In the limit ε → 0, the solutions converge to piecewise smooth functions which are discontinuous accross a characteristic surface. Such solutions have source terms which are also piecewise smooth. Discontinuous sources are idealizations of smooth sources with a thin (∼ ε) transition layer. The fundamental problem addressed here is to describe the dynamics of solutions with sources with such tra...

HJ equations have a long history, dating back at least to the calculus of variations of the 19th century, and HJ equations find wide application in science, engineering, etc. Perhaps surprisingly, it was only relatively recently that a satisfactory general notion of solutions for (1) became available, with the introduction of the concept of viscosity solution (Crandall-Lions, c. 1980). The diff...

We construct a discrete family of smooth non-supersymmetric three charge geometries carrying D1 brane, D5 brane and Kaluza-Klein monopole charges in Type IIB supergravity compactified on a six-torus, which can be interpreted as the geometric description of some special states of the brane system. These solutions are asymptotically flat in four dimensions, and generalise previous supersymmetric ...

Periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation are investigated. Attention is restricted to solutions that, for the appropriate choice of certain constant parameters, reduce to solutions of the reduced Ostrovsky equation. It is shown how the nature of the waves may be categorized in a simple way by considering the value of a certain single combination ...

In this paper, we are concerned with the following fractional Schrödinger-Poisson system:    (−∆s)u + V (x)u + φu = m(x)|u|q−2|u|+ f(x,u), x ∈ Ω, (−∆t)φ = u2, x ∈ Ω, u = φ = 0, x ∈ ∂Ω, where s,t ∈ (0,1], 2t + 4s > 3, 1 < q < 2 and Ω is a bounded smooth domain of R3, and f(x,u) is linearly bounded in u at infinity. Under some assumptions on m, V and f we obtain the existence of non-trivial so...

We consider the Vlasov-Einstein system in a spherically symmetric setting and prove the existence of static solutions which are asymptotically flat and have finite total mass and finite extension of the matter. Among these there are smooth, singularity-free solutions, which have a regular center and may have isotropic or anisotropic pressure, and also solutions, which have a Schwarzschild-singu...

GNSS/INS (Global Navigation Satellite System/Inertial Navigation System) systems have found widespread usage in industry, especially in automated agriculture. Automated agriculture requires high frequency, precise, steady and smooth attitude solutions. This necessitates smooth error patterns in navigation solutions. Given that navigation solutions can be derived from different information sourc...

‎The main purpose of this paper is to study the numerical solution of nonlinear Volterra integral equations with constant delays, based on the multistep collocation method. These methods for approximating the solution in each subinterval are obtained by fixed number of previous steps and fixed number of collocation points in current and next subintervals. Also, we analyze the convergence of the...

As two-dimensional coupled system of nonlinear partial differential equations does not give enough smooth solutions, when approximated by linear, quadratic and cubic polynomials and gives poor convergence or no convergence. In such cases, approximation by zero degree polynomials like Haar wavelets (continuous functions with finite jumps) are most suitable and reliable. Therefore, modified numer...