Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 9672314, 6 pages
http://dx.doi.org/10.1155/2016/9672314

Research Article

On the Solution of Local Fractional Differential Equations Using Local Fractional Laplace Variational Iteration Method

Pranay Goswami¹ and Rubayyi T. Alqahtani²

¹School of Liberal Studies, Ambedkar University Delhi, Delhi 110006, India
²Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11566, Saudi Arabia

Received 24 August 2015; Revised 19 November 2015; Accepted 24 December 2015

Academic Editor: Laura Gardini

Copyright © 2016 Pranay Goswami and Rubayyi T. Alqahtani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We present iteration formulae of a fractional space-time telegraph equation using the combination of fractional variational iteration method and local fractional Laplace transform.

1. Introduction

Fractional differential equations have number of properties and play master role in different fields of study. These equations and their solutions gained remarkable interest due to its striking role in every science and technology [1–4]. Various fields like diffusion, transport theory, scattering theory, rheology, quantitative biology, and so forth are successfully applying fractional differential equations to define and explain number of phenomena but fractional calculus is not perfectly applicable in the case of fractal functions. In order to deal with fractal problems in various fields, the concept of local fractional derivative was developed. The local fractional calculus was introduced by Yang and further applications of this derivative can be found in the references [5–8]. There are variety of analytical methods for solving them like Adomian decomposition method, generalized differential transform method, homotopy perturbation method, separating variables, local fractional Laplace method, local fractional Sumudu method, variational iteration method, and fractional variational iteration method [5, 6, 9–14].

In this paper, we develop an iteration formula to solve generalized fractional space-time telegraph equation by using the combination of fractional variational iteration method and local fractional Laplace transform.

2. Definitions and Mathematical Preliminaries

Definition 1 (see [7, 8]). The partition of interval [] is denoted as , , , and with and Local fractional integral of in the interval [] is given by [7, 8]:

Definition 2. The local fractional derivative of of order at is defined [see [7, 8]] as

where .

Definition 3. The local fractional Laplace transform of of order is defined [15] as

where and is Mittage Leffler function .

And inverse local fractional Laplace transform is defined as

On using the above definition of local fractional Laplace transform, the following result can easily be obtained [15]:

3. The Local Fractional Laplace Variational Iteration Method

This section introduces the idea of local fractional Laplace variational method for the following fractional space-time telegraph equation:

where , , is a linear operator, is nonlinear operator, and is a source term.

Taking local fractional Laplace transform of (6), both sides with respect to “” are as follows:

For an algebraic equation, the iteration formula can be constructed as

The optimality condition for the extreme , leads to

where is the classical variational operator.

By the formula of (8), we get the iteration formula for (7) as follows:

Put , the Lagrange multiplier [16] in (10):

Taking inverse local fractional Laplace transform into account, we arrived at

This is the iteration formula for (6).

Example 4. To illustrate the above method, we can consider the following linear equation:

We construct the following iteration formula with the help of (12):

Now, applying local fractional Laplace transform to the above equation, find

This is the iteration formula for (13).
Let us start from .
Now, by putting the values of “,” we get the iterations; for , put in (15), and solving inverse local fractional Laplace transform, we have

For , is given by the following iteration:

Using (16) and after easy calculations, we get

Similarly for , is given by the following iteration:

This is the exact solution of (13).

Example 5. The following space-time fractional homogeneous telegraph equation can also be solved by the above introduced local fractional Laplace variational iteration method:

with initial conditions

We can find the iteration formula for the above with the help of (12) as

Consider initial iteration as follows:

Now, by putting the values of “,” we get the iterations; for , put in (22); we have

Using (23) and applying local fractional Laplace and inverse local fractional Laplace transform, we get

Similarly, we can find

Consequently, we obtain

Remark 6. The result in (27) is the same as the result obtained by Jafari and Jassim [17].

Example 7. The following space-time fractional homogeneous telegraph equation can also be solved by the above introduced local fractional Laplace variational iteration method:

with initial conditions

Then, we can find the iteration formula for the above with the help of (12) as

Consider initial iteration as follows:

Now, by putting the values of “,” we get the iterations; for , put in (30); we have

Using (31) and applying local fractional Laplace and inverse local fractional Laplace transform, we get

Similarly, we can find

Consequently, we obtain

Example 8. We consider the following local fractional Laplace equation:

with the initial condition

Then, we can find the iteration formula for the above with the help of (12) as

Consider initial iteration as follows:

Now, by putting the values of “,” we get the iterations; for , put in (38); we have

Using (39) and applying local fractional Laplace and inverse local fractional Laplace transform, we get

Similarly, we can find

Consequently, we obtain

Example 9. We consider the following local fractional Laplace equation:

with the initial condition

Then, we can find the iteration formula for the above with the help of (12) as

Consider initial iteration as follows:

Now, by putting the values of “,” we get the iterations; for , put in (46); we have

Using (47) and applying local fractional Laplace and inverse local fractional Laplace transform, we get

Similarly, we can find

Consequently, we obtain

4. Conclusion

In this work, we have presented successful demonstration of the local fractional Laplace variational iteration method for solutions of a wide class of problems. Analytical solutions of the telegraph and Laplace equations on Cantor sets involving local fractional derivatives are efficiently developed.

Conflict of Interests

The authors declare no conflict of interests.

Authors’ Contribution

Both authors have worked equally in this paper. Both authors have read and approved the final paper.

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