4. New Laplace Variational iteration method

To illustrate the idea of new Laplace variational iteration method, we consider the following general differential equations in physics.

$$L\left[\mu\left(\mathbf{x},t\right)\right] + N\left[\mu\left(\mathbf{x},t\right)\right] = h\left(\mathbf{x},t\right)\tag{28}$$

where L is a linear partial differential operator given by , N is nonlinear operator and is a known analytical function. According to the variational iteration method, we can construct a correction functional for Eq. (28) as follows:

$$\begin{aligned} u\_{n+1}(\mathbf{x},t) &= u\_n(\mathbf{x},t) + \int\_0^t \overline{\lambda}(\mathbf{x},\boldsymbol{\varepsilon}) [Lu\_n(\mathbf{x},\boldsymbol{\varepsilon}) + N\ddot{u}\_n(\mathbf{x},\boldsymbol{\varepsilon}) - h(\mathbf{x},\boldsymbol{\varepsilon})] d\boldsymbol{\varepsilon}, \\ &\quad n \ge 0, \end{aligned} \tag{29}$$

ð34Þ

165

ð35Þ

ð37Þ

(38)

(39)

The extreme condition of requires that . This means that the right

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

ð Þ t � ς ½ � Lunð Þþ x; ς Nu~nðx; ςÞ � hðx; ςÞ dς

In this section, we apply the Laplace variational iteration method to solve some linear and

The Laplace variational iteration correction functional will be constructed in the following

<sup>λ</sup>ð Þ <sup>x</sup>; <sup>t</sup> � <sup>ς</sup> ð Þ un ttðx; <sup>ς</sup>Þ � ð Þ un xxðx; <sup>ς</sup>Þ þ unðx; <sup>ς</sup><sup>Þ</sup> � �d<sup>ς</sup>

<sup>¼</sup> <sup>ℓ</sup>½ �þ unð Þ <sup>x</sup>; <sup>t</sup> <sup>ℓ</sup> <sup>λ</sup>ð Þ <sup>x</sup>; <sup>t</sup> � �<sup>ℓ</sup> ð Þ un ttðx; <sup>t</sup>Þ � ð Þ un xxðx; <sup>t</sup>Þ þ unðx; <sup>t</sup><sup>Þ</sup> � �

<sup>s</sup><sup>2</sup>ℓunð Þ� <sup>x</sup>; <sup>t</sup> sunð Þ� <sup>x</sup>; <sup>0</sup> <sup>∂</sup>un

�ℓð Þ un xxð Þþ <sup>x</sup>; <sup>t</sup> <sup>ℓ</sup>unð Þ <sup>x</sup>; <sup>t</sup>

<sup>þ</sup><sup>ℓ</sup> <sup>λ</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>∗</sup> ð Þ un ttðx; <sup>t</sup>Þ � ð Þ un xxðx; <sup>t</sup>Þ þ unðx; <sup>t</sup><sup>Þ</sup> � � � �

2 6 4 3

<sup>5</sup>, n <sup>≥</sup> <sup>0</sup>, (36)

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

3 5

<sup>∂</sup><sup>t</sup> ð Þ <sup>x</sup>; <sup>0</sup>

3 7 5

hand side of Eq. (34) should be set to zero; then, we have the following condition:

Then, we have the following iteration formula

2 4

�ℓ ðt

5. Applications

Example (5.1)

manner:

or

0

nonlinear partial differential equations in physics.

þℓ ðt

0

2 4

Consider the initial linear partial differential equation

<sup>ℓ</sup>½ �¼ unþ<sup>1</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>ℓ</sup>½ � unð Þ <sup>x</sup>; <sup>t</sup>

<sup>ℓ</sup>½ �¼ unþ<sup>1</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>ℓ</sup>½ � unð Þ <sup>x</sup>; <sup>t</sup>

<sup>¼</sup> <sup>ℓ</sup>½ �þ unð Þ <sup>x</sup>; <sup>t</sup> <sup>ℓ</sup> <sup>λ</sup>ð Þ <sup>x</sup>; <sup>t</sup> � �

<sup>ℓ</sup>½ �¼ unþ<sup>1</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>ℓ</sup>½ � unð Þ <sup>x</sup>; <sup>t</sup>

where is a general Lagrange multiplier, which can be identified optimally via the variational theory, the subscript denotes the nth approximation, is considered as a restricted variation, that is, .

Also, we can find the Lagrange multipliers, by using integration by parts of Eq. (28), but in this chapter, the Lagrange multipliers are found to be of the form , and in such a case, the integration is basically the single convolution with respect to t, and hence, Laplace transform is appropriate to use.

Take Laplace transform of Eq. (29); then the correction functional will be constructed in the form:

$$\begin{aligned} \ell[\boldsymbol{u}\_{n+1}(\mathbf{x},t)] &= \ell[\boldsymbol{u}\_n(\mathbf{x},t)] \\ &+ \ell \left[ \int\_0^t \overline{\lambda}(\mathbf{x},\boldsymbol{\varepsilon}) [L\boldsymbol{u}\_n(\mathbf{x},\boldsymbol{\varepsilon}) + N\tilde{\boldsymbol{u}}\_n(\mathbf{x},\boldsymbol{\varepsilon}) - h(\mathbf{x},\boldsymbol{\varepsilon})] d\boldsymbol{\varepsilon} \right], n \ge 0, \end{aligned} \tag{30}$$

Therefore

$$\begin{aligned} \ell[\boldsymbol{\mu}\_{n+1}(\mathbf{x},t)] &= \ell[\boldsymbol{\mu}\_{n}(\mathbf{x},t)] \\ + \ell\left[\overline{\boldsymbol{\Lambda}}(\mathbf{x},t) \ast \left[\boldsymbol{\Lambda}\boldsymbol{\mu}\_{n}(\mathbf{x},t) + N\widetilde{\boldsymbol{\mu}}\_{n}(\mathbf{x},t) - h(\mathbf{x},t)\right]\right] \\ = \ell[\boldsymbol{\mu}\_{n}(\mathbf{x},t)] + \ell\left[\overline{\boldsymbol{\Lambda}}(\mathbf{x},t)\right]\ell[\boldsymbol{\Lambda}\boldsymbol{\mu}\_{n}(\mathbf{x},t) + N\widetilde{\boldsymbol{\mu}}\_{n}(\mathbf{x},t) - h(\mathbf{x},t)] \end{aligned} \tag{31}$$

where \* is a single convolution with respect to t.

To find the optimal value of , we first take the variation with respect to . Thus:

$$\begin{aligned} \frac{\delta}{\delta u\_n} \ell[\mu\_{n+1}(\mathbf{x}, t)] &= \frac{\delta}{\delta u\_n} \ell[\mu\_n(\mathbf{x}, t)] + \\\\ \frac{\delta}{\delta u\_n} \ell\left[\overline{\Lambda}(\mathbf{x}, t)\right] \ell[\mathrm{L}\mu\_n(\mathbf{x}, t) + \mathrm{N}\tilde{\mu}\_n(\mathbf{x}, t) - \hbar(\mathbf{x}, t)] \end{aligned} \tag{32}$$

Then, Eq. (32) becomes

$$\ell \left[ \delta u\_{n+l}(\mathbf{x}, t) \right] = \ell \left[ \delta u\_n(\mathbf{x}, t) \right] + \delta \ell \left[ \overline{\mathcal{A}}(\mathbf{x}, t) \right] \ell \left[ L u\_n(\mathbf{x}, t) \right] \tag{33}$$

In this chapter, we assume that L is a linear partial differential operator given by , then, Eq. (33) can be written in the form:

Solution of Nonlinear Partial Differential Equations by New Laplace Variational Iteration Method http://dx.doi.org/10.5772/intechopen.73291 165

$$\ell \left[ \delta \boldsymbol{u}\_{n+l}(\mathbf{x},t) \right] = \ell \left[ \delta \boldsymbol{u}\_{n}(\mathbf{x},t) \right] + \ell \left[ \overline{\mathcal{A}}(\mathbf{x},t) \right] \left[ \mathrm{s}^{2} \ell \, \delta \boldsymbol{u}\_{n}(\mathbf{x},t) \right] \tag{34}$$

The extreme condition of requires that . This means that the right hand side of Eq. (34) should be set to zero; then, we have the following condition:

$$\ell \left[ \overline{\mathcal{A}} \left( \mathbf{x}, t \right) \right] = \frac{-1}{s^2} \quad \Rightarrow \quad \overline{\mathcal{A}} \left( \mathbf{x}, t \right) = -t \tag{35}$$

Then, we have the following iteration formula

$$\begin{aligned} \ell[\boldsymbol{u}\_{n+1}(\mathbf{x},t)] &= \ell[\boldsymbol{u}\_n(\mathbf{x},t)] \\ -\ell\left[\int\_0^t (t-\boldsymbol{\varsigma})[\boldsymbol{L}\boldsymbol{u}\_n(\mathbf{x},\boldsymbol{\varsigma}) + \boldsymbol{N}\ddot{\boldsymbol{u}}\_n(\mathbf{x},\boldsymbol{\varsigma}) - \boldsymbol{h}(\mathbf{x},\boldsymbol{\varsigma})]d\boldsymbol{\varsigma}\right], n \ge 0,\end{aligned} \tag{36}$$
