Numerical solution of functional integral equations by the variational iteration method

AbstractIn the present article, we apply the variational iteration method to obtain the numerical solution of the functional integral equations. This method does not need to be dependent on linearization, weak nonlinearity assumptions or perturbation theory. Application of this method in finding the approximate solution of some examples confirms its validity. The results seem to show that the method is very effective and convenient for solving such equations

Application of a new variational functional for electron-molecule collisions: an extension of the Schwinger variational principle

Discusses a variational functional for scattering theory which has been recently proposed by Takatsuka and McKoy (1980). It is shown that this functional can provide results with a purely discrete set of functions which are approximately equivalent to those obtained by Lucchese et al. (1980) from the first iteration of the iterative Schwinger method. Applications to the scattering of electrons by systems including CO+ and LiH illustrate this relationship and other features of the method

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

Fractional Variational Iteration Method for Fractional Nonlinear Differential Equations

Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to overcome such shortcomings, a fractional variational iteration method is proposed. The Lagrange multipliers can be identified explicitly based on fractional variational theory.Comment: 12 pages, 1 figure

Variational Approximations in a Path-Integral Description of Potential Scattering

Using a recent path integral representation for the T-matrix in nonrelativistic potential scattering we investigate new variational approximations in this framework. By means of the Feynman-Jensen variational principle and the most general ansatz quadratic in the velocity variables -- over which one has to integrate functionally -- we obtain variational equations which contain classical elements (trajectories) as well as quantum-mechanical ones (wave spreading).We analyse these equations and solve them numerically by iteration, a procedure best suited at high energy. The first correction to the variational result arising from a cumulant expansion is also evaluated. Comparison is made with exact partial-wave results for scattering from a Gaussian potential and better agreement is found at large scattering angles where the standard eikonal-type approximations fail.Comment: 35 pages, 3 figures, 6 tables, Latex with amsmath, amssymb; v2: 28 pages, EPJ style, misprints corrected, note added about correct treatment of complex Gaussian integrals with the theory of "pencils", matches published versio