We prove a convergence result for a natural discretization of the Dirichlet problem of the elliptic Monge-Ampère equation using finite dimensional spaces of piecewise polynomial C functions. Discretizations of the type considered in this paper have been previously analyzed in the case the equation has a smooth solution and numerous numerical evidence of convergence were given in the case of non...

In this paper we show that non-smooth functions which are distributional traveling wave solutions to the two component Camassa-Holm equation provided set $u^{-1}(c)$, where $c$ is speed of wave, measure zero. particular there no new peakon or cuspon beyond those already satisfying equation. However, has distinct from smooth as well when $u^{-1}(c)$ not We provide examples such solutions.

The fundamental “two-fluid” model for describing plasma dynamics is given by the EulerMaxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. We prove global stability of a constant neutral background, in the sense that irrotational, smooth and localized perturbations of a constant background with small amplitude lead to globa...

The article first studies the propagation of well prepared high frequency waves with small amplitude ε near constant solutions for entropy solutions of multidimensional nonlinear scalar conservation laws. Second, such oscillating solutions are used to highlight a conjecture of Lions, Perthame, Tadmor, ([23]), about the maximal regularizing effect for nonlinear conservation laws. For this purpos...

The modified Szabo wave equation is one of the various models that have been developed to model the power law frequency-dependent attenuation phenomena in lossy media. The purpose of this study is to develop two different efficient numerical methods for the two-dimensional Szabo equation and to compare the relative merits of each method. In both methods we employ the ADI scheme to split directi...

Piecewise quasilinearization methods for singular boundary-value problems in second-order ordinary differential equations are presented. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus yielding piecewise analytical solutions. The accuracy of the globally smooth piecewise quasilinear method is assessed by comparisons...

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharactersitic C H-minimal graphs whose components of the unit horizontal Gauss map are in W 1,1 are ruled surfaces with C seed curves. In a different direction, we investigate ways in which patches of C H-minimal graphs can be glued together to form continuous piecewise C Hminimal surfaces. We apply these descri...

By using the bifurcation theory of dynamical systems to study the dynamical behavior of the Green–Naghdi equations, the existence of solitary wave solutions along with smooth periodic traveling wave solutions is obtained. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact and explicit parametric rep...

in this paper we study a two-phase free boundary problem for a semilinear elliptic equation on a bounded domain $dsubset mathbb{r}^{n}$ with smooth boundary‎. ‎we give some results on the growth of solutions and characterize the free boundary points in terms of homogeneous harmonic polynomials using a fundamental result of caffarelli and friedman regarding the representation of functions whose ...