Generalized Hyers–Ulam stability of n -dimensional wave equations in the L 2 -norm

A theoretical analysis of a new second order finite volume approximation based on a low-order scheme using general admissible spatial meshes for the one dimensional wave equation

Properties of the density for a three-dimensional stochastic wave equation

This paper deals withtt-periodicity and regularity of solutions to the one dimensional nonlinear wave equation withxx-dependent coefficients

Periodic solutions for one dimensional wave equation with bounded nonlinearity

In this paper, we propose a direct approach to obtain blow-up solution to a one dimensional nonlinear wave equation. The key point, which is different from the previous works, is that the dissipation of the energy will not be used. So our method can be applied to the non-dissipative system.

In this paper, the collision dynamics of breather and lump-type localized waves in the (3+1)-dimensional shallow water wave equation are investigated in detail. Firstly, the auto-Bäcklund transformation and the linear representation of the (3+1)-dimensional shallow water wave equation are derived in virtue of the truncated Painlevé expansion method, which provide convenience in solving the (3+1)-dimensional shallow water wave equation. Secondly, based on the linear representation and the principle of linear superposition, the rational solutions in the exponential and polynomial forms are constructed. Tuning the free parameters of the rational solutions, localized waves of various patterns are obtained such as breather, lump-type localized waves and their hybrid structure. The anomalous inelastic interaction phenomenons of breather and lump-type localized waves are exhibited. Thirdly, combining the large-time behaviors of solution with the velocity relationship of localized waves, the dynamical properties and the classification of localized wave solutions are discussed in detail. Finally, we discuss the bound state of breather and lump-type localized waves under the velocity resonance condition, three different types of lump-breather molecules are displayed. The obtained results further enrich the structures and dynamical behaviors of localized waves. It is expected that the interaction phenomena taking place in the (3+1)-dimensional shallow water wave equation will be helpful in predicting or controlling some related shallow water wave phenomena.

A superconvergent isogeometric formulation is presented to compute the eigenvalues for three dimensional wave equation. This three dimensional superconvergent isogeometric formulation is characterized by a higher order mass matrix formulation with particular reference to the quadratic basis functions. The three dimensional higher order mass matrix is built upon an optimal combination of the reduced bandwidth mass matrix and the consistent mass matrix. The frequency error associated with the isogeometric discretization of three dimensional wave equation is derived in detail. In particular, the optimal mass combination parameter for higher order mass matrix is devised as a function of the two spatial wave propagation angles, which enables that arbitrary frequency corresponding to a given wave propagation direction can be computed in a superconvergent way. Two extra orders of accuracy, i.e., 6th order of accuracy, are attained by the proposed higher order mass matrix than the consistent mass matrix for the frequency computation of three dimensional wave equation. The dispersion property of the present three dimensional higher order mass matrix formulation is examined as well. The accuracy of the proposed three dimensional superconvergent isogeometric formulation is testified by several numerical examples.

On computation of solution for [formula omitted] dimensional fractional order general wave equation

The analysis of a generalised (3+1)-dimensional nonlinear wave equation that simulates a variety of nonlinear processes that occur in liquids including gas bubbles will be performed. After some cosmetic adjustments to the underlying equation, this generalised (3+1)-dimensional nonlinear wave equation naturally degenerates into the (3+1)-dimensional Kadomtsev-Petviashvili equation, the (3+1)-dimensional nonlinear wave equation, and the Korteweg-de Vries equation. To completely investigate this fundamental equation, a clear and rigorous technique is used. In order to obtain innovative symmetry reductions, group invariant solutions, conservation laws, and eventually kink wave solutions, the Lie symmetry, multiplier, and simplest equation methods are used. Complex waves and their dealing dynamics in fluids can be well imitated by the verdicts.

A weighted numerical algorithm for two and three dimensional two-sided space fractional wave equations

Exact solutions of the generalized [formula omitted]-dimensional shallow water wave equation

Construction of new soliton-like solutions of the (2 + 1) dimensional dispersive long wave equation

On the structure of conservation laws of (3+1)-dimensional wave equation

A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a (1+1)-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.

We construct a family of compact fourth order accurate finite difference schemes for the three dimensional scalar wave (d’Alembert) equation with constant or variable propagation speed. High order accuracy is of key importance for the numerical simulation of waves as it reduces the dispersion error (i.e., the pollution effect). The schemes that we propose are built on a stencil that has only three nodes in any coordinate direction or in time, which eliminates the need for auxiliary initial or boundary conditions. These schemes are implicit in time and conditionally stable. A particular scheme with the maximum Courant number can be chosen within the proposed class. The inversion at the upper time level is done by FFT for constant coefficients and multigrid for variable coefficients, which keeps the overall complexity of time marching comparable to that of a typical explicit scheme.