M.A. Zahran

Modified extended tanh-function method for solving nonlinear partial differential equations

Abstract Based on an extended tanh-function method, a general method is suggested to obtain multiple travelling wave solutions for nonlinear partial differential equations (PDEs). The validity and reliability of the method is tested by its application to some nonlinear PDEs. The obtained results are compared with that of an extended tanh-function method and hyperbolic-function method. New exact solutions are found.

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.

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.

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.

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.

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.

Fractional Integral Representation of Master Equation

Abstract Using the definition of Liouville–Riemann (L–R) fractional integral operator, master equation can be represented in the domain of fractal time evolution with a critical exponent a (0 a ⩽1) . The relation between the continuous time random walks (CTRW) and fractional master equation (FME) has been achieved by obtaining the corresponding waiting time density (WTD) ψ ( t ) . The latter is obtained in a closed form in terms of the generalized Mittag–Leffler (M–L) function. The asymptotic expansion of the (M–L) function show the same behavior considered in the theory of random walk. Applying the Fourier and Laplace–Mellin transforms to (FME) , one obtains the solution, in closed form, in terms of the Fox function.

Fractional representation of Fokker–Planck equation

Abstract From 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 β (0 [El-Wakil SA, Zahran MA. Chaos, Solitons & Fractals 11 (2000) 791–98]. The solutions of Fokker–Planck equation will establish in three different cases of mean-square displacement as follows: (i) 〈(x(t+τ)−x(t))2〉∼τ, (ii) 〈(x(t+τ)−x(t)) 2 〉τ β , 0 (iii) 〈(x(t+τ)−x(t)) 2 〉∼x −θ τ β , θ=d w −2. The distribution function of each case can be obtained in a closed form of Fox-function.