The   ) ( exp    -expansion method is a promising method for finding exact traveling wave solutions to nonlinear evolution equations in physical sciences. In this article, we use the   ) ( exp    -expansion method to find the exact solutions for the nonlinear Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation and the good Boussinesq equations. Many solitary wave solutions are formally d...

Nowadays NLEEs have been the subject of allembracing studies in various branches of nonlinear sciences. Most of the phenomena in real world can be described using non-linear equations. A nonlinear phenomenon plays a vital role in applied mathematics, physics and engineering branches. Many complex nonlinear phenomenons in plasma physics, fluid dynamics, chemistry, biology, mechanics, elastic med...

The exact traveling wave solutions of the nonlinear variable coefficients Burgers-Fisher equation and the generalized Gardner equation with forced terms can be found in this article using the generalized ( ′ G )-expansion method. As a result, hyperbolic, trigonometric and rational function solutions with parameters are obtained. When these parameters are taken special values, the solitary wave ...

We consider the spectral problem for a perturbed two-dimensional oscillator. The role of perturbation is played by an integral Hartree type nonlinearity with self-action potential depending on distance between points and possessing Coulomb singularity. find asymptotic eigenvalues eigenfunctions near boundaries clusters appearing unperturbed operator. we construct expansion circle, where solutio...

A method to compute the bound state eigenvalues and eigenfunctions of a Schrödinger equation or a spinless Salpeter equation with central interaction is presented. This method is the generalization to the three-dimensional case of the Fourier grid Hamiltonian method for one-dimensional Schrödinger equation. It requires only the evaluation of the potential at equally spaced grid points and yield...

Let H = −d/dx + V be a Schrödinger operator on the real line, where V ∈ L ∩ L. We define the perturbed Fourier transform F for H and show that F is an isometry from the absolute continuous subspace onto L(R). This property allows us to construct a kernel formula for the spectral operator φ(H). Schrödinger operator is a central subject in the mathematical study of quantum mechanics. Consider the...

We consider the method of particular solutions for numerically computing eigenvalues and eigenfunctions of the Laplacian on a smooth, bounded domain Ω in Rn with either Dirichlet or Neumann boundary conditions. This method constructs approximate eigenvalues E, and approximate eigenfunctions u that satisfy ∆u = Eu in Ω, but not the exact boundary condition. An inclusion bound is then an estimate...

We present a novel method for flexible and efficient simulation of example-based elastic deformation. The geometry of all input shapes is projected into a common shape space spanned by the Laplace-Beltrami eigenfunctions. The eigenfunctions are coupled to be compatible across shapes. Shape representation in the common shape space is scale-invariant and topology-independent. The limitation of pr...

This paper introduces a numerical method for computing the spectrum of adjoint (left) eigenfunctions of spiral wave solutions to reaction-diffusion systems in arbitrary geometries. The method is illustrated by computing over a hundred eigenfunctions associated with an unstable time-periodic single-spiral solution of the Karma model on a square domain. We show that all leading adjoint eigenfunct...

In this paper, we consider the problem of estimating the eigenvalues and eigenfunctions of the covariance kernel (i.e., the functional principal components) from sparse and irregularly observed longitudinal data. We approach this problem through a maximum likelihood method assuming that the covariance kernel is smooth and finite dimensional. We exploit the smoothness of the eigenfunctions to re...