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

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 transformmethod, homotopy perturbationmethod, separating variables, local fractional Laplace method, local fractional Sumudumethod, variational iterationmethod, 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


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][2][3][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][6][7][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][10][11][12][13][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.
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]: (5)

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 0 <  ≤ 1, ,  ≥ 0,   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  +1 /  = 0, 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 V 0 (, ) = V(0, ) = V 0 .Now, by putting the values of "," we get the iterations; for V 1 (, ), put  = 0 in (15), and solving inverse local fractional Laplace transform, we have For  = 1, V 2 () is given by the following iteration: Using ( 16) and after easy calculations, we get Similarly for  = 2, V 2 () 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 V (0, ) =   (−  ) , 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 V 1 (, ), put  = 0 in (22); we have Using ( 23) and applying local fractional Laplace and inverse local fractional Laplace transform, we get Similarly, we can find Mathematical Problems in Engineering Consequently, we obtain V (, ) =   (−  ) cos  (  ) +   (−  ) sin  (  ) . ( 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: 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 V 1 (, ), put  = 0 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 V (, ) = cosh  (  ) −   (−  ) .(39)