In this dissertation, we study the Karhunen-Loève (KL) expansion and the exact L small ball probability for Gaussian processes. The exact L small ball probability is connected to the Laplace transform of the Gaussian process via Sytaja Tauberian theorem. Using this technique, we solved the problem of finding the exact L small ball estimates for the Slepian process S(t) defined as S(t) = W (t+a)...

In this work decay estimates are derived for the solutions of 1-D linear parabolic PDEs with disturbances at both boundaries and distributed disturbances. The decay estimates are given in the 2 L and 1 H norms of the solution and discontinuous disturbances are allowed. Although an eigenfunction expansion for the solution is exploited for the proof of the decay estimates, the estimates do not re...

We analyze the Ritz–Galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates. For a simple eigenvalue, we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants. For a multiple eigenvalue we prove, in addition, apparently the first truly a priori error estimates th...

The acoustic attenuation performance of perforated dissipative circular expansion chambers with inlet/outlet extensions is investigated. The eigenvalues and eigenfunctions of the sound field are analytically determined in the extended inlet/outlet circular ducts, upstream/downstream end annular dissipative chambers, and the central perforated dissipative expansion chamber. Utilizing the continu...

In this work we study the application of the Method of Fundamental Solutions (MFS) to the numerical calculation of the eigenvalues and eigenfunctions for the 2D Bilaplacian in simply connected plates. This problem was considered in [20] using wave-type functions and in [14] using radial basis function for a circular and rectangular domain. The MFS is a meshfree method that was already applied t...

Perron-Frobenius operators and their eigendecompositions are increasingly being used as tools of global analysis for higher dimensional systems. The numerical computation of large, isolated eigenvalues and their corresponding eigenfunctions can reveal important persistent structures such as almostinvariant sets, however, often little can be said rigorously about such calculations. We attempt to...

We demonstrate that the G ′ /G expansion method which is often used in finding exact solutions of nonlinear differential equation is equivalent to the well – known tanh method and application of these methods gives the same exact solutions of nonlinear differential equations.

This is a survey on eigenfunctions of the Laplacian on Riemannian manifolds (mainly compact and without boundary). We discuss both local results obtained by analyzing eigenfunctions on small balls, and global results obtained by wave equation methods. Among the main topics are nodal sets, quantum limits, and L norms of global eigenfunctions. The emphasis is on the connection between the behavio...

Marseille Cedex 9 France One-dimensional quantum scattering for potentials de ned as measures Charles-Antoine GUERIN1 Abstract We generalize the basic one-dimensional scattering formalism to potentials de ned as measures and retrieve the classical results that hold for smooth potentials. We introduce a set of generalized eigenfunctions for the corresponding Schr odinger operator and study thei...