Application of He's variational iteration method for solving the Cauchy reaction–diffusion problem

Abstract

In this paper, the solution of Cauchy reaction-diffusion problem is presented by means of variational iteration method. Reactiondiffusion equations have special importance in engineering and sciences and constitute a good model for many systems in various fields. Application of variational iteration technique to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution. Moreover, this technique does not require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computations.

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.…”

On the Reconstruction of the First Term in the Variational Iteration Method for Solving Differential Equations

“…From the given data, the effect of the reconstruction of the initial guess is clear. In the next part of this example we consider the Klein-Gordon equation with cubic nonlinearity [20] …”

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

“…This scheme is applied to solve the Lane-Emden equation which arises in astronomy [29]. Authors of [30] employed the variational iteration procedure to solve the Cauchy reactiondiffusion equation.…”