K. R. Raslan

Adomian decomposition method for Burger's-Huxley and Burger's-Fisher equations

The approximate solutions for the Burgers-Huxley and Burgers-Fisher equations are obtained by using the Adomian decomposition method [Solving Frontier Problems of Physics: the Decomposition Method, Kluwer, Boston, 1994]. The algorithm is illustrated by studying an initial value problem. The obtained results are presented and only few terms of the expansion are required to obtain the approximate solution which is found to be accurate and efficient.

A computational method for the regularized long wave (RLW) equation

A numerical solution of the regularized long wave (RLW) equation, based on collocation method using cubic B-spline finite elements is used to simulate the migration and interaction of solitary waves. Interaction of solitary waves with different amplitudes are shown. The three invariants of the motion are evaluated to determine the conservation properties of the system. The temporal evaluation of a Maxwellian initial pulse is then studied, and then we prove that the number of solitons which are generated from Maxwellian initial condition are determined.

A finite difference scheme for the MRLW and solitary wave interactions

The modified regularized long wave (MRLW) equation is solved numerically using the finite difference method. Fourier stability analysis of the linearized scheme shows that it is a marginally stable. Also, the local truncation error of the method is investigated. Three invariants of motion are evaluated to determine the conservation properties of the problem, and the numerical scheme leads to accurate and efficient results. Moreover, interaction of two and three solitary waves is shown. The development of the Maxwellian initial condition into solitary waves is also shown and we show that the number of solitons which are generated from the Maxwellian initial condition can be determined. Numerical results show also that a tail of small amplitude appears after the interactions.

Collocation method using quartic B-spline for the equal width (EW) equation

The equal width (EW) equation is solved numerically by giving a new algorithm based on collocation method using quartic B-splines at the mid knots points as element shape. Also, we use the fourth Runge-Kutta method for solving the system of first order ordinary differential equations instead of finite difference method. Our test problems, including the migration and interaction of solitary waves, are used to validate the algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. The temporal evaluation of a Maxwellian initial pulse is then studied.

Solitary wave solutions for the general KDV equation by Adomian decomposition method

In recent publications [Chaos, Solitons Fractals 12 (2001) 2283; Int. J. Appl. Math. 3 (4) (2000) 361], we have dealt with the numerical solutions of the Korteweg-De-Vries (KDV) and modified Korteweg-De-Vries (MKDV) equations. We extend this study to a more general nonlinear equation, which is the general Korteweg-De-Vries (GKDV) equation, in which the previous studies is a special case of it. The method applied here is Adomian decomposition method, which has been developed by George Adomian [Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, MA, 1994]. Numerical examples are tested to illustrate the pertinent feature of the proposed algorithm.

Comparison study between restrictive Taylor, restrictive Padé approximations and Adomian decomposition method for the solitary wave solution of the General KdV equation

Some accurate finite difference sophisticated methods for solving initial boundary value problem for partial differential equations gives its exact solution with certain values of the mesh sizes of space and time as done [H.N.A. Ismail, On the convergence of the restrictive Pade approximation to the exact solutions of IBVP of parabolic and hyperbolic types, Appl. Math. Comput., accepted; H.N.A. Ismail, G.S.E. Salem, On the convergence of the restrictive Taylor approximation to the exact solutions of IBVP for parabolic, hyperbolic, convection diffusion, and KdV equations, Appl. Math. Comput., in press]. The restrictive Pade and restrictive Taylor approximations are very promising methods. In recent publications [H.N.A. Ismail, K.R. Raslan, G.S.E. Salem, Solitary wave solutions for the General KdV equation by Adomian decomposition method, 28th International Conference for Statistics and Computer Science and its Applications, Cairo, Appl. Math. Comput., June, accepted; A.M. Wazwaz, Construction of solitary wave solutions and rational solutions for the KdV equation by Adomian decomposition method, Chaos, Soliton. Fract. 12 (2001) 2283-2293; A.M. Wazwaz, Solitary wave solutions for the modified KdV equation by Adomian decomposition method, Int. J. Appl. Math. 3(4) (2000) 361-368], we have dealt with the numerical solutions of the Korteweg-de-Vries (KdV) and modified Korteweg-de-Vries (MKdV) equations. We extend this study to a more general nonlinear equation, which is the General Korteweg-de-Vries (GKdV) equation. The method applied here is Adomian decomposition method, which has been developed by George Adomian [Solving Frontier Problems of Physics: the Decomposition Method, Kluwer Academic Publishers, Boston, MA, 1994], restrictive Taylor and restrictive Pade approximations. Numerical examples are tested to illustrate the pertinent feature of the proposed algorithms.

Numerical Study of Rosenau-KdV Equation Using Finite Element Method Based on Collocation Approach

In the present paper, a numerical method is proposed for the numerical solution of Rosenau-KdV equation with appropriate initial and boundary conditions by using collocation method with septic B-spline functions on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To check accuracy of the error norms L2 and L∞ are computed. Interaction of two and three solitary waves are used to discuss the effect of the behavior of the solitary waves during the interaction. Furthermore, evolution of solitons is illustrated by undular bore initial condition. These results show that the technique introduced here is suitable to investigate behaviors of shallow water waves.