It is proven that every sequence from the Askey scheme of hypergeometric polynomials satisfies differentials or difference equations of first order of the form $T p_{n}(x) = A_{n}(x) p_{n-1}(x) - B_{n}(x) p_{n}(x)$, where T is a linear degree reducing operator, which leeds to the fact that these polynomial sets satisfy a relation of the form $p^{'}_{n}(x) = A_{n}(x) p_{n-1}(x) - B_{n}(x) p_{n}(x)$.

We study the existence, uniqueness, regularity and continuous dependence on initial data of solutions for the Cauchy problem associated with the coupled system of Boussinesq type equations forced by highly oscillatory smooth coefficients. The model considered describes two-way propagation of long water waves with small amplitude on the surface of a one-dimensional channel with rough bottom (disordered topography). The dependent variables in this model are the wave n elevation and the potential velocity u measured at the fixed depth Z₀ = square root of 2/3.

In this research, we consider a nonhomogeneous linear difference equation with constant coefficients of order k greater than or equal to 2 subject to boundary conditions that split into two groups: initial and final conditions. We establish necessary and / or sufficient conditions on the distribution(s) of the zeros of the associated characteristic polynomial that ensure the existence of a unique solution.

We consider a fourth order nonlinear partial differential equation in n-dimensional space introduced by Abreu in the context of Kahler metrics on toric varieties. Rotation invariant similarity solutions, depending only on the radial coordinate in Rⁿ, are determined from the solutions of a second order ordinary differential equation (ODE), with a non-autonomous Lagrangian formulation. A local asymptotic analysis of solutions of the ODE in the neighbourhood of singular points is carried out, and the existence of a class of solutions on an interval of the positive real semi-axis is proved using a nonlinear integral equation. The integrability (or otherwise) of Abreu's equation is discussed.

Using the Morawetz' Radial Identity, we show that the local energy of a solution is integrable in time and the local L² norm of the solution approaches zero as time approaches the infinity for a nonlinear beam equation with the spatial dimension > 5.

We give sufficent conditions ensuring existence and regularity of a radial solution to the Darboux equation in exterior domains.

We construct exact, entropy satisfying shock wave solutions of the Einstein equations for a perfect fluid which extend the Oppeheimer-Snyder (OS) model to the case of non-zero pressure, inside the Black Hole. These solutions put forth a new Cosmological Model in which the expanding Friedmann-Robertson-Walker (FRW) universe emerges from the Big Bang with a shock wave at the leading edge of the expansion, analogous to a classical shock wave explosion. This explosion is large enough to account for the enormous scale on which the galaxies and the background radiation appear uniform. In these models, the shock wave must lie beyond one Hubble length from the FRW center, this threshhold being the boundary across which the bounded mass lies inside its own Schwarzshild radius, 2M/r > 1, and in this sense the shock wave solution evolves inside a Black Hole. The entropy condition, which breaks the time symmetry by selecting the explosion over the implosion, also implies that the shock wave must weaken until it eventually settles down to a zero pressure OS interface, bounding a finite total mass, that emerges from the White Hole event horizon of an ambient Schwarzschild spacetime. However, unlike shock matching outside a Black Hole, the equation of state p = (c²/3) p, the equation of state at the earliest stage of Big Bang physics, is distinguished at the instant of the Big Bang--for this equation of state alone, the shock wave emerges from the Big Bang at a finite nonzero speed, the speed of light, decelerating to a subluminous wave from that time onward. These shock wave solutions indicate a new cosmological model in which the Big Bang arises from a localized White Hole explosion occurring inside a matter filled universe that eventually evolves outward through the White Hole event horizon of an asymptotically flat Schwarzschild spacetime.

We investigate the eigenvalues and eigenfunctions of the Schur complement operator of the first kind of Stokes problem posed on a plane domain and give results on the number, multiplicity and stability of these eigenvalues in dependence on the problem domain.