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Advances in Difference Equations

, 2012:204 | Cite as

An efficient new perturbative Laplace method for space-time fractional telegraph equations

  • Yasir KhanEmail author
  • Josef Diblík
  • Naeem Faraz
  • Zdeněk Šmarda
Open Access
Research
Part of the following topical collections:
  1. Progress in Functional Differential and Difference Equations

Abstract

In this paper, we propose a new technique for solving space-time fractional telegraph equations. This method is based on perturbation theory and the Laplace transformation. Fractional Taylor series and fractional initial conditions have been introduced. However, all the previous works avoid the term of fractional initial conditions in the space-time telegraph equations. The results of introducing fractional order initial conditions and the Laplace transform for the studied cases show the high accuracy, simplicity and efficiency of the approach.

Keywords

Fractional Order Fractional Derivative Fractional Differential Equation Fractional Brownian Motion Homotopy Analysis Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Telegraph equations are hyperbolic partial differential equations that are applicable in several fields such as wave propagation [1], signal analysis [2], random walk theory [3], etc. In recent years, there has been a great deal of interest in fractional differential equations [4, 5]. Time-fractional telegraph equations have been studied by Orsingher and Zhao [6] and Orsingher and Beghin [7]. Telegraph equations apply to high-frequency transmission lines such as telegraph wires and radio frequency conductors. They are also applicable to designing high-voltage transmission lines.

In this paper, we consider two different types of telegraph equations. The first one is the space-fractional telegraph equation
α u x α = 2 u t 2 + a u t + b u + f ( x , t ) , 0 < α 2 , Open image in new window
subject to the initial and boundary conditions
u ( 0 , t ) = φ ( t ) , u x ( 0 , t ) = ψ ( t ) , u ( x , 0 ) = ϕ ( x ) , 0 < x < 1 ; Open image in new window
and the second equation is the classical time-fractional telegraph equation
α u t α + α 1 u t α 1 + u = 2 u x 2 + f ( x , t ) , 1 < α 2 , Open image in new window
subject to the initial conditions
u ( x , 0 ) = φ ( t ) , u t ( x , 0 ) = ψ ( x ) , Open image in new window

where u can be considered as a function depending on distance (x) and time (t), a and b are constants depending on a given problem and f, φ, ϕ, ψ are known continuous functions. The second equation has been solved by Das et al. [8] using the homotopy analysis method.

The aim of this paper is to introduce a new method for fractional space-time telegraph equations. This new technique is a combined form of the perturbation method [9, 10, 11, 12, 13, 14] with the Laplace transform. This method is called the perturbation Laplace method (PLM). Moreover, we have introduced fractional order initial conditions for space-time telegraph equations. Point to be noted regarding fractional differential equations is that one should use fractional Taylor series. To make the calculation easy and simple, for the first time, we have used the Laplace transform to solve the systems of equations formed after applying homotopy perturbation instead of applying an inverse operator. Through the Laplace transform of fractional order term, it is easy to judge that one must use fractional order initial conditions. It is easy to judge, by applying the Laplace transformation, that it is essential to use a fractional order initial condition to analyze any physical phenomenon which has been expressed in terms of fractional differential equations. To the best of authors’ knowledge, in the literature on space-fractional telegraph equations [15], there is no closed form solution for different values of α except for the standard case, i.e., for α = 2 Open image in new window. The elegance of this article can be attributed to its endeavor of finding the solution in a simple way by considering only the PLM. Two examples which show that only a few iterations are needed to obtain accurate approximate solutions are solved.

2 Fractional calculus theory

We give some basic definitions and properties of the fractional calculus theory proposed by Jumarie [16] which are used further in this paper.

Definition 1 Let f : R R Open image in new window, x f ( x ) Open image in new window, denote a continuous (but not necessarily differentiable) function. Then its fractional derivative of order α, α < 0 Open image in new window, is defined by the following expression:
f ( α ) ( x ) = 1 Γ ( α ) 0 x ( x ξ ) α 1 f ( ξ ) d ξ . Open image in new window
For positive α, we define
and
f ( α ) ( x ) = ( f ( α n ) ( x ) ) ( n ) , n α < n + 1 , n 1 . Open image in new window
With this definition, the Laplace transform L { } Open image in new window of the fractional derivative is defined as follows:
L { f ( α ) ( x ) } = s α L { f ( x ) } s α 1 f ( 0 ) , 0 < α < 1 . Open image in new window

Proposition 1 (On the decomposition of fractional derivatives)

Let α be such that 0 < 3 α < 1 Open image in new window. There are two different ways to obtain D 3 α f ( x ) Open image in new window (see [16]). One can calculate D α D α D α f ( x ) Open image in new window to obtain the Laplace transform
L { D α D α D α f ( x ) } = s 3 α f ( s ) s 3 α 1 f ( 0 ) s 2 α 1 f ( α ) ( 0 ) s α 1 f ( 2 α ) ( 0 ) . Open image in new window
(1)
Proposition 2 Assume that the continuous function f : R R Open image in new window, x f ( x ) Open image in new window has a fractional derivative of order for any positive integer k and any α, 0 < α 1 Open image in new window. Then the following equality holds:
f ( x + h ) = k = 0 h α k α k ! f ( α k ) ( x ) , 0 < α 1 . Open image in new window
On making x = 0 Open image in new window and the substitution h x Open image in new window, we obtain the fractional Maclaurin series
f ( x ) = k = 0 x α k α k ! f ( α k ) ( 0 ) , 0 < α 1 . Open image in new window

3 Perturbative Laplace method

In order to elucidate the solution procedure of the perturbative Laplace method (PLM), we consider the following fractional differential equation:
where D n α = n α t n α Open image in new window, Open image in new window is generally the linear differential operator with respect to the variable x and f ( x ) Open image in new window, q ( x , t ) Open image in new window are continuous functions. In view of HPM [9, 10, 11, 12, 13, 14, 15], we can construct a homotopy for Eq. (2) as follows:
where p [ 0 , 1 ] Open image in new window is an embedding parameter. If p = 0 Open image in new window, Eq. (3) and Eq. (4) become
D n α u ( x , t ) = 0 , Open image in new window
(5)

and when p = 1 Open image in new window, both Eq. (3) and Eq. (4) turn out to be the original fractional differential equation (2).

The homotopy perturbation method [9, 10, 11, 12, 13, 14, 15] admits a solution in the form
u = p 0 u 0 + p 1 u 1 + p 2 u 2 + . Open image in new window
(6)
Setting p = 1 Open image in new window in the solution of Eq. (6), we get
u = u 0 + u 1 + u 2 + . Open image in new window
(7)
Invoking Eq. (6) into Eq. (4) and collecting the terms with the same powers of p, we can obtain a series of equations of the following form:
By using the definition given in Eq. (1), we get
p 0 : s n α u 0 ( x , s ) s n α 1 u 0 ( x , 0 ) s ( n 1 ) α 1 u 0 ( α ) ( x , 0 ) p 0 : s ( n 2 ) α 1 u 0 ( 2 α ) ( x , 0 ) s α 1 u 0 ( ( n 1 ) α ) ( x , 0 ) = 0 , p 1 : s n α u 1 ( x , s ) = R u 0 ( x , s ) + q ( x , s ) , p 2 : s n α u 2 ( x , s ) = R u 1 ( x , s ) , p 3 : s n α u 3 ( x , s ) = R u 2 ( x , s ) , Open image in new window
(9)
Solving Eq. (9) for u 0 , u 1 , u 2 , u 3 , Open image in new window respectively, by using the fractional initial value conditions, we get
p 0 : u 0 ( x , t ) = L 1 { 1 s n α ( s n α 1 u 0 ( x , 0 ) + s ( n 1 ) α 1 u 0 ( α ) ( x , 0 ) p 0 : + s ( n 2 ) α 1 u 0 ( 2 α ) ( x , 0 ) + + s α 1 u 0 ( n α ) ( x , 0 ) ) } , p 1 : u 1 ( x , t ) = L 1 { 1 s n α ( R u 0 ( x , s ) + q ( x , s ) ) } , p 2 : u 2 ( x , t ) = L 1 { 1 s n α ( R u 1 ( x , s ) ) } , p 3 : u 3 ( x , t ) = L 1 { 1 s n α ( R u 2 ( x , s ) ) } , Open image in new window
(10)

Substituting successive iterations in Eq. (7) will give the required result.

4 Space-time fractional telegraph equations

Example 1 Let us consider the space-fractional telegraph equation
α u x α = 2 u t 2 + u t + u , 0 < α 2 , Open image in new window
(11)
subject to the initial and boundary conditions
u ( 0 , t ) = e t , u x ( 0 , t ) = e t , u ( x , 0 ) = e x , 0 < x < 1 . Open image in new window
In order to illustrate the efficiency of our method, we replace the fractional order α, 0 < α 2 Open image in new window, by the order 2α, 0 < α 1 Open image in new window, in Eq. (11)
2 α u x 2 α = 2 u t 2 + u t + u , 0 < α 1 , Open image in new window
(12)
subject to the initial and boundary conditions
u ( 0 , t ) = e t , u ( α ) ( 0 , t ) = e t , u ( x , 0 ) = e x , 0 < x < 1 . Open image in new window
Using the procedure in Section 2, we can write Eq. (12) in the form of recurrence equations as follows:
In view of Eq. (9), Eq. (13) can be written in the following form:
Solving Eq. (14) for u 0 , u 1 , u 2 , u 3 , Open image in new window , we get the following form:
Equation (15) in the most refined form can be written as
The solution in a series form can be expressed as
u ( x , t ) = e t ( 1 + x α Γ ( 1 + α ) + x 2 α Γ ( 2 α + 1 ) + x 3 α Γ ( 3 α + 1 ) + x 4 α Γ ( 4 α + 1 ) + x 5 α Γ ( 5 α + 1 ) + ) = k = 0 e t x k α Γ ( 1 + k α ) = e t E α ( x α ) , Open image in new window

where E α Open image in new window denotes the Mittag-Leffler function.

Example 2 We consider the nonhomogeneous fractional time telegraph equation
α u t α + α 1 u t α 1 + u = 2 u x 2 + f ( x , t ) , 1 < α 2 , Open image in new window
(17)
subject to the initial conditions
u ( x , 0 ) = 0 , u t ( x , 0 ) = 0 , Open image in new window
with
f ( x , t ) = t n Γ ( n + 1 ) sinh x , Open image in new window
and for the fractional initial condition
α u t α + α 1 u t α 1 + u = 2 u x 2 + f ( x , t ) , 1 < α 2 , Open image in new window
(18)
subject to the initial conditions
u ( x , 0 ) = 0 , u ( α ) ( x , 0 ) = 0 , Open image in new window
with
f ( x , t ) = t n Γ ( n + 1 ) sinh x . Open image in new window
By applying the aforesaid method, we can write Eq. (18) in the form of recurrence equations as follows:
p 0 : D α u 0 ( x , t ) = 0 , u 0 ( x , 0 ) = 0 , u 0 ( α ) ( x , 0 ) = 0 , p 1 : D α u 1 ( x , t ) = u 0 ( α 1 ) ( x , t ) u 0 ( x , t ) + 2 u 0 ( x , t ) x 2 + f ( x , t ) , p 1 : u 1 ( x , 0 ) = u 1 ( α ) ( x , 0 ) = 0 , p 2 : D α u 2 ( x , t ) = u 1 ( α 1 ) ( x , t ) u 1 ( x , t ) + 2 u 1 ( x , t ) x 2 , p 2 : u 2 ( x , 0 ) = u 2 ( α ) ( x , 0 ) = 0 , p 3 : D α u 3 ( x , t ) = u 2 ( α 1 ) ( x , t ) u 2 ( x , t ) + 2 u 2 ( x , t ) x 2 , p 3 : u 3 ( x , 0 ) = u 3 ( α ) ( x , 0 ) = 0 , Open image in new window
(19)
By using Eq. (9), we can write Eq. (19) in the following form:
Proceeding as before, we obtain
p 0 : u 0 ( x , t ) = 0 , p 1 : u 1 ( x , t ) = L 1 { 1 s α ( sinh x s n + 1 ) } , p 2 : u 2 ( x , t ) = L 1 { 1 s α ( sinh x s n + 2 ) } , Open image in new window
(20)
Equation (20) can also be written as
and so on. In this way, the rest of components of the homotopy perturbation series can be obtained. Finally, we obtain the series solution as
u ( x , t ) = ( t n + α Γ ( n + α + 1 ) t n + α + 1 Γ ( n + α + 2 ) + t n + α + 2 Γ ( n + α + 3 ) + ) sinh x . Open image in new window
Figure 1 shows the approximate solution of Eq. (18) for different values of α using the perturbative Laplace method. The numerical result of the probability density function u ( x , t ) Open image in new window for different fractional Brownian motions α = 1.80 , 1.90 , 1.99 Open image in new window, and also for standard motion α = 2 Open image in new window, is calculated for n = 1 Open image in new window, x = 2 Open image in new window with respect to the variable t. The three-dimensional variation of u ( x , t ) Open image in new window vs. x and t at α = 1.5 Open image in new window and n = 1 Open image in new window is shown in Figure 2. It is seen from the figures that u ( x , t ) Open image in new window increases with increasing t but decreases with increasing α, which assures the exponential decay of regular Brownian motion.
Figure 1

Plot of u ( x , t ) Open image in new window with respect to t at x = 2 Open image in new window , where n = 1 Open image in new window .

Figure 2

Plot of u ( x , t ) Open image in new window with respect to x and t at n = 1 Open image in new window and α = 1.5 Open image in new window .

5 Conclusion

In this paper, we have introduced a combination of perturbation and Laplace methods for space-time fractional problem which we called the PLM. We described the method and used it in some fractional telegraph equations in order to show its applicability and validity. We achieved accurate approximations by using only a few numbers of iterations, which reveals efficiency of the new method. The solution very rapidly converges by utilizing the perturbation Laplace method. The PLM is also valid for other fractional differential equations, and this paper can be used as a standard paradigm for other applications.

Notes

Acknowledgements

The second author is supported by Grant P201/11/0768 of the Czech Grant Agency (Prague). The fourth author is supported by Grant FEKT-S-11-2-921 of the Faculty of Electrical Engineering and Communication, Brno University of Technology.

Supplementary material

13662_2012_312_MOESM1_ESM.jpeg (54 kb)
Authors’ original file for figure 1
13662_2012_312_MOESM2_ESM.jpeg (123 kb)
Authors’ original file for figure 2

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© Khan et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Yasir Khan
    • 1
    Email author
  • Josef Diblík
    • 2
  • Naeem Faraz
    • 3
  • Zdeněk Šmarda
    • 2
  1. 1.Department of MathematicsZhejiang UniversityHangzhouChina
  2. 2.Department of MathematicsBrno University of TechnologyBrnoCzech Republic
  3. 3.Modern Textile InstituteDonghua UniversityShanghaiChina

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