Hassan A. Zedan

Exact solutions for the generalized KdV equation by using Backlund transformations

Abstract In this paper, the Backlund transformations for the generalized KdV equation are constructed through Ablowitz–Kaup–Newell–Segur (AKNS) system in using Ricattis form of the inverse method. The derived Backlund transformations are used to generate new classes of exact solutions. The technique developed here is based on the construction of wave functions which are the solutions of the associated AKNS. From known simple solutions we shall construct other solitons and wave solutions as well.

Solution of Davey-Stewartson equations by homotopy perturbation method

In this paper, we extend the homotopy perturbation method to solve the Davey-Stewartson equations. The homotopy perturbation method is employed to compute an approximation to the solution of the equations. Computation the absolute errors between the exact solutions of the Davey-Stewartson equations and the HPM solutions are presented. Some plots are given to show the simplicity the method.

On the solution of the nonlinear Schrodinger equation

Abstract In this paper we study the nonlinear Schrodinger equation with respect to the unknown function S ( x , t ). New dimensional reduction and exact solution for a nonlinear Schrodinger equation are presented and a complete group classification is given with respect to the function S ( x , t ). Moreover, specializing the potential function S ( x , t ), new classes of invariant solution and group classification are obtained in the cases of physical interest.

New Solutions for System of Fractional Integro-Differential Equations and Abel’s Integral Equations by Chebyshev Spectral Method

Chebyshev spectral method based on operational matrix is applied to both systems of fractional integro-differential equations and Abel’s integral equations. Some test problems, for which the exact solution is known, are considered. Numerical results with comparisons are made to confirm the reliability of the method. Chebyshev spectral method may be considered as alternative and efficient technique for finding the approximation of system of fractional integro-differential equations and Abel’s integral equations.

Haar Wavelet Method for the System of Integral Equations

We employed the Haar wavelet method to find numerical solution of the system of Fredholm integral equations (SFIEs) and the system of Volterra integral equations (SVIEs). Five test problems, for which the exact solution is known, are considered. Comparison of the results is obtained by the Haar wavelet method with the exact solution.

Applications of the New Compound Riccati Equations Rational Expansion Method and Fan's Subequation Method for the Davey-Stewartson Equations

We used what we called extended Fans sub-equation method and a new compound Riccati equations rational expansion method to construct the exact travelling wave solutions of the Davey-Stewartson (DS) equations. The basic idea of the proposed extended Fans subequation method is to take fulls advantage of the general elliptic equations, involving five parameters, which have many new solutions and whose degeneracies lead to special subequations involving three parameters like Riccati equation, first-kind elliptic equation, auxiliary ordinary equation and generalized Riccati equation. Many new exact solutions of the Davey-Stewartson (DS) equations including more general soliton solutions, triangular solutions, and double-periodic solutions are constructed by symbolic computation.

Equivalence groups for system of nonlinear partial differential equations

Abstract The nonlinear system of partial differential equations is considered, a general procedure to obtain the equivalence algebra is established. A special case is presented. Moreover, using a semidirect approach, a partial classification of the given system is obtained.

Direct approach for group classification for nonlinear filtration problem

Abstract In this paper, we investigate the group classification for all possible prosities K ( t , x , y , h ) for the filtration problem modeled by Rouse [1] . Moreover, some classes of invariant solutions are obtained.

Description of the Magnetic Field and Divergence of Multisolenoid Aharonov-Bohm Potential

Explicit formulas for the magnetic field and divergence of multisolenoid Aharonov-Bohm potential are obtained; the mathematical essence of this potential is explained. It is shown that the magnetic field and divergence of this potential are very singular generalized functions concentrated at a finite number of thin solenoids. Deficiency index is found for the minimal operator generated by the Aharonov-Bohm differential expression.

Q-Symmetry and Conditional Q-Symmetries for Boussinesq Equation

We study in this paper the Q-symmetry and conditional Q-symmetries of Boussinesq equation. The solutions which we obtain, in this case, are in the form of convergent power series with easily computable components.