Fractional variational iteration method for solving the

Journal of Physics: Conference Series

PAPER • OPEN ACCESS

Fractional variational iteration method for solving the hyperbolic telegraph equation To cite this article: H K Jassim and W A Shahab 2018 J. Phys.: Conf. Ser. 1032 012015

View the article online for updates and enhancements.

Related content - Variational iteration method used to solve the one-dimensional acoustic equations Paskalia Siwi Setianingrum and Sudi Mungkasi - Variational iteration method for solving the population dynamics model of two species Benedictus Dwi Yuliyanto and Sudi Mungkasi - Application of the three-dimensional telegraph equation to cosmic-ray transport Robert C. Tautz and Ian Lerche

This content was downloaded from IP address 181.214.211.158 on 05/06/2018 at 06:07

The Sixth Scientific Conference “Renewable Energy and its Applications” IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 1032 (2018) 1234567890 ‘’“” 012015 doi:10.1088/1742-6596/1032/1/012015

Fractional variational iteration method for solving the hyperbolic telegraph equation H K Jassim, W A Shahab Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq

Hassan Kamil Jassim: [email protected] Abstract. In this article, an analytical solution procedure is described for solving one dimensional second order hyperbolic telegraph equation using a reliable semi-analytic method so called the local fractional variational iteration method (LFVIM) subject to the appropriate initial conditions. Various illustrate examples are carried out to check the accuracy, efficiency, and convergence of the described method.

1. Introduction The fractional telegraph equations (FTEs) have been investigated by many authors in recent years. Garg et al. [1] derived a solution of space-time fractional telegraph function in a bounded domain by the method of generalized differential transform and obtained the solution in terms of Mittage-Leffler function. Chen et al. [2] obtained the solutionkof nonhomogeneous FTE. Ansari [3] derived a formal solution of the FTE by applying a fractional exponential operator. Huang [4] considered the timefractional telegraph equation for the Cauchy problem and signaling problem, he solved the problem by the combined Fourier-Laplace transforms. Also, Huang derived the solution for the bounded problem in a bounded-space domain by means of Sine-Laplace transforms methods. Das et al. [5] used a homotopy analysis method in approximating an analytical solution for the (FTE). and different particular cases have been derived. VIM proposed in [6-8] was utilized to deal with the following differential equations: Helmholtz [9], Burger’s and coupled Burger’s [10], Klein- Gordon [11], KdV [12], the oscillation [13], Schrodinger [14], reaction-diffusion [15], diffusion equation [16], Bernoulli equation [17], and others. The extended VIM, called the fractional VIM, was developed and applied to handle some fractional differential equations within the modified Riemann-Liouville derivative [18] and local fractional derivative operator [19,20]. The present paper describes an analytical scheme, the LFVIM to provide approximate analytical results of the one dimensional telegraph equations. The accuracy and efficiency of the proposed method are demonstrated by several test examples. The biggest benefit of the described method is that it finds the solution of FTE. directly without using any transformation, linearization, discretization or any other restrictive conditions. Further, the method can be easily implemented in multidimensional problems arising in many areas of science and engineering. The paper has the following organization. In Section 2 the concepts of local fractional derivatives and integrals are Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1


briefly reviewed. In Section 3 the LFVIM is recalled. In Section 4 the analyticalksolutions for local fractionaljtelegraph equations are presented. Finally, Section 5 outlines the main conclusions. 2. A Brief Review of the Local Fractional Calculus Definition 1 [19-22]: The LF derivative of f (x) of order  at x  x0 is f ( ) ( x0 )  lim

 ( f ( x)  f ( x0 )) ( x  x0 )

x  x0

,

(2.1)

where  ( f ( x)  f ( x0 ))  (  1) ( f ( x)  f ( x0 )). The formulas of LF derivatives are: Dx( ) ag ( x)  aDx( ) g ( x), d dx

(2.2) 

 x n  x ( n 1)    (1  n )  1  (n  1)  , n  N  

(2.3)

Definition 2 [19-22]: The LF integral of f (x) in the interval [a, b] is given by ( ) f ( x)  a Ib

1 (1   )

b



f (t ) (dt ) 

a

N 1 1 lim f (t j ) (t j ) . (1   ) t 0 j  0



(2.5)

The formulas of LF integrals are as: ( ) 0 I x ag ( x)

 a 0 I x( ) g ( x),

(2.6) 

x n  x ( n 1) ( )   I  , n N 0 t    (1  n )  1  (n  1) 

(2.7)

3. Analysis of the LFVIM We consider a generalmnonlinear PDEs: L u( x, t )  R u( x, t )  N u( x, t )  f ( x, t ) , 0    1 ,

(3.1)

subject to fractal initial conditions  u ( x,0)  2 ( x), t  and the fractal boundary conditions u ( x,0)  1 ( x) ,

(3.2)

 u (0, t ) (3.3)   2 (t ) . x where L and R are linear LFDOs of order 2 and N denotes nonlinear LFDOs and f ( x, t ) is the u (0, t )   1 (t ) ,

nondifferentiable function. According to the theory of LFVI algorithm [19,20], we can write the iterationkformula for (3.1) is constructed as:

  ( )   u n1 (t )  u n (t ) 0 I t( )  L u n ( )  R u~n ( )  N  u~n ( )  f ( ) .  (1   )    Making the LF variation of (3.4), we have

2

(3.4)


  ( )  L u n ( )  R u~n ( )  N u~n ( )  f ( ) .  (1   ) 

  u n1 (t )    u n (t ) 0 I t( )   

(3.5)

The extremum condition of un 1 is given by

  u n1  0 .

(3.6)

In view of (3.6), we get:

  ( )   1     ( 1   )  

( )

 0,  t

 ( ) 0 , (1   )  t

  ( )     (1   )   

( 2 )

 0.

(3.7)

 t

So, from (3.7), we get

 ( ) (  t )  . (1   ) (1   ) Thus, we have  (  t )  L u ( )  R u n ( )  N u n ( )  f ( ) . u n1 (t )  u n (t ) 0 I t( )   (1   )  n   and t 2 ( x). (1   ) Finally, from (3.8), the solution of (3.1) is: u  lim u n . u0 ( x, t )  1 ( x) 

(3.8)

(3.9)

(3.10)

n

4. Applications Example 1. Let us consider the linear telegraph equation with LFDOs:

 2 u x 2



 2 u t 2



 u

 u , 0   1

(4.1)

 u ( x,0)  2 E ( x ),  t

(4.2)

t 

and u ( x,0)  E ( x ) ,

 u (0, t )  E (2t  ) .  x The correctionhfunctional is given by u (0, t )  E (2t  ) ,

(4.3)

2 2      ( )   un ( x,  )  u~n ( x,  )  u~n ( x,  ) ~  un 1 ( x, t )  un ( x, t ) 0 I t( )     u ( x ,  ) . (4.4) n 2  2     x     (1   )   

where

 ( ) (  t )  . (1   ) (1   )

Therefore, 2 2      (  t )   u n ( )  u n ( )  u n ( )  u n1 (t )  u n (t ) 0 I t( )     u (  ) . n 2  2     (1   )    x       Now, we get

3

(4.5)


 2t   u 0  E ( x  ) 1  .  (1   )  

(4.6)

2 2      (  t )   u 0 ( x,  )  u 0 ( x,  )  u 0 ( x,  )  u1  u 0  0 I t( )     u ( x ,  ) 0 2  2     (1   )   x     

 2t  4t 2 8t 3   E ( x ) 1      (1   ) (1  2 ) (1  3 ) 

(2) t  .   0 (1   ) 3

 E ( x ) 

(4.7)

2 2      (  t )   u1 ( x,  )  u1 ( x,  )  u1 ( x,  )  u 2  u1  0 I t( )     u ( x ,  ) 1 2 2     ( 1   ) x       

 2t  4t 2 8t 3 16t 4 32t 5   E ( x ) 1        (1   ) (1  2 ) (1  3 ) (1  4 ) (1  5 ) 

(2) t  .   0 (1   ) 5

 E ( x ) 

(4.8)

2 2      (  t )   u 2 ( x,  )  u 2 ( x,  )  u 2 ( x,  )  u 3  u 2 ( x, t ) 0 I t( )     u ( x ,  ) 2 2  2     x     (1   )   

   4t 2 8t 3 16t 4 1  2t      (1   ) (1  2 ) (1  3 ) (1  4 )   E ( x )   32t 5 64t 6 128t 7    (1  5 )  (1  6  (1  7 )   

(2) t  .   0 (1   ) 7

 E ( x ) 

(4.9)



u n  E ( x  )

2 n 1



 0

(2)  t  . (1   )

(4.10)

u( x, t )  E ( x ) E (2t  ) .

(4.11)

The plots ofkthe FVIM solutions of (4.1) with initialgconditions (4.2) and the boundarynconditions (4.3) 1 ln⁡(2)

are shown in theffigures 1,2,3 for differentkvalues of 𝛼 = 1. 2 . ln⁡(3).

4


4

x 10 2.5 2 1.5 1 0.5 0 3

2.5

60 2

40 1.5

20 1

0

Figure 1: The plot of solution to local fractional telegraph equation with fractal dimension   1 .

25 20 15 10 5 0 3 2.5

60 2

40 1.5

20 1

0

Figure 2: The plot of solution to local fractional telegraph equation with fractal dimension  

1 . 2

80

60

40

20

0 3 2.5

60 2

40 1.5

20 1

0

Figure 3: The plot of solution to local fractional telegraph equation with fractal dimension  

5

ln 2 . ln 3


Example 2. Consider the nonlinear telegraph equation with LFDOs:

 2 u x

2



 2 u t

2



 u t



 u 2  E ( x   t  )  E (2 x  ) E (2t  )

, 0   1

(4.12)

and

  u ( x,0)

u ( x,0)  E ( x  ) ,

t





 u (0, t )

u (0, t )  E (t  ) ,

x

  E ( x  ),

(4.13)

 E (t  )

. Applying Eqs. (3.8) and (4.12), we arrive at the following iteration formula:

(4.14)

2 2      (  t )   u n  u n  u n 2      u n1  u n  0 I t( )     u  E ( x   )  E ( 2 x ) E (  2  ) n     (1   )   2 x 2       (4.15) By using Eqs. (3.9) and (4.13), we obtain   t  u 0  E ( x  ) 1   (1   )   . (4.16) Therefore, we obtain: 2 2      (  t )   u 0  u 0  u 0 2      u1  u 0  0 I t( )     u  E ( x   )  E ( 2 x ) E (  2  ) 0     (1   )   2 x 2      

  t t 2 t 3   E ( x ) 1      ( 1   )  ( 1  2  )  ( 1  3  )  .

(4.17)

    (  t )   u1  u1  u1 2      u 2  u1  0 I t( )     u  E ( x   )  E ( 2 x ) E (  2  ) 1    2   x 2      (1   )    2

2



  t t 2 t 3 t 4 t 5   E ( x ) 1        ( 1   )  ( 1  2  )  ( 1  3  )  ( 1  4  )  ( 1  5  )  .

(4.18)



u n  E ( x  )

2 n 1



 0

(1)  t  (1   ) .

(4.19)

u  lim u n ( x, t ) n

(1) t    0 (1   ) 

 E ( x ) 

 E ( x  t ) .

(4.20)

The plots ofkthe FVIM solutions of (4.12) with initialgconditions (4.13) and the boundarynconditions 1 ln⁡(2) (4.14) are shown in theffigures 1,2,3 for differentkvalues of 𝛼 = 1. 2 . ln⁡(3).

6


4

x 10 2.5 2 1.5 1 0.5 0 3

2.5

60 2

40 1.5

20 1

0

Figure 4: The plot of solution to problem (4.12) with fractal dimension   1 .

25 20 15 10 5 0 3 2.5

60 2

40 1.5

20 1

0

Figure 5: The plot of solution to problem (4.12) with fractal dimension



1 . 2

80

60

40

20

0 3 2.5

60 2

40 1.5

20 1

0

Figure 6: The plot of solution to problem (4.12) with fractal dimension

7



ln 2 . ln 3


5. Conclusion In this manuscript, utilizing the LFDOs, we investigated the telegraph equations. Based on the LFVIM, the solutions of the localdfractional telegraph equationswwere presented. The iteration functions, which is localgfractional continuous, is obtained easily within the fractalkLagrange multipliers, which can be optimally determined by the localkfractional variational theory. It is shown that the LFVIM is an efficient and simple tool for handling PDEs with localofractional differential operator. References [1] M. Garg, P. Manohar, and S. L. Kalla, Generalized differential Transform method to space-time fractional telegraph equation, International Journal of Differential Equations, vol. 2011, Article ID 548982, (2011) pp. 1-8. [2] J. Chen, F. Liu, and V. Anh, Analytical solution for the time fractional telegraph equation by the method of separating variables, Journal of Mathematical Analysis and Applications, vol. 338, no. 2, (2008), pp. 1364-1377. [3] A. Ansari, Fractional exponential operators and time fractional telegraph equation,” Boundary Value Problems, vol. 2012, Article 125, (2012), pp. 1-7. [4] F. Huang, Analytical solution for the time-fractional telegraph equation,” Journal of Applied Mathematics, vol. 2009, Article ID 890158, (2009), pp. 1-10. [5] S. Das, K. Vishal, P. K. Gupta, and A. Yildirim, An approximate analytical solution of timefractional telegraph equation, Applied Mathematics and Computation, vol. 217, no. 18, (2011), pp. 7405–7411. [6] J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, vol. 167, no.1-2, (1998), pp. 57– 68. [7] J.-H. He, Variational iteration method a kind of non-linear analytical technique: some examples, International Journal of Non-Linear Mechanics, vol. 34, no. 4, (1999), pp. 699–708. [8] J. H. He, G. C. Wu, and F. Austin, The variational iteration method which should be followed, Nonlinear Science Letters A, vol. 1, no. 1, (2010), pp. 1–30. [9] S. Momani and S. Abuasad, Application of He’s variational iteration method to Helmholtz equation, Chaos, Solitons & Fractals, vol. 27, no. 5, (2006), pp. 1119–1123. [10] M. A. Abdou and A. A. Soliman, Variational iteration method for solving Burger’s and coupled Burger’s equations, Journal of Computational and Applied Mathematics, vol. 181, no. 2, (2005), pp. 245– 251. [11] S. Abbas bandy, Numerical solution of non-linear Klein- Gordon equations by variational iteration method, International Journal for Numerical Methods in Engineering, vol.70, no. 7, (2007) pp. 876-881. [12] S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, Traveling wave solutions of seventh-order generalized KdV equations using He’s polynomials, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 2, (2009), pp. 227–234. [13] V. Marinca, N. Heris¸ and C. Bota, Application of the variational iteration method to some nonlinear one-dimensional oscillations, Meccanica, vol. 43, no. 1, (2008) pp. 75–79. [14] A. M. Wazwaz, A study on linear and nonlinear Schrodinger equations by the variational iteration method, Chaos, Solitons & Fractals, vol. 37, no. 4, (2008), pp. 1136–1142. [15] M. Dehghan and F. Shakeri, Application of He’s variational iteration method for solving the Cauchy reaction-diffusion problem, Journal of Computational and Applied Mathematics, vol. 214, no. 2, (2008) pp. 435–446. [16] S. Das, Approximate solution of fractional diffusion equation revisited, International Review of Chemical Engineering, vol. 4, no. 5, (2012), pp. 501–504.

8


[17] J. Hristov, An exercise with the He’s variation iteration method to a fractional Bernoulli equation arising in a transient conduction with a non-linear boundary heat flux, International Review of Chemical Engineering, vol. 4, no. 5, (2012) pp. 489–497. [18] G.-C. Wu and E. W. M. Lee, Fractional variational iteration method and its application, Physics Letters A General, Atomic and Solid State Physics, vol. 374, no. 25, (2010), pp. 2506–2509. [19] S. Xu, X. Ling, Y. Zhao, H. K. Jassim, A Novel Schedule for Solving the Two-Dimensional Diffusion in Fractal Heat Transfer, Thermal Science, Vol. 19, Suppl. 1, pp. S99-S103, 2015. [20] H. Jafari, H. K. Jassim, S. P. Moshokoa, V. M. Ariyan and F. Tchier, Reduced differential transform method for partial differential equations within local fractional derivative operators, Advances in Mechanical Engineering, 8(4), 1-6, 2016. [21] S. Q. Wang, Y. J. Yang, and H. K. Jassim, Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative, Abstract and Applied Analysis, vol. 2014, Article ID 176395, (2014), pp. 1-7. [22] S. P. Yan, H. Jafari, and H. K. Jassim, Local Fractional Adomian Decomposition and Function Decomposition Methods for Solving Laplace Equation within Local Fractional Operators, Advances in Mathematical Physics, vol. 2014, Article ID 161580, (2014), pp. 1-8.

9