Definition (2.1)

Let f xð Þ,g xð Þ be integrable functions, then the convolution of f xð Þ,g xð Þ is defined as:

$$f(\mathbf{x})^\ast \mathbf{g}(\mathbf{x}) = \int\_0^\mathbf{x} f(\mathbf{x} - \mathbf{t}) \mathbf{g}(\mathbf{t}) \mathbf{d}\mathbf{t}$$

and the Laplace transform is defined as:

$$\ell\left[f(\mathbf{x})\right] = \mathbf{F}(\mathbf{s}) = \int\_0^\infty e^{-s\mathbf{x}} f(\mathbf{x})d\mathbf{x}$$

where Re s > 0, where s is complex valued and ℓ is the Laplace operator.

Further, the Laplace transform of first and second derivatives are given by:

$$\begin{aligned} \text{(i)} & \ell \left[ \mathbf{f}'(\mathbf{x}) \right] = \mathbf{s} \ell \left[ \mathbf{f}(\mathbf{x}) \right] - \mathbf{f}(0) \\ \text{(ii)} & \ell \left[ \mathbf{f}''(\mathbf{x}) \right] = \mathbf{s}^2 \ell \left[ \mathbf{f}(\mathbf{x}) \right] - \mathbf{s} \mathbf{f}(0) - \mathbf{f}'(0) \end{aligned}$$

More generally:

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,

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

In this chapter, we have applied the modified variational iteration method (VIM) and Laplace

2. Combine Laplace transform and variational iteration method to solve

In this section, we combine Laplace transform and modified variational iteration method to figure out a new case of differential equation called convolution differential equations; it is possible to obtain the exact solutions or better approximate solutions of these equivalences. This method is utilized for solving a convolution differential equation with given initial conditions.

The results obtained by this method show the accuracy and efficiency of the method.

Let f xð Þ,g xð Þ be integrable functions, then the convolution of f xð Þ,g xð Þ is defined as:

g xð Þ¼

<sup>ℓ</sup> ½ �¼ <sup>f</sup>ð Þ<sup>x</sup> F sð Þ¼

ð Þ<sup>x</sup> � � <sup>¼</sup> <sup>s</sup>ℓ½ �� f xð Þ f 0ð Þ

ð Þ<sup>x</sup> � � <sup>¼</sup> <sup>s</sup><sup>2</sup>ℓ½ �� f xð Þ <sup>s</sup>f 0ð Þ� <sup>f</sup>

ð x

fð Þ x � t g tð Þdt

�sxfð Þ<sup>x</sup> dx

0 ð Þ0

0

ð ∞

0 e

<sup>f</sup>ð Þ<sup>x</sup> <sup>∗</sup>

where Re s > 0, where s is complex valued and ℓ is the Laplace operator.

ð Þ<sup>i</sup> <sup>ℓ</sup> <sup>f</sup> 0

ð Þii <sup>ℓ</sup> <sup>f</sup> 00

Further, the Laplace transform of first and second derivatives are given by:

and it is employed in a straightforward manner.

154 Differential Equations - Theory and Current Research

transform to solve convolution differential equations.

convolution differential equations

and the Laplace transform is defined as:

accuracy of the result.

Definition (2.1)

$$\ell\left[\mathbf{f}^{(n)}(\mathbf{x})\right] = \mathbf{s}^n \ell[\mathbf{f}(\mathbf{x})] - \mathbf{s}^{n-1} \mathbf{f}(0) - \mathbf{s}^{n-2} \mathbf{f}'(0) - \dots - \mathbf{s} \mathbf{f}^{(n-2)}(0) - \mathbf{f}^{(n-1)}(0)$$

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

$$\ell^{-1}[\mathcal{F}(\mathbf{s})] = f(\mathbf{x}) = \frac{1}{2\pi i} \int\_{\alpha - i^{\infty}}^{\alpha + i^{\infty}} \mathcal{F}(\mathbf{s}) e^{\mathbf{s}\mathbf{x}} d\mathbf{s}$$

where the integration is within the regions of convergence. The region of convergence is halfplane α < Ref gs .
