We establish the uniqueness and local existence of weak solutions for a system of partial differential equations which describes non-linear motions of viscous stratified fluid in a homogeneous gravity field. Due to the presence of the stratification equation for the density, the model and the problem are new and thus different from the classical Navier-Stokes equations. Impasse singularities of...

We derive a lower bound on the large-time exponential behavior of the solution to a stochastic parabolic partial differential equation on R+ × R in the case of a spatially homogeneous Gaussian potential that is white-noise in time, and study the relation between the lower bound and the potential’s spatial modulus of continuity.

In this paper, we apply the homotopy perturbation method (HPM) for solving the Cauchy problems of the first-order partial differential equation. We obtain the exact solutions to the homogeneous or inhomogeneous linear and nonlinear equations. Numerical results are presented to confirm the efficiency of the HPM. Mathematics Subject Classification: 35K15, 35C05, 65D99, 65M99

The Richards equation is widely used as a model for the flow of water in unsaturated soils. For modelling one-dimensional flow in a homogeneous soil, this equation can be cast in the form of a specific nonlinear partial differential equation with a time derivative and one spatial derivative. This paper is a survey of recent progress in the pure mathematical analysis of this last equation. The e...

in this paper, a new fractional sub-equation method is proposed for finding exact solutions of fractional partial differential equations (fpdes) in the sense of modified riemann-liouville derivative. with the aid of symbolic computation, we choose the space-time fractional zakharov-kuznetsov-benjamin-bona-mahony (zkbbm) equation in mathematical physics with a source to illustrate the validity a...

the present study introduces a new technique of homotopy perturbation method for the solution of systems of fractional partial differential equations. the proposed scheme is based on laplace transform and new homotopy perturbation methods. the fractional derivatives are considered in caputo sense. to illustrate the ability and reliability of the method some examples are provided. the results ob...

We extend Walsh’s theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulatio...

This paper describes a new comparison principle that can be used for the comparison of space-time estimates for dispersive equations. In particular, results are applied to the global smoothing estimates for several classes of dispersive partial differential equations.