Related papers: Discrete Boussinesq-type equations
This paper introduces a second-order time discretization for solving the incompressible Boussinesq equation. It uses the generalized scalar auxiliary variable (GSAV) and a backward differentiation formula (BDF), based on a Taylor expansion…
This paper presents a finite-dimensional approximation for a class of partial differential equations on the space of probability measures. These equations are satisfied in the sense of viscosity solutions. The main result states the…
This article presents a concise survey of basic discrete and semi-discrete nonlinear models which produce two- and three-dimensional (2D and 3D) solitons, and a summary of main theoretical and experimental results obtained for such…
We present particular solutions of the discrete Painlev\'e III (d-P$\rm_{III}$) equation of rational and special function (Bessel) type. These solutions allow us to establish a close parallel between this discrete equation and its…
{We provide a probabilistic approach in order to investigate the smoothness of the solution to the Poisson and Dirichlet problems in $L$-shaped domains. In particular, we obtain (probabilistic) integral representations for the solution. We…
This work oversees with the coupled Schr\"odinger-Boussinesq equations with conformable derivative, which have lots of applications in laser and plasma. The said equations are reduced to a coupled stationary form using complex travelling…
We establish a priori Lipschitz estimates for equations with mixed local and nonlocal diffusion, coercive gradient terms and unbounded right-hand side in Lebesgue spaces through an integral refinement of the Bernstein method. This relies on…
We develop a fully discrete, semi-implicit mixed finite element method for approximating solutions to a class of fourth-order stochastic partial differential equations (SPDEs) with non-globally Lipschitz and non-monotone nonlinearities,…
This paper is concerned with the numerical approximation of initial-boundary-value problems of a three-parameter family of Bona-Smith systems, derived as a model for the propagation of surface waves under a physical Boussinesq regime. The…
In this paper, we first address the space-time decay properties for higher order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a…
A class of backward doubly stochastic differential equations (BDSDEs in short) with continuous coefficients is studied. We give the comparison theorems, the existence of the maximal solution and the structure of solutions for BDSDEs with…
Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus we generalize a notion of Braess and Sch\"oberl, originally studied for a posteriori error estimation. We construct…
A class of discrete equations is considered from three perspectives corresponding to three measures of the complexity of solutions: the (hyper-) order of meromorphic solutions in the sense of Nevanlinna, the degree growth of iterates over a…
We study compact polyhedral surfaces as Riemann surfaces and their discrete counterparts obtained through quadrilateral cellular decompositions and a linear discretization of the Cauchy-Riemann equation. By ensuring uniformly bounded…
In this paper, we introduce the reverse-space and reverse-space-time nonlocal discrete derivative nonlinear Schr\"odinger (DNLS) equations through the nonlocal symmetry reductions of the semi-discrete Gerdjikov-Ivanov equation. The…
Mixed type (bright-dark) soliton solutions of the integrable N-coupled nonlinear Schr{\"o}dinger (CNLS) equations with mixed signs of focusing and defocusing type nonlinearity coefficients are obtained by using Hirota's bilinearization…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
In this paper, we present the two-dimensional generalized nonlinear Schr\"odinger equations with the Lax pair. These equations are related to many physical phenomena in the Bose-Einstein condensates, surface waves in deep water and…
Three weakly nonlinear but fully dispersive Whitham-Boussinesq systems for uneven bathymetry are studied. The derivation and discretization of one system is presented. The numerical solutions of all three are compared with wave gauge…
In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of ill-posed partial differential equations. As a case study, our proposed method is applied to the Boussinesq equation, which has been investigated…