Essam M. Abulwafa

The Pomraning-Eddington approximation to diffusion of light in turbid materials

Abstract The source-free diffusion problem of light in turbid media with generalized boundary conditions is considered. The intensity of light is considered as a sum of collimated and diffused radiance. In this way the problem is transformed to a source problem with a collimated source (problem 1). This problem is solved in terms of the corresponding source-free problem of simple boundary conditions (problem 2). The Pomraning-Eddington method is used to solve problem 2. Two coupled first-order differential equations are obtained involving the energy density and the radiation net flux. Weight functions are introduced in order to force the boundary conditions to be fulfilled. Numerical results are given and compared with previous calculations. The calculations show that the accuracy depends on the choice of the weight function.

Time-fractional KdV equation for plasma of two different temperature electrons and stationary ion

Using the time-fractional KdV equation, the nonlinear properties of small but finite amplitude electron-acoustic solitary waves are studied in a homogeneous system of unmagnetized collisionless plasma. This plasma consists of cold electrons fluid, non-thermal hot electrons, and stationary ions. Employing the reductive perturbation technique and the Euler-Lagrange equation, the time-fractional KdV equation is derived and it is solved using variational method. It is found that the time-fractional parameter significantly changes the soliton amplitude of the electron-acoustic solitary waves. The results are compared with the structures of the broadband electrostatic noise observed in the dayside auroral zone.

The fractional Fokker–Planck equation on comb-like model

The fractional Fokker–Planck equation, were used to describe the anomalous diffusion in external fields, is derived using a comb-like structure as a background model. For the force-free case, the distribution function associated with space dependence diffusion coefficient along the backbone of the structure are obtained in a closed form of H-function. The operator method has been used to solve the fractional Fokker–Planck equation taking the external field into account.

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.

Radiative-transfer in a linearly-anisotropic spherical medium

Abstract The radiative-transfer problem in an absorbing, emitting, anisotropically scattering, inhomogenous, spherical shell is considered. The medium is taken to have diffusely reflecting boundaries and an internal energy source. The Galerkin method is used to calculate the partial heat fluxes at the boundaries, the radiation density and the net radiant heat flux through the medium. The calculations are carried out for isotropic scattering, forward and backward anisotropic scattering, for linear and quadratic space-dependent, single-scattering albedos, and for different internal sources.

Conductive-radiative heat transfer in an inhomogeneous slab with directional reflecting boundaries

Conductive-radiative heat transfer through an inhomogeneous, anisotropic scattering, turbid plane-parallel medium is considered. The medium is taken to be of diffuse and specular reflecting boundaries. The specular reflecting coefficients of the boundaries are considered to be angular dependent. The variational technique is used to solve the radiative problem while an iterative method is taken to include the nonlinearity effect of the temperature distribution of the medium from the conductive energy equation. The dimensionless temperature distribution and conductive, radiative and total net fluxes through the medium are calculated. The calculations for matched boundaries and a homogeneous, isotropic scattering slab are calculated and compared with other calculations to show good agreement. The calculations are carried out for mismatched boundaries, with homogeneous and inhomogeneous media. The results are given for isotropic and forward linear anisotropic scattering slabs.

Time-fractional study of electron acoustic solitary waves in plasma of cold electron and two isothermal ions

In this paper, a homogeneous system of unmagnetized collisionless plasma consisting of a cold electron fluid, low-temperature ion obeying Boltzmann-type distribution and high-temperature ion obeying non-thermal distribution is considered. The perturbation method with two different forms of stretching will be considered to drive the KdV and modified KdV (mKdV) equations. The Agrawals method is applied to formulate the time-fractional KdV and mKdV equations. A variational iteration method is used to solve these equations. The results show that the fractional order of the differential equations can be used to modify the shape of the solitary pulse instead of adding higher order dissipation terms to the equations. This study may be useful to construct the compressive and rarefactive electrostatic potential pulses associated with the broadband electrostatic noise type emissions.

Ion-acoustic waves in plasma of warm ions and isothermal electrons using time-fractional KdV equation

The ion-acoustic solitary wave in collisionless unmagnetized plasma consisting of warm ions-fluid and isothermal electrons is studied using the time fractional KdV equation. The reductive perturbation method has been employed to derive the Korteweg-de Vries equation for small but finite amplitude ion-acoustic wave in warm plasma. The Lagrangian of the time fractional KdV equation is used in a similar form to the Lagrangian of the regular KdV equation with fractional derivative for the time differentiation. The variation of the functional of this Lagrangian leads to the Euler—Lagrange equation that gives the time fractional KdV equation. The variational-iteration method is used to solve the derived time fractional KdV equation. The calculations of the solution are carried out for different values of the time fractional order. These calculations show that the time fractional can be used to modulate the electrostatic potential wave instead of adding a higher order dissipation term to the KdV equation. The results of the present investigation may be applicable to some plasma environments, such as the ionosphere plasma.

Transient radiative heat transfer through thin films using Laguerre–Galerkin method

Heat transfer through a semiconductor or dielectric thin film is investigated by using the single relaxation time approximation to the Boltzmann equation. The radiance is expanded in terms of the Laguerre polynomial with time as argument, and the ensuing time-independent equation is solved with the aid of the Galerkin technique. Films of different thicknesses, ranging from 0.01 to 10 mean free paths, have been considered. The results, calculated for different time, ranging from 0.01 to 10 relaxation times, are presented in the forms of the following quantities (in dimensionless units): the temperature (normalized dimensionless internal energy), the heat flux, and the irradiance. Differences between the results obtained by this approach and those found by solving partial differential equations of heat conduction (Fouriers Law and Cattaneos equation) are noted.

Radiative transfer in a spherical inhomogeneous medium with anisotropic scattering

Abstract The radiative heat flux at the boundary of a sphere containing an internal energy source and subject to general boundary conditions is obtained in terms of the albedo of the corresponding source-free problem with isotropic boundary condition. The relations obtained apply to the general case of anisotropic scattering in an inhomogeneous medium. The advantage of these relations is the result of the fact that there is no need to obtain a particular solution for specified internal sources. Therefore, calculations can be done easily for a non-uniform source distribution. The phase function is approximated by using a linear anisotropic relation. The linear coefficient is taken to be the sum of the coefficients of the Legendre expansion of the phase function. The resulting relations are used to calculate the partial heat flux and emissivity for a given internal energy source and inhomogeneous medium.