In this paper we examine a spatially two–dimensional forced, damped Boussinesq equation with quadratic nonlinearity. This equation describes small, nonlinear oscillations of a circular, elastic membrane in the presence of viscosity. The eigenfunction expansion method is used for the purpose of constructing solutions for small initial data analytically. The constructive method enables us to perf...

On compact manifolds (M, g) without boundary of dimension n ≥ 2, the gradient estimates for unit band spectral projection operators χλ is proved for any second order elliptic differential operators L by maximum principle. A new proof of Hörmander Multiplier Theorem on the eigenfunction expansion of the operator L is given in this setting by using the gradient estimates and the Calderón-Zygmund ...

This paper proposes a new approach to solve finite-horizon optimal stopping problems for a class of Markov processes that includes one-dimensional diffusions, birth-death (BD) processes, and jump-diffusions and continuous-time Markov chains obtained by time changing diffusions and BD processes with Lévy subordinators. When the expectation operator has a purely discrete spectrum in the Hilbert s...

Abstract The purpose of this paper is to give a direct proof an eigenfunction expansion formula for one-dimensional two-state quantum walks, which analog that Sturm–Liouville operators due Weyl, Stone, Titchmarsh, and Kodaira. In the context theory CMV matrices, it had been already established by Gesztesy–Zinchenko. Our approach restricted class walks mentioned above, whereas gives some importa...

A recent result of Muckenhoupt concerning the convergence of the expansion of an arbitrary function in terms of the Hermite series of orthogonal polynomials is generalised to a class of orthogonal expansions which arise from an eigenfunction problem associated with a second-order linear differential equation.

We derive the biorthogonality condition for axisymmetric Stokes flow in a region between two concentric spheres. This biorthogonality condition is a property satisfied by the eigenfunctions and adjoint eigenfunctions, which is needed to compute the coefficients of the eigenfunction expansion solution of the corresponding creeping flow problem.

Light transport in superdiffusive media of finite size is studied theoretically. The intensity Green's function for a slab geometry is found by discretizing the fractional diffusion equation and employing the eigenfunction expansion method. Truncated step length distributions and complex boundary conditions are considered. The profile of a coherent backscattering cone is calculated in the super...