A. Elhanbaly

Adomian decomposition method for solving fractional nonlinear differential equations

In this article, we have discussed a new application of Adomian decomposition method on time fractional nonlinear fractional differential equations. Three models with fractional-time derivative of order @a, 0<@a<1 are considered and solved by means of Adomian decomposition method. The behaviour of Adomian solutions and the effects of different values of @a are investigated. Numerical examples are tested to illustrate the pertinent feature of the proposed algorithm.

Adomian decomposition method for solving the diffusion–convection–reaction equations

Abstract Adomian decomposition method is used to solve the explicit and numerical solutions of three types of the diffusion–convection–reaction (DECR) equations. The calculations are carried out for three different types of the DCRE such as, the Black–Scholes equation used in financial market option pricing, Fokker–Planck (FP) equation for lazer filed and Fokker–Planck equation from plasma physics. The behaviour of the approximate solutions of the distribution functions is shown graphically and compared with that obtained by other theories such as the variational iteration method.

Fractional (space-time) Fokker–Planck equation

Abstract By using Kramers–Moyal forward expansion and the definition of characteristic function (CF) with some consideration related to derivatives of fractional order, one can obtain the fractional space-time Fokker–Planck equation (FFPE) ∂ β p(x,t) ∂ t β =(−i) γ D γ x σ(x,t) p(x,t), 0 0 The obtained equation could be related to a dynamical system subject to fractional Brownian motion. Therefore, the solution of FFPE will be established on three different cases that correspond to different physical situations related to the mean-square displacement, 〈(x(t+τ)−x(t))2〉∼σ(x,t)τβ.

Maximum-entropy approach with higher moments for solving Fokker–Planck equation

The maximum-entropy method with higher number of moments is used to solve the Fokker–Planck equation. An adopted Newton method is used to iterate the maximum entropy set of equations. The method is used to calculate the probability density function of the Fokker–Planck equation. The calculations are carried out for three examples. (1) The bistable systems of double well potential that is used in many problems related to the fluctuation and relaxation processes in far from equilibrium systems. (2) The Malthus–Verhulst model, which is used to study the evolution of the number of individuals of an ecological species and the evolution of the intensity of the laser light. (3) The Black–Scholes equation used in financial market option pricing. Although the maximum-entropy approach has several advantages, it is not convergent at large times and so cannot be used to calculate the steady state solution.

Solution of Fokker–Planck equation by means of maximum entropy approach

Abstract Maximum entropy approach is used to find the exact solution of the one-dimensional Fokker–Planck equation with variable coefficients. The exact solutions of the probability density of generalized Fokker–Planck equation are explicitly obtained. Three classes of special interest of Fokker–Planck equations are discussed in details.

Maximum entropy method for solving the collisional Vlasov equation

By means of the maximum entropy method the reduced Vlasov–Fokker–Planck equation has been solved explicitly. The behaviour of the approximate maximum entropy distribution functions are shown graphically, and compared with that obtained by using Lie group method. The results reported in this article provide further evidence of the usefulness of both maximum entropy and Lie group methods for obtaining time dependent solution in compact form.

New application of Adomian decomposition method on Fokker-Planck equation

Adomian decomposition method has been used for finding the explicit and numerical solutions of the reduced Fokker Planck equation in a force field. The obtained results are compared with those found by other theories such as Lie group theory. The behaviour of the approximate distribution functions are shown graphically. The results of Lie group method confirm the correctness of those obtained by Adomian decomposition method.

On the solution of Spencer–Lewis equation in an infinite medium

Abstract Two different techniques have been used to solve the Spencer–Lewis (S–L) equation in an infinite medium plane geometry namely, maximum entropy and flux-limited. The two solutions for the total electron density by using the flux-limited as well by means of the maximum entropy are explicitly obtained.

On the solution of Fokker-Planck equation for electron transport

Abstract Two different techniques have been used to solve the Fokker–Planck equation for electron transport in infinite homogeneous medium namely, maximum entropy and flux-limited approach. The solutions obtained for the scalar flux function φ0(x,s) by both methods are numerically compared.

Solution of Spencer–Lewis equation in an infinite medium plane and spherical geometries

Abstract The total electron density and energy deposition profile for electrons incident on an aluminium slab have been studied. This has been achieved by using moment technique and Legendre expansion for solving Spencer–Lewis (S–L) equation. The solution has been obtained in two different geometries, i.e., in an infinite medium plane and spherical geometries. Then, numerical calculations have been done for the total electron density and energy deposition profile for electrons incident on a 0.04 ( g / cm 2 ) thick aluminium slab. The obtained results are satisfactory when compared with that found previously (Filippone, Nucl Sci Eng 95 (1987) 22).