy decay like [Formula: see text] is established with the help of potential. The proofs of main results rely on a multiplier method and modified techniques adopted in [R. Ikehata and Y. Inoue, Total energy decay for semilinear wave equations with a critical potential type of damping, Nonlinear Anal. 69(4) (2008) 1396–1401].

Application of subordination principle to coefficient inverse problem for multi-term time-fractional wave equation

Application of subordination principle to coefficient inverse problem for multi-term time-fractional wave equation

An analytical approach for Yang transform on fractional-order heat and wave equation

A novel approach to locate the approximate analytical solutions for non-linear partial differential equations is presented in this paper: the Yang transformation method combined with the Caputo derivative. In the current work, we determine the fractional Heat and Wave equation’s approximate analytical solutions. This current work addresses the Yang transformation approach in addition with the Caputo derivative. The suggested method yields approximately analytical solutions in the form of series with a simple, straightforward mechanics and a proportionality dependent on values of the fractional-order derivative. A few numerical heat equation and wave equation problems are solved to show the usefulness and reliability of the method. The tabular form [tables 7–12] makes the claim that the absolute error decreased as the number of terms in the series increased. It is also confirmed that the results are graphical compatible.

Inverse problems for real principal type operators

abstract: We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. We also give two different boundary determination methods for general operators, and prove global uniqueness results for determining coefficients in nonlinear real principal type equations. The article presents a unified approach for treating inverse boundary problems for transport and wave equations, and highlights the role of propagation of singularities in the solution of related inverse problems.

A nonlinear damped transmission problem as limit of wave equations with concentrating nonlinear terms away from the boundary

In this paper we study an initial and boundary value problem for damped wave equations with nonlinear singular terms concentrating away from the boundary of the domain, on an interior neighbourhood of a hyper-surface M that collapses to M as ɛ goes to zero. We describe the conditions for well posedness of both the approximating and limit problems, as well as the convergence, at the singular limit, of the solutions of the former to solutions of the latter, when the parameter ɛ goes to zero.

The Cauchy problem for general nonlinear wave equations with doubly dispersive

The Cauchy problem for general nonlinear wave equations with doubly dispersive

Coefficient inverse problem for anisotropic time-domain wave equation

In this paper, we investigate an inverse problem of determining coefficient via partial boundary data in anisotropic media. More precisely, we study a scheme for asymptotic reconstruction on coefficient inverse problem for anisotropic time-domain wave equation by a geometric control method. As an initial attempt, it seems natural to treat the case of two dimensional Euclidean space in analogy with many other well-known inverse boundary problems.

Several different ways to increase the accuracy of the numerical solution of the first order wave equation

The article presents several different ways to increase the accuracy of the numerical solution of differential equations. The comparison of schemes with different accuracy for the first order wave equation problem is presented. The schemes of the first order of upwind scheme, the second order of accuracy of McCormack, the third order of accuracy of Warming–Cutler–Lomax, and the scheme of the fourth order of accuracy of Abarbanel–Gotlieb–Turkel were applied. The condensed computational grids for the McCormack scheme are used, and the results using an adaptive grid for the McCormack scheme were compared.

Inverse Coefficient Problem for Fractional Wave Equation with the Generalized Riemann–Liouville Time Derivative

Inverse Coefficient Problem for Fractional Wave Equation with the Generalized Riemann–Liouville Time Derivative

Solving the backward problem for time-fractional wave equations by the quasi-reversibility regularization method

Solving the backward problem for time-fractional wave equations by the quasi-reversibility regularization method

Deep neural networks learning forward and inverse problems of two-dimensional nonlinear wave equations with rational solitons

In this paper, we investigate the forward problems on the data-driven rational solitons for the (2+1)-dimensional Kadomtsev–Petviashvili-I (KP-I) equation and spin-nonlinear Schrödinger (spin-NLS) equation via the deep neural networks learning. Moreover, the inverse problems of the (2+1)-dimensional KP-I equation and spin-NLS equation are studied via deep learning. The main idea of the data-driven forward and inverse problems is to use the deep neural networks with the activation function to approximate the solutions of the considered (2+1)-dimensional nonlinear wave equations by optimizing the chosen loss functions related to the considered nonlinear wave equations.

A Globally Stable Self-Similar Blowup Profile in Energy Supercritical Yang-Mills Theory

This paper is concerned with the Cauchy problem for an energy-supercritical nonlinear wave equation in odd space dimensions that arises in equivariant Yang-Mills theory. In each dimension, there is a self-similar finite-time blowup solution to this equation known in closed form. It will be proved that this profile is stable in the whole space under small perturbations of the initial data. The blowup analysis is based on a recently developed coordinate system called hyperboloidal similarity coordinates and depends crucially on growth estimates for the free wave evolution, which will be constructed systematically for odd space dimensions in the first part of this paper. This allows to develop a nonlinear stability theory beyond the singularity.