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...

we investigate the boundary-value problem generated by the sturm-liouville equation withdiscontinuous coefficients, eigenparameter dependent boundary conditions and transmission conditionsat the point of discontinuity. with a different approach we introduce an adequate hilbert spaceformulation, investigate some properties of eigenvalues, green’s function and resolvent operator, andfind simple c...

This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation are discussed. In particular, the asymptotic behaviour of the first eigenfunction is studied sinc...

In recent years several high quality Sturm-Liouville software packages have been written (e.g., SLEDGE, SLEIGN, the NAG routine SL02FM). All excel at providing eigenfunction estimates at a nite set of points but only the NAG routine provides interpolation capability, and this by a fairly crude method (especially when compared with the sophistication of its eigenvalue solver). We describe an acc...

The motion of a circular elastic plate floating on the surface is investigated in time-domain. solution found from single frequency solutions, and method to solve for given using eigenfunction matching method. Simple plane incident waves with Gaussian profile wavenumber space are considered, more complex focused wave group considered. Results range parameters. Code provided show how simulate mo...

The article describes the eigenvalue and eigenfunction problems. Basic properties, some applications and examples in system analysis are provided.

The propagation of light beams through astigmatic lens systems is accompanied by a transfer of orbital angular momentum. We develop a method to describe propagating light beams by operators of which the field is an eigenfunction. This method is applied to determine when an astigmatic lens system transforms gaussian beams into other gaussian beams and where in the system angular momentum is tran...