Semianalytical Solutions of Some Nonlinear-Time Fractional Models Using Variational Iteration Laplace Transform Method

Khurram Shabbir,Liliana Guran,Alberto Lastra,Javed Iqbal

doi:10.1155/2021/8345682

Abstract

In this work, we combined two techniques, the variational iteration technique and the Laplace transform method, in order to solve some nonlinear-time fractional partial differential equations. Although the exact solutions may exist, we introduced the technique VITM that approximates the solutions that are difficult to find. Even a single iteration best approximates the exact solutions. The fractional derivatives being used are in the Caputo-Fabrizio sense. The reliability and efficiency of this newly introduced method is discussed in details from its numerical results and their graphical approximations. Moreover, possible consequences of these results as an application of fixed-point theorem are placed before the experts as an open problem.

Highlights

Almost all the phenomena in science and engineering are naturally modeled in the form of nonlinear differential equations, like Korteweg-de Vries equation [1, 2], nonlinear Schrödinger equation [3,4,5], alternating current power flow model [6], Richards equation for unsaturated water flow [7,8,9,10], Burger equation [11], and gravitational general theory [12]
We show how the proposed technique approaches the exact solution; see Figures 2(a), 2(b), and 1(a), which are the approximations of Figure 1(b)
The proposed method variational iteration transform method (VITM) being the combination of two basic techniques, variational iteration method (VIM) and Laplace transform, is understandable by just having the formal knowledge of advanced calculus; it is understandable even for the reader who has no strong background and base in calculus of variations

Summary

Introduction and Preliminaries

Almost all the phenomena in science and engineering are naturally modeled in the form of nonlinear differential equations, like Korteweg-de Vries equation [1, 2], nonlinear Schrödinger equation [3,4,5], alternating current power flow model [6], Richards equation for unsaturated water flow [7,8,9,10], Burger equation [11], and gravitational general theory [12]. Much attention is being paid in combining more than one technique to solve a model especially nonlinear models, to get better and rapid results. Two techniques, variational iteration technique and the Laplace transform, are being utilized, and the combined technique, the variational iteration transform method (VITM), is employed to handle the nonlinear fractional order partial differential equations, like the Kortewegde Vries equation [26], Schrödinger equation [27], and Burger. The Riemann-Liouville fractional derivative of a function f ðtÞ is defined to be. Let us recall one of the most recent definitions of the fractional derivative Caputo-Fabrizio derivative, as follows. Let FðtÞ∈H1 ða, bÞ, b > a; the Caputo-Fabrizio time fractional derivative of FðtÞ is defined as. Let f ðtÞ be a function; its Laplace transform is defined as ð∞.

Methodology of VITM

Applications of VITM on Various FODE Types

Discussions and Concluding Remarks