and reducing the problem to one of iteratively solving a linear equation for un once the previous iterates have been determined. However, Adomian partitions (1.4) into a sequence of linear ODEs in either x or t whose solutions cannot generally be made to satisfy the boundary and initial conditions. Even when applied to an initial value problem with boundary conditions, the convergence of the so...

The Karhunen-Loève expansion of a Gaussian process, a common tool on finite element methods for differential equations with stochastic coefficients, is based on the spectral decomposition of its covariance function. The eigenpairs of the covariance are expressed as a Fredholm integral equation of second kind, which can be readily approximated with piecewise-constant finite elements. In this wor...

We report a suboptimal wavelet packet (WP) representation of signals emanating from a chaotic attractor contaminated by low levels of noise. Our method—geared towards choosing a suboptimal scaling function to parsimoniously represent the signalinvolves extracting local eigenfunctions using artificial ensembles generated from a pseudo-probability space, and using the extracted local eigenfunctio...

The paper proposes a stable method for constructing normal to surface given approximately. is calculated as the gradient of function in equation. As known, problem calculating derivative ill-posed. In paper, an approach adopted solving this values unbounded operator. To construct its solution, principle minimum smoothing functional Morozov’s formulation used. obtained form Fourier series expans...

PT-Symmetry is one of the interesting topics in quantum mechanics and optics. One of the demonstration of PT-Symmetric effects in optics is appeared in the nonlinear directional coupler (NLDC). In the paper we numerically investigate the stability of temporal bright solitons propagate in a PT-Symmetric NLDC by considering gain in bar and loss in cross. By using the analytical solutions of pertu...

In the article, a nonlocal boundary value problem has been investigated for fourth-order mixed-type equation degenerating inside and on of domain. Applying method separation variables to under study, spectral an ordinary differential is obtained. The Green function last constructed, with help which it equivalently reduced Fredholm integral second kind symmetric kernel, implies existence eigenva...

In this work, we deal with the autonomy issue in the perturbation expansion for the eigenfunctions of a compact Hilbert–Schmidt integral operator. Here, the autonomy points to the perturbation expansion coefficients of the relevant eigenfunction not depending on the perturbation parameter explicitly, but the dependence on this parameter arises from the coordinate change at the zero interval lim...

We study the semigroup generated by hypoelliptic Laplacian on circle and maximal bounded holomorphic extension of this semigroup. Using an orthogonal decomposition into harmonic oscillators with complex shifts, we describe domain show that boundedness in a half-plane corresponds to absolute convergence expansion eigenfunctions. This relies novel integral formula for spectral projections which a...

The approximation of the eigenvalues and eigenfunctions of an elliptic operator is a key computational task in many areas of applied mathematics and computational physics. An important case, especially in quantum physics, is the computation of the spectrum of a Schrödinger operator with a disordered potential. Unlike plane waves or Bloch waves that arise as Schrödinger eigenfunctions for period...