S.A. El-Wakil

Modified extended tanh-function method and its applications to nonlinear equations

New exact travelling wave solutions for the generalized shallow water wave equation, the improved Boussinesq equation and the coupled system for the approximate equations for water waves are found using a modified extended tanh-function method. The obtained results include rational, periodic, singular and solitary wave solutions.

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.

Fractional Fokker–Planck equation

Abstract By using the definition of the characteristic function and Kramers–Moyal Forward expansion, one can obtain the Fractional Fokker–Planck Equation (FFPE) in the domain of fractal time evolution with a critical exponent α (0 α ⩽1). Two different classes of fractional differential operators, Liouville–Riemann (L–R) and Nishimoto (N) are used to represent the fractal differential operators in time. By applying the technique of eigenfunction expansion to get the solution of FFPE, one finds that the time part of eigenfunction expansion in terms of L–R represents the waiting time density Ψ ( t ), which gives the relation between fractal time evolution and the theory of continuous time random walk (CTRW). From the principle of maximum entropy, the structure of the distribution function can be known.

New exact solutions for a generalized variable coefficients 2D KdV equation

Abstract Using homogeneous balance method an auto-Backlund transformation for a generalized variable coefficients 2D KdV equation is obtained. Then new exact solutions are found which include solitary one. Also, we have found certain new analytical soliton-typed solution in terms of the variable coefficients of the studied 2D KdV equation.

Exact travelling wave solutions for the generalized shallow water wave equation

Abstract Using homogeneous balance method an auto-Backlund transformation for the generalized shallow water wave equation is obtained. Then solitary wave solutions are found. Also, modified extended tanh-function method is applied and new exact travelling wave solutions are obtained. The obtained solutions include rational, periodical, singular and solitary wave solutions.

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.

Nonlinear electron-acoustic solitary waves in a relativistic electron-beam plasma system with non-thermal electrons

The nonlinear properties of small amplitude electron-acoustic solitary waves (EASWs) have been investigated in an unmagnetized collisionless four-component plasma system consisting of a cold relativistic electron fluid, non-thermal hot electrons obeying a non-thermal distribution, a relativistic electron beam and stationary ions. It is found that the basic set of fluid equations is reduced to a Korteweg–de Vries (KdV) equation, which governs the nonlinear characteristics of EASWs. The effects of relativistic electrons and energetic population parameter δ on the nature of EASWs are discussed.

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.

Contribution of higher order dispersion to nonlinear dust-acoustic solitary waves in dusty plasma with different sized dust grains and nonthermal ions

The propagation of nonlinear dust-acoustic waves (DAWs) in an unmagnetized, collisionless dusty plasma consisting of dust grains obeying the power-law dust size distribution and nonthermal ions are investigated. For nonlinear DAWs, a reductive perturbation method was employed to obtain a Korteweg?de Vries (KdV) equation for the first-order potential. As the wave amplitude increases, the width and the velocity of the soliton deviate from the prediction of the KdV equation, i.e. the breakdown of the KdV approximation occurs. To overcome this weakness, we extended our analysis to obtain the KdV equation with the fifth-order dispersion term. After that, the higher order solution for the resulting equation has been achieved via what is called the perturbation technique. The effects of dust size distribution, dust radius and nonthermal distribution of ions on the higher order soliton amplitude, width and energy of electrostatic solitary structures are presented.