1. Introduction

In the recent years, many authors have devoted their attention to study solutions of nonlinear partial differential equations using various methods. Among these attempts are the Adomian decomposition method, homotopy perturbation method, variational iteration method (VIM) [1–5], Laplace variational iteration method [6–8], differential transform method and projected differential transform method.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Many analytical and numerical methods have been proposed to obtain solutions for nonlinear PDEs with fractional derivatives such as local fractional variational iteration method [9], local fractional Fourier method, Yang-Fourier transform and Yang-Laplace transform and other methods. Two Laplace variational iteration methods are currently suggested by Wu in [10–13]. In this chapter, we use the new method termed He's variational iteration method, and it is employed in a straightforward manner.

More generally:

plane α < Ref gs .

or equivalently,

follows:

If

then:

ℓ f ð Þ <sup>n</sup> ð Þ<sup>x</sup>

h i <sup>¼</sup> <sup>s</sup><sup>n</sup>ℓ½ �� f xð Þ <sup>s</sup>

Theorem (2.2) (Convolution Theorem)

Consider the differential equation,

With the initial conditions

and the one-sided inverse Laplace transform is defined by:

ℓ�<sup>1</sup>

n�1

½ �¼ F sð Þ fð Þ¼ x

f 0ð Þ� s

n�2 f 0

> 1 2πi

where the integration is within the regions of convergence. The region of convergence is half-

<sup>ℓ</sup>½ �¼ f xð Þ F sð Þ, <sup>ℓ</sup>½ �¼ g xð Þ G sð Þ,

<sup>ℓ</sup> f xð Þ<sup>∗</sup> ½ �¼ g xð Þ <sup>ℓ</sup>½ �¼ f xð Þg xð Þ f sð Þg sð Þ

½ �¼ F sð ÞG sð Þ f xð Þ<sup>∗</sup>

yð Þ¼ 0 h xð Þ, y<sup>0</sup>

; <sup>y</sup>00; ::…; <sup>y</sup>ð Þ <sup>n</sup> � �<sup>∗</sup>

operator and <sup>N</sup><sup>∗</sup>½ � y xð Þ is the nonlinear convolution term which is defined by:

<sup>N</sup><sup>∗</sup> y xð Þ � � <sup>¼</sup> <sup>f</sup> <sup>y</sup>; <sup>y</sup><sup>0</sup>

ð x

0

Rynð Þ <sup>ξ</sup> , Ny~nð Þ <sup>ξ</sup> and <sup>N</sup><sup>∗</sup>y~nð Þ <sup>ξ</sup> are considered as restricted variations, that is,

ynþ<sup>1</sup>ð Þ¼ <sup>x</sup> <sup>y</sup>nð Þþ <sup>x</sup>

where L is a linear second-order operator, R is a linear first-order operator, N is the nonlinear

According to the variational iteration method, we can construct a correction functional as

g xð Þ

<sup>L</sup> y xð Þ � � <sup>þ</sup> Ryxð Þ � � <sup>þ</sup> Nyxð Þ � � <sup>þ</sup> <sup>N</sup><sup>∗</sup> y xð Þ � � <sup>¼</sup> <sup>0</sup> (1)

g y; y<sup>0</sup>

; <sup>y</sup>00; :…; <sup>y</sup>ð Þ <sup>n</sup> � �

λ ξð Þ Lynð Þþ <sup>ξ</sup> Ry~nð Þþ <sup>ξ</sup> <sup>N</sup>y~nð Þþ <sup>ξ</sup> <sup>N</sup><sup>∗</sup>y~nð Þ <sup>ξ</sup> � �d<sup>ξ</sup> (3)

ð Þ¼ 0 k xð Þ (2)

ℓ�<sup>1</sup>

αþ ð i∞

Solution of Nonlinear Partial Differential Equations by New Laplace Variational Iteration Method

α�i∞

F sð Þe sxds

ð Þ� <sup>0</sup> … � sfð Þ <sup>n</sup>�<sup>2</sup> ð Þ� <sup>0</sup> <sup>f</sup>

ð Þ <sup>n</sup>�<sup>1</sup> ð Þ<sup>0</sup>

http://dx.doi.org/10.5772/intechopen.73291

155

Also, the main aim of this chapter is to introduce an alternative Laplace correction functional and express the integral as a convolution. This approach can tackle functions with discontinuities as well as impulse functions effectively. The estimation of the VIM is to build an iteration method based on a correction functional that includes a generalized Lagrange multiplier. The value of the multiplier is chosen using variational theory so that each iteration improves the accuracy of the result.

In this chapter, we have applied the modified variational iteration method (VIM) and Laplace transform to solve convolution differential equations.
