He’s Variational Iteration Method for Solving a Partial Differential Equation Arising in Modelling of theWater Waves

Abstract

The variational iteration method is applied to solve the Kawahara equation. This method produces the solutions in terms of convergent series and does not require linearization or small perturbation. Some examples are given. The comparison with the theoretical solution shows that the variational iteration method is an efficient method.

Numerical Study of Second Painlevé Equation

“…The VIM plays an important role in recent researches for solving various kinds of problems. see for example [23,34,1,26,27,3] and the references therein. See also [22,19].…”

The use of variational iteration method and Adomian decomposition method to solve the Eikonal equation and its application in the reconstruction problem

“…In this method the linear and nonlinear structures are handled in a similar manner without any need to restrictive assumptions. This approach is successfully and effectively applied to various equations such as autonomous ordinary differential equations [32], delay differential equations [33], Fokker-Planck equation [34], nonlinear fractional differential equation with Riemann-Liouville's fractional derivatives [35], quadratic Riccati differential equation [36], one-phase direct and inverse Stefan problems [37], one-dimensional elastic half-space model subjected initially to a prescribed harmonic displacement [38], the linear and the nonlinear Goursat problems [39], Klein-Gordon equation [40], shock-peakon and shockcompacton solutions for K ( p, q) equation [41], generalized Burgers-Huxley equation [42], wave equation [43], some problems in calculus of variations [44], nonlinear PDEs arising in the process of understanding the role of nonlinear dispersion and in the forming of structures like liquid drops and exhibiting compactons [45], generalized Zakharov equation [46], Jaulent-Miodek equation [47], characteristic and non-characteristic Cauchy reaction-diffusion problems [48], two-point boundary value problems [49], cubic nonlinear Schrödinger equation [50], parabolic integro-differential equations arising in heat conduction in materials with memory [51], nonlinear integro-differential equations which arise in biology [52], generalized pantograph equation [53], Kawahara PDE arising in the modelling of water waves [54], systems of differential equations [55], differential equation arising in astrophysics [56], the telegraph and fractional telegraph equations [57], identifying an unknown function in a parabolic equation with overspecified data [58], etc.…”

Solution of a nonlinear time-delay model in biology via semi-analytical approaches

“…He's variational iteration method is employed in [62] for solving the modified Camassa-Holm and Degasperis-Procesi equations. In [72] the variational iteration method was employed to solve the Kawahara equation that models the plasma waves, the capillary gravity water and describes water waves with surface tension. Authors of [64] presented an idea for accelerating the convergence of the resulted sequence to the solution of the problem by choosing a suitable initial term and examined the proposed idea for ordinary, partial differential, and integro-differential equations.…”