In this thesis, we study several stochastic partial differential equations (SPDE’s) in the spatial domain R, driven by multiplicative space-time white noise. We are interested in how rough and unbounded initial data affect the random field solution and the asymptotic properties of this solution. We first study the nonlinear stochastic heat equation. A central special case is the parabolic Ander...

It is seen how to write the standard form of the four partial differential equations in four unknowns of anisotropic thermoelasticity as a single equation in one variable, in terms of isothermal and isentropic wave operators. This equation, of diffusive type, is of the eighth order in the space derivatives and seventh order in the time derivatives and so is parabolic in character. After having ...

This paper deals with the one-dimensional vertical propagation of linear adiabatic waves in plane atmospheres. In the literature there are various representations of the standard form of the wave equation from which different forms of the so called cut-off frequency are inferred. It is not uncommon that statements concerning the propagation behavior of waves are made which are based on the heig...

in this paper, the coupled dispersive (2+1)-dimensional long wave equation is studied. the traveling wave hypothesis yields complexiton solutions. subsequently, the wave equation is studied with power law nonlinearity where the ansatz method is applied to yield solitary wave solutions. the constraint conditions for the existence of solitons naturally fall out of the derivation of the soliton so...

In this paper, the Lakshmanan-Porsezian-Daniel (LPD) equation is studied. New analytical rational solitons to LPD are presented by an ansatz method. Wave solutions of three types, such as parabolic, trigonometric and hyperbolic function have been retrieved. All plotted in 3D enhance understanding their physical characteristics. These simulations, which represent behavior resulting hyperbolic, p...

The parabolic or forward scattering approximation has been used extensively in the study of wave propagation. This approximation is combined with a gaussian white noise approximation for waves propagating in a random medium. The validity of this approximation is proved for stratiied weakly uctuating random media in the high frequencies regime. The limiting distribution of the wave eld is charac...

We develop a representation of reverse time migration in terms of Fourier integral operators the canonical relations of which are graphs. Through the dyadic parabolic decomposition of phase space, we obtain the solution of the wave equation with a boundary source and homogeneous initial conditions using wave packets. On this basis, we develop a numerical procedure for the reverse time continuat...

We study the numerical recovery of Manning’s roughness coefficient for the diffusive wave approximation of the shallow water equation. We describe a conjugate gradient method for the numerical inversion. Numerical results for one-dimensional model are presented to illustrate the feasibility of the approach. Also we provide a proof of the differentiability of the weak form with respect to the co...