11,592 research outputs found
An analytical approximation to the solution of a wave equation by a variational iteration method
AbstractIn this work, a variational iteration method, which is a well-known method for solving functional equations, has been employed to solve the general form of a wave equation which governs numerous scientific and engineering experimentations. Some special cases of wave equations are solved as examples to illustrate the capability and reliability of the method. The results reveal that the method is very effective. The restrictions of the method are mentioned
Arbitrary l-state solutions of the rotating Morse potential by the asymptotic iteration method
For non-zero values, we present an analytical solution of the radial
Schr\"{o}dinger equation for the rotating Morse potential using the Pekeris
approximation within the framework of the Asymptotic Iteration Method. The
bound state energy eigenvalues and corresponding wave functions are obtained
for a number of diatomic molecules and the results are compared with the
findings of the super-symmetry, the hypervirial perturbation, the
Nikiforov-Uvarov, the variational, the shifted 1/N and the modified shifted 1/N
expansion methods.Comment: 15 pages with 1 eps figure. accepted for publication in Journal of
Physics A: Mathematical and Genera
Time-Fractional KdV Equation: Formulation and Solution using Variational Methods
In this work, the semi-inverse method has been used to derive the Lagrangian
of the Korteweg-de Vries (KdV) equation. Then, the time operator of the
Lagrangian of the KdV equation has been transformed into fractional domain in
terms of the left-Riemann-Liouville fractional differential operator. The
variational of the functional of this Lagrangian leads neatly to Euler-Lagrange
equation. Via Agrawal's method, one can easily derive the time-fractional KdV
equation from this Euler-Lagrange equation. Remarkably, the time-fractional
term in the resulting KdV equation is obtained in Riesz fractional derivative
in a direct manner. As a second step, the derived time-fractional KdV equation
is solved using He's variational-iteration method. The calculations are carried
out using initial condition depends on the nonlinear and dispersion
coefficients of the KdV equation. We remark that more pronounced effects and
deeper insight into the formation and properties of the resulting solitary wave
by additionally considering the fractional order derivative beside the
nonlinearity and dispersion terms.Comment: The paper has been rewritten, 12 pages, 3 figure
Any -state solutions of the Hulth\'en potential by the asymptotic iteration method
In this article, we present the analytical solution of the radial
Schr\"{o}dinger equation for the Hulth\'{e}n potential within the framework of
the asymptotic iteration method by using an approximation to the centrifugal
potential for any states. We obtain the energy eigenvalues and the
corresponding eigenfunctions for different screening parameters. The wave
functions are physical and energy eigenvalues are in good agreement with the
results obtained by other methods for different values. In order to
demonstrate this, the results of the asymptotic iteration method are compared
with the results of the supersymmetry, the numerical integration, the
variational and the shifted 1/N expansion methods.Comment: 14 pages and 1 figur
Formal analytical solutions for the Gross-Pitaevskii equation
Considering the Gross-Pitaevskii integral equation we are able to formally
obtain an analytical solution for the order parameter and for the
chemical potential as a function of a unique dimensionless non-linear
parameter . We report solutions for different range of values for the
repulsive and the attractive non-linear interactions in the condensate. Also,
we study a bright soliton-like variational solution for the order parameter for
positive and negative values of . Introducing an accumulated error
function we have performed a quantitative analysis with other well-established
methods as: the perturbation theory, the Thomas-Fermi approximation, and the
numerical solution. This study gives a very useful result establishing the
universal range of the -values where each solution can be easily
implemented. In particular we showed that for , the bright soliton
function reproduces the exact solution of GPE wave function.Comment: 8 figure
Finite-well potential in the 3D nonlinear Schroedinger equation: Application to Bose-Einstein condensation
Using variational and numerical solutions we show that stationary
negative-energy localized (normalizable) bound states can appear in the
three-dimensional nonlinear Schr\"odinger equation with a finite square-well
potential for a range of nonlinearity parameters. Below a critical attractive
nonlinearity, the system becomes unstable and experiences collapse. Above a
limiting repulsive nonlinearity, the system becomes highly repulsive and cannot
be bound. The system also allows nonnormalizable states of infinite norm at
positive energies in the continuum. The normalizable negative-energy bound
states could be created in BECs and studied in the laboratory with present
knowhow.Comment: 8 pages, 12 figure
A quasi-optimal coarse problem and an augmented Krylov solver for the Variational Theory of Complex Rays
The Variational Theory of Complex Rays (VTCR) is an indirect Trefftz method
designed to study systems governed by Helmholtz-like equations. It uses wave
functions to represent the solution inside elements, which reduces the
dispersion error compared to classical polynomial approaches but the resulting
system is prone to be ill conditioned. This paper gives a simple and original
presentation of the VTCR using the discontinuous Galerkin framework and it
traces back the ill-conditioning to the accumulation of eigenvalues near zero
for the formulation written in terms of wave amplitude. The core of this paper
presents an efficient solving strategy that overcomes this issue. The key
element is the construction of a search subspace where the condition number is
controlled at the cost of a limited decrease of attainable precision. An
augmented LSQR solver is then proposed to solve efficiently and accurately the
complete system. The approach is successfully applied to different examples.Comment: International Journal for Numerical Methods in Engineering, Wiley,
201
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