On nonlinear fractional KleinâGordon equation
Signal Processing 91 (2011) 446–451
Contents lists available at ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
On nonlinear fractional Klein–Gordon equation Alireza K. Golmankhaneh a,b,, Ali K. Golmankhaneh c, Dumitru Baleanu d,e a
Department of Physics, University of Pune, Pune 411007, India Department of Physics, Islamic Azad University, Urmia Branch, Oromiyeh, PO Box 969, Iran c Department of Physics, Islamic Azad University, Mahabad Branch, Mahabad, Iran d Department of Mathematics and Computer Science, C - ankaya University, 06530 Ankara, Turkey e Institute of Space Sciences, PO Box MG-23, 76900 Magurele-Bucharest, Romania b
a r t i c l e i n f o
abstract
Available online 20 April 2010
Numerical methods are used to find exact solution for the nonlinear differential equations. In the last decades Iterative methods have been used for solving fractional differential equations. In this paper, the Homotopy perturbation method has been successively applied for finding approximate analytical solutions of the fractional nonlinear Klein–Gordon equation can be used as numerical algorithm. The behavior of solutions and the effects of different values of fractional order a are shown graphically. Some examples are given to show ability of the method for solving the fractional nonlinear equation. Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved.
Keywords: Caputo fractional derivative Fractional Klein Gordon Homotopy perturbation method Numerical algorithm Iteration method
1. Introduction In the last decades, fractional calculus found many applications in various fields of physical sciences such as viscoelasticity, diffusion, control, relaxation processes and so on [1–11]. Fractional differential equations are extensively used in modeling phenomena in various fields of science and engineering [12–34]. In order to show the superiority and efficiency of the fractional calculus in above-mentioned areas various methods have been developed to solve linear/nonlinear fractional differential equations (FDE) [35–37]. For example, Rida et al. [38] ¨ have studied nonlinear Schrodinger equation of fractional order. The investigation of the exact solutions of nonlinear evolution equations play an important role in the study of nonlinear physical phenomena. It is well known that except a limited number of these problems, most of them do not have analytical solution. Therefore, these nonlinear Corresponding author at: Department of Physics, University of Pune, Pune 411007, India. E-mail addresses: [email protected] (A.K. Golmankhaneh), [email protected] (A.K. Golmankhaneh), [email protected] (D. Baleanu).
equations have been solved using other methods. Some of them are solved using numerical techniques and some are solved using the analytical method of perturbation. In the numerical method, stability and convergence must be considered so as to avoid divergence or inappropriate results. In the analytical perturbation method, we exert the small parameter in the equation. So, finding the small parameter and exerting it into the equation are difficulties of this method, and also because the basis of the common perturbation method is upon the existence of a small parameter, developing the method for different applications is very difficult. Therefore, many different methods have recently introduced some ways to eliminate the small parameter, such as Homotopy perturbation method. Homotopy perturbation method (HPM) has been introduced by He which possesses a great potential in solving different kinds of differential and functional equations. Both linear and nonlinear and systems of such types are amenable to the method. In the nonlinear case for differential equation and partial differential equation. The method has the advantage of dealing directly withthe problem. That is, the equations are solved without transforming them also avoids linearization, discretization or any unrealistic assumption. In dealing with
0165-1684/$ - see front matter Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2010.04.016
A.K. Golmankhaneh et al. / Signal Processing 91 (2011) 446–451
nonlinear equations the nonlinearity terms is replaced by a series. Then it is an easy algorithm for computing the solution. As a result, it yields a very rapidly convergent series solution, and usually a few iterations lead to very accurate approximation of the exact solution [39–44]. Some other numerical methods [45–47] can be done on the same examples and the results are comparable. As it is well known, linear and nonlinear Klein–Gordon equations model many problems in classical and quantum mechanics, solitons and condensed matter physics. For example, nonlinear sine Klein–Gordon equation models a Josephson junction, the motion of rigid pendula attached to a stretched wire, and dislocations in crystals [48–52]. A non-local version of these equations are properly described by the fractional version of them. Chowdhury and Hashim have employed HPM for solving Klein–Gordon equations [44]. The main aim of this work is to apply the HPM to solve the nonlinear Klein–Gordon equations of fractional order. A comparison with either the Adomian decomposition method (ADM) or variational iteration method (VIM) shows that HPM is easier than both. Since using ADM we need to calculate Adomian’s polynomials for nonlinear terms. Also applying variational iteration method needs to find Lagrange multiplier by making correction functional stationary in the fractional case [51]. The plan of our paper is as follows: In Section 2 the HPM is briefly reviewed. Further, some basic definitions and properties of the fractional Riemann–Liouville integral and Caputo fractional derivatives are briefly mentioned. Section 3 deals with solving the fractional nonlinear Klein–Gordon equation using HPM. In Section 4 we have presented some examples. Section 5 is dedicated to conclusion.
447
The approximate solution of Eq. (1), therefore can be readily obtained: u ¼ lim U ¼ U0 þ U1 þ U2 þ : p-1
ð3Þ
The convergence of the series equation (3) has been proved in [39,40]. 2.2. Fractional calculus If a 40 then left sided Riemann–Liouville fractional integral of order a is defined [1–10] Z x 1 f ðtÞ dt, 0 o x oa: ð4Þ Ixa f ðxÞ ¼ GðaÞ 0 ðxtÞ1a Let u(x,t) and n1 r a o n, then partial Caputo fractional derivatives is defined as follows: Z t @a uðx,tÞ 1 @n ¼ ðttÞna1 n uðx, tÞ dt, ð5Þ a @t GðnaÞ 0 @t The Caputo derivative is defined as follows: n Z x 1 d a na n D f ðxÞ ¼ ðxtÞna1 f ðtÞ dt: a Dx f ðxÞ ¼ a Ix dt GðnaÞ a ð6Þ By inspection we observe that a a a a Dt ðf ðtÞ þ gðtÞÞ ¼ a Dt f ðtÞ þ a Dt gðtÞ
ð7Þ
and a c ¼ 0,
a Dt
where c is a constant:
ð8Þ
In addition of this we have Ixa
m 1 k X @a uðx,tÞ @ uðx,0Þ t k : ¼ uðx,tÞ @t a @t k k! k¼0
ð9Þ
2. Basic tools 3. Fractional nonlinear Klein–Gordon equation 2.1. Homotopy perturbation method The idea of the HPM and its application in various differential equations are given in [39–43]. For convenience of the reader, we present a short review of the HPM. Consider nonlinear differential equation LðuÞ þNðuÞ ¼ f ðrÞ,
r 2 O,
ð1Þ
uðx,0Þ ¼ u0 ,
r 2 G,
where L is a linear operator, while N is nonlinear operator, G is the boundary of the domain O and f ðrÞ is a known analytic function. He’s Homotopy perturbation method defines the Homotopy Uðr,pÞ : O ½0,1-R which satisfies HðU,pÞ ¼ ð1pÞ½LðUÞLðu0 Þ þ p½LðUÞ þ NðUÞf ðrÞ ¼ 0,
ð10Þ
x 2 R,
where a,b and c are real constants. To solve Eq. (10) using HPM, we construct the following Homotopy: a a @ U @ U @2 Uðx,tÞ HðU,pÞ ¼ ð1pÞ aUðx,tÞ þ p a @t @t a @x2 2 bU ðx,tÞcU 3 ðx,tÞ ¼ 0: ð11Þ
ð2Þ
where p 2 ½0,1 is an impeding parameter, u0 is an initial approximation which satisfies the boundary conditions. The changing process of p from zero to unity is just that of U(r,p) from u0 to u(r). The basic assumption is that the solution of Eq. (2) can be expressed as a power series in p: U ¼ U0 þ pU 1 þ p2 U2 þ :
@a uðx,tÞ @2 uðx,tÞ 2 ¼ þ auðx,tÞ þ bu ðx,tÞ þ cu3 ðx,tÞ, a @t @x2 with the initial condition
with boundary condition Bðu,@u=@nÞ ¼ 0,
It is well known the nonlinear Klein–Gordon equation has many application in physics. In this section we solve the fractional nonlinear Klein–Gordon equation using HPM. We consider Klein–Gordon equation:
Let the solution of Eq. (11) be such that U ¼ U0 þ pU 1 þ p2 U2 þ :
ð12Þ
Substituting Eq. (12) into Eq. (11), and equating the coefficients of the terms with identical powers of p, p0 :
@a U0 ¼ 0, @t a
ð13Þ
448
p1 :
A.K. Golmankhaneh et al. / Signal Processing 91 (2011) 446–451
@a U1 @2 U0 2 aU 0 bU 0 cU 30 ¼ 0, @t a @x2
U1 ðx,0Þ ¼ 0,
ð14Þ
@a U2 @2 U1 aU 1 2bU 0 U1 @t a @x2 2 cðU0 U1 þ 2U1 U02 Þ ¼ 0, U2 ðx,0Þ ¼ 0,
4. Illustrative examples
p2 :
Example 1. Consider fractional linear Klein–Gordon equation with initial condition:
^
@a uðx,tÞ @2 u 2 u ¼ 0, @t a @x
pj :
j1 X @a Uj @2 Uj1 aU j1 b Ui Uji1 a 2 @t @x i¼0
c
j1 ji1 X X
Ui Uk Ujki1 ¼ 0,
t Z 0, uðx,0Þ ¼ 1þ sinðxÞ:
ð18Þ
Substituting (a =1, b= 0 and c =0) into Eq. (11) using the corresponding Homotopy, Eqs. (16) and (17) will have
Uj ðx,0Þ ¼ 0:
ð15Þ
i¼0 k¼0
We suppose U0 =u0 = u(x,0). Eqs. (15) yield the following relations: Z t j1 X @2 Uj1 1 Uj ¼ ðttÞa1 þaU j1 þ b Ui Uji1 GðaÞ 0 @x2 i¼0 ! j1 ji1 X X þc Ui Uk Ujki1 dt: ð16Þ
U1 ðx,tÞ ¼
U2 ðx,tÞ ¼
ta
Gða þ 1Þ
,
t 2a
Gð2a þ 1Þ
,
^ Un ðx,tÞ ¼
t na
Gðna þ1Þ
:
i¼0 k¼0
Consequently, the solution of Eq. (10) may be obtained by choosing p ¼ 1,
Finally, exact solution will be
u ¼ lim U ¼ U0 þU1 þU2 þ :
uðx,tÞ ¼ 1 þ sinðxÞ þ
p-1
0.8
2
0.6
0 0
0.4 1 2 x
3 4
1 X
t na : Gðna þ1Þ n¼1
2
0.8
1
0.6
0 0
t
0.4 1
0.2
u (x, t)
u (x, t)
4
u (x, t)
ð17Þ
0.2
2 x
0
3 4
2 1.5 1 0.5 0 0
0.8 0.6 0.4 1 2 x
t
0.2 3 4
0
Fig. 1. Graph of the u(x,t) corresponding to the values a ¼ 0:5, 1:5 and 2.5 from left to right.
0
t
A.K. Golmankhaneh et al. / Signal Processing 91 (2011) 446–451
In Fig. 1 we have presented the graphs of u(x,t) corresponding to the values a ¼ 0:5,1:5 and 2.5.
t Z0, uðx,0Þ ¼ 1 þ sinðxÞ:
t
Gð2a þ 1Þ
ð19Þ
@a uðx,tÞ @2 u 2 þ uu3 ¼ 0, @t a @x
U1 ðx,tÞ ¼
160 sin2 ðxÞ82 sin3 ðxÞ10sinð4xÞÞ,
U2 ðx,tÞ ¼
^ Thus, the approximate solution is
U3 ðx,tÞ ¼
ta ð1 þ 3sinðxÞ uðx,tÞ ¼ 1 þsinðxÞ Gða þ1Þ t 2a þsin2 ðxÞÞ þ ð11sinðxÞ þ 12 sin2 ðxÞ Gð2a þ 1Þ
x
t 2a
3
5
ð3 sechðxÞ34 sech ðxÞ18 sech ðxÞÞ,
t 3a
3
5
ð64 sech ðxÞ288 sech ðxÞ
Gð3a þ 1Þ
ð22Þ
^
u (x, t)
2 1.5 1 0.5 0 -2
0.01 0.008 0.006 0.004
-1
0.002
1
3
ð2 sechðxÞ3 sech ðxÞÞ,
Gð2a þ 1Þ
0 x
2 0
0.002
1 2
2 1.5 1 0.5 0 -2
u (x, t)
u (x, t)
ta
Gða þ 1Þ
7
0.01 0.008 0.006 0.004 t 0
ð21Þ
þ 240 sech ðxÞÞ,
0
-1
t Z 0,
with the initial condition u(x,0)= sech(x). Substituting (a= 1, b= 0 and c= 1) into Eqs. (11), (16) and (17) we get the first few terms
t 3a ð1857sinðxÞ U3 ðx,tÞ ¼ Gð3a þ 1Þ
-200 -2
ð20Þ
Example 3. Consider the following fractional nonlinear Klein–Gordon equation as
ð11sinðxÞ þ 12 sin2 ðxÞ þ 2 sin3 ðxÞÞ,
-100
ð1857sinðxÞ160 sin2 ðxÞ
In Fig. 2 we have shown the graphs of u(x,t) corresponding to the values a ¼ 0:01, 0:5 and 1.
ta ð1 þ 3sinðxÞ þ sin2 ðxÞÞ, U1 ðx,tÞ ¼ Gða þ 1Þ U2 ðx,tÞ ¼
Gð3a þ 1Þ
82 sin3 ðxÞ10sinð4xÞÞ . . . :
We construct a Homotopy substituting (a = 0, b= 1 and c= 0) into Eq. (11). In view of Eqs. (16) and (17) and corresponding Homotopy we obtain
2a
t 3a
þ2 sin3 ðxÞÞ þ
Example 2. Consider fractional nonlinear Klein–Gordon differential equation @a uðx,tÞ @2 u 2 þ u2 ¼ 0, @t a @x
449
0.01 0.008 0.006 0.004
-1 0 x
t
0.002
1 2 0
Fig. 2. Graph of the u(x,t) corresponding to the values a ¼ 0:01,0:5 and 1 from left to right.
0
t
450
A.K. Golmankhaneh et al. / Signal Processing 91 (2011) 446–451
30 20 10 0 -2
u (x, t)
u (x, t)
0.01 0.008 0.006 t
0.004
-1 0 x
-0.4 -0.6 -0.8 -1 -2
0.01 0.008 0.006 0.004 t
-1 0
0.002
0.002
x
1
1 2 0
u (x, t)
2 0
-0.4 -0.6 -0.8 -1 -2
0.01 0.008 0.006 0.004
-1 0 x
t
0.002 1 2 0
Fig. 3. Graph of the u(x,t) corresponding to the values a ¼ 0:01,0:5 and 1 from left to right.
Hence the approximation solution is ta 3 ð2 sechðxÞ3 sech ðxÞÞ uðx,tÞ ¼ sechðxÞ Gða þ1Þ t 2a 3 5 ð3 sechðxÞ34 sech ðxÞ18 sech ðxÞÞ Gð2a þ1Þ t 3a 3 5 ð64 sech ðxÞ288 sech ðxÞ Gð3a þ1Þ 7
þ 240 sech ðxÞÞ . . . :
ð23Þ
In Fig. 3 we have sketched the plots of u(x,t) corresponding to the values a ¼ 0:01, 0:5 and 1. 5. Conclusion In this study, the Homotopy perturbation method with new strategies has employed to obtain approximate analytical solution of nonlinear Klein–Gordon equations. It is quite important to notice that higher number of iteration and higher orders of p are needed to gain more accuracy. The number of iterations and the order of p depend on the nonlinear term of the equation. This algorithm provides accurate numerical solutions without discretization for nonlinear differential equations. In this method, in spite of Adomian decomposition method we do not need to calculate Adomian polynomials or variational iteration method needs to find Lagrange multiplier by making correction functional stationary in the fractional case. The result obtained appears to be
general solutions since if we choose a ¼ 1 in the solution we arrive at the classical one. Mathematica has been used for presenting graph of solution in the present paper. References [1] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic, New York, 1974. [2] K.S. Miller, B. Ross, An Introduction to the Fractional Integrals and Derivatives—Theory and Application, Wiley, New York, 1993. [3] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York, 1993. [4] R. Hilfer, Application of Fractional Calculus in Physics, World Scientific, 2000. [5] I. Podlubny, Fractional Differential Equations, Academic, New York, 1999. [6] G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, 2005. [7] A.A. Kilbas, H.H. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, The Netherlands, 2006. [8] R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Inc., Connecticut, 2006. [9] R. Gorenflo, F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Orders, Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, New York, 1997. [10] B.J. West, M. Bologna, P. Grigolini, Physics of Fractal Operators, Springer, New York, 2003. [11] A. Oustaloup, La de´rivation non entie re: the´orie, synthe se et applications, Editions Hermes, Paris, France, 1995. [12] F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E 55 (1997) 3581. [13] N. Laskin, Fractional quantum mechanics, Phys. Rev. E 62 (2000) 3135.
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Contents lists available at ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
On nonlinear fractional Klein–Gordon equation Alireza K. Golmankhaneh a,b,, Ali K. Golmankhaneh c, Dumitru Baleanu d,e a
Department of Physics, University of Pune, Pune 411007, India Department of Physics, Islamic Azad University, Urmia Branch, Oromiyeh, PO Box 969, Iran c Department of Physics, Islamic Azad University, Mahabad Branch, Mahabad, Iran d Department of Mathematics and Computer Science, C - ankaya University, 06530 Ankara, Turkey e Institute of Space Sciences, PO Box MG-23, 76900 Magurele-Bucharest, Romania b
a r t i c l e i n f o
abstract
Available online 20 April 2010
Numerical methods are used to find exact solution for the nonlinear differential equations. In the last decades Iterative methods have been used for solving fractional differential equations. In this paper, the Homotopy perturbation method has been successively applied for finding approximate analytical solutions of the fractional nonlinear Klein–Gordon equation can be used as numerical algorithm. The behavior of solutions and the effects of different values of fractional order a are shown graphically. Some examples are given to show ability of the method for solving the fractional nonlinear equation. Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved.
Keywords: Caputo fractional derivative Fractional Klein Gordon Homotopy perturbation method Numerical algorithm Iteration method
1. Introduction In the last decades, fractional calculus found many applications in various fields of physical sciences such as viscoelasticity, diffusion, control, relaxation processes and so on [1–11]. Fractional differential equations are extensively used in modeling phenomena in various fields of science and engineering [12–34]. In order to show the superiority and efficiency of the fractional calculus in above-mentioned areas various methods have been developed to solve linear/nonlinear fractional differential equations (FDE) [35–37]. For example, Rida et al. [38] ¨ have studied nonlinear Schrodinger equation of fractional order. The investigation of the exact solutions of nonlinear evolution equations play an important role in the study of nonlinear physical phenomena. It is well known that except a limited number of these problems, most of them do not have analytical solution. Therefore, these nonlinear Corresponding author at: Department of Physics, University of Pune, Pune 411007, India. E-mail addresses: [email protected] (A.K. Golmankhaneh), [email protected] (A.K. Golmankhaneh), [email protected] (D. Baleanu).
equations have been solved using other methods. Some of them are solved using numerical techniques and some are solved using the analytical method of perturbation. In the numerical method, stability and convergence must be considered so as to avoid divergence or inappropriate results. In the analytical perturbation method, we exert the small parameter in the equation. So, finding the small parameter and exerting it into the equation are difficulties of this method, and also because the basis of the common perturbation method is upon the existence of a small parameter, developing the method for different applications is very difficult. Therefore, many different methods have recently introduced some ways to eliminate the small parameter, such as Homotopy perturbation method. Homotopy perturbation method (HPM) has been introduced by He which possesses a great potential in solving different kinds of differential and functional equations. Both linear and nonlinear and systems of such types are amenable to the method. In the nonlinear case for differential equation and partial differential equation. The method has the advantage of dealing directly withthe problem. That is, the equations are solved without transforming them also avoids linearization, discretization or any unrealistic assumption. In dealing with
0165-1684/$ - see front matter Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2010.04.016
A.K. Golmankhaneh et al. / Signal Processing 91 (2011) 446–451
nonlinear equations the nonlinearity terms is replaced by a series. Then it is an easy algorithm for computing the solution. As a result, it yields a very rapidly convergent series solution, and usually a few iterations lead to very accurate approximation of the exact solution [39–44]. Some other numerical methods [45–47] can be done on the same examples and the results are comparable. As it is well known, linear and nonlinear Klein–Gordon equations model many problems in classical and quantum mechanics, solitons and condensed matter physics. For example, nonlinear sine Klein–Gordon equation models a Josephson junction, the motion of rigid pendula attached to a stretched wire, and dislocations in crystals [48–52]. A non-local version of these equations are properly described by the fractional version of them. Chowdhury and Hashim have employed HPM for solving Klein–Gordon equations [44]. The main aim of this work is to apply the HPM to solve the nonlinear Klein–Gordon equations of fractional order. A comparison with either the Adomian decomposition method (ADM) or variational iteration method (VIM) shows that HPM is easier than both. Since using ADM we need to calculate Adomian’s polynomials for nonlinear terms. Also applying variational iteration method needs to find Lagrange multiplier by making correction functional stationary in the fractional case [51]. The plan of our paper is as follows: In Section 2 the HPM is briefly reviewed. Further, some basic definitions and properties of the fractional Riemann–Liouville integral and Caputo fractional derivatives are briefly mentioned. Section 3 deals with solving the fractional nonlinear Klein–Gordon equation using HPM. In Section 4 we have presented some examples. Section 5 is dedicated to conclusion.
447
The approximate solution of Eq. (1), therefore can be readily obtained: u ¼ lim U ¼ U0 þ U1 þ U2 þ : p-1
ð3Þ
The convergence of the series equation (3) has been proved in [39,40]. 2.2. Fractional calculus If a 40 then left sided Riemann–Liouville fractional integral of order a is defined [1–10] Z x 1 f ðtÞ dt, 0 o x oa: ð4Þ Ixa f ðxÞ ¼ GðaÞ 0 ðxtÞ1a Let u(x,t) and n1 r a o n, then partial Caputo fractional derivatives is defined as follows: Z t @a uðx,tÞ 1 @n ¼ ðttÞna1 n uðx, tÞ dt, ð5Þ a @t GðnaÞ 0 @t The Caputo derivative is defined as follows: n Z x 1 d a na n D f ðxÞ ¼ ðxtÞna1 f ðtÞ dt: a Dx f ðxÞ ¼ a Ix dt GðnaÞ a ð6Þ By inspection we observe that a a a a Dt ðf ðtÞ þ gðtÞÞ ¼ a Dt f ðtÞ þ a Dt gðtÞ
ð7Þ
and a c ¼ 0,
a Dt
where c is a constant:
ð8Þ
In addition of this we have Ixa
m 1 k X @a uðx,tÞ @ uðx,0Þ t k : ¼ uðx,tÞ @t a @t k k! k¼0
ð9Þ
2. Basic tools 3. Fractional nonlinear Klein–Gordon equation 2.1. Homotopy perturbation method The idea of the HPM and its application in various differential equations are given in [39–43]. For convenience of the reader, we present a short review of the HPM. Consider nonlinear differential equation LðuÞ þNðuÞ ¼ f ðrÞ,
r 2 O,
ð1Þ
uðx,0Þ ¼ u0 ,
r 2 G,
where L is a linear operator, while N is nonlinear operator, G is the boundary of the domain O and f ðrÞ is a known analytic function. He’s Homotopy perturbation method defines the Homotopy Uðr,pÞ : O ½0,1-R which satisfies HðU,pÞ ¼ ð1pÞ½LðUÞLðu0 Þ þ p½LðUÞ þ NðUÞf ðrÞ ¼ 0,
ð10Þ
x 2 R,
where a,b and c are real constants. To solve Eq. (10) using HPM, we construct the following Homotopy: a a @ U @ U @2 Uðx,tÞ HðU,pÞ ¼ ð1pÞ aUðx,tÞ þ p a @t @t a @x2 2 bU ðx,tÞcU 3 ðx,tÞ ¼ 0: ð11Þ
ð2Þ
where p 2 ½0,1 is an impeding parameter, u0 is an initial approximation which satisfies the boundary conditions. The changing process of p from zero to unity is just that of U(r,p) from u0 to u(r). The basic assumption is that the solution of Eq. (2) can be expressed as a power series in p: U ¼ U0 þ pU 1 þ p2 U2 þ :
@a uðx,tÞ @2 uðx,tÞ 2 ¼ þ auðx,tÞ þ bu ðx,tÞ þ cu3 ðx,tÞ, a @t @x2 with the initial condition
with boundary condition Bðu,@u=@nÞ ¼ 0,
It is well known the nonlinear Klein–Gordon equation has many application in physics. In this section we solve the fractional nonlinear Klein–Gordon equation using HPM. We consider Klein–Gordon equation:
Let the solution of Eq. (11) be such that U ¼ U0 þ pU 1 þ p2 U2 þ :
ð12Þ
Substituting Eq. (12) into Eq. (11), and equating the coefficients of the terms with identical powers of p, p0 :
@a U0 ¼ 0, @t a
ð13Þ
448
p1 :
A.K. Golmankhaneh et al. / Signal Processing 91 (2011) 446–451
@a U1 @2 U0 2 aU 0 bU 0 cU 30 ¼ 0, @t a @x2
U1 ðx,0Þ ¼ 0,
ð14Þ
@a U2 @2 U1 aU 1 2bU 0 U1 @t a @x2 2 cðU0 U1 þ 2U1 U02 Þ ¼ 0, U2 ðx,0Þ ¼ 0,
4. Illustrative examples
p2 :
Example 1. Consider fractional linear Klein–Gordon equation with initial condition:
^
@a uðx,tÞ @2 u 2 u ¼ 0, @t a @x
pj :
j1 X @a Uj @2 Uj1 aU j1 b Ui Uji1 a 2 @t @x i¼0
c
j1 ji1 X X
Ui Uk Ujki1 ¼ 0,
t Z 0, uðx,0Þ ¼ 1þ sinðxÞ:
ð18Þ
Substituting (a =1, b= 0 and c =0) into Eq. (11) using the corresponding Homotopy, Eqs. (16) and (17) will have
Uj ðx,0Þ ¼ 0:
ð15Þ
i¼0 k¼0
We suppose U0 =u0 = u(x,0). Eqs. (15) yield the following relations: Z t j1 X @2 Uj1 1 Uj ¼ ðttÞa1 þaU j1 þ b Ui Uji1 GðaÞ 0 @x2 i¼0 ! j1 ji1 X X þc Ui Uk Ujki1 dt: ð16Þ
U1 ðx,tÞ ¼
U2 ðx,tÞ ¼
ta
Gða þ 1Þ
,
t 2a
Gð2a þ 1Þ
,
^ Un ðx,tÞ ¼
t na
Gðna þ1Þ
:
i¼0 k¼0
Consequently, the solution of Eq. (10) may be obtained by choosing p ¼ 1,
Finally, exact solution will be
u ¼ lim U ¼ U0 þU1 þU2 þ :
uðx,tÞ ¼ 1 þ sinðxÞ þ
p-1
0.8
2
0.6
0 0
0.4 1 2 x
3 4
1 X
t na : Gðna þ1Þ n¼1
2
0.8
1
0.6
0 0
t
0.4 1
0.2
u (x, t)
u (x, t)
4
u (x, t)
ð17Þ
0.2
2 x
0
3 4
2 1.5 1 0.5 0 0
0.8 0.6 0.4 1 2 x
t
0.2 3 4
0
Fig. 1. Graph of the u(x,t) corresponding to the values a ¼ 0:5, 1:5 and 2.5 from left to right.
0
t
A.K. Golmankhaneh et al. / Signal Processing 91 (2011) 446–451
In Fig. 1 we have presented the graphs of u(x,t) corresponding to the values a ¼ 0:5,1:5 and 2.5.
t Z0, uðx,0Þ ¼ 1 þ sinðxÞ:
t
Gð2a þ 1Þ
ð19Þ
@a uðx,tÞ @2 u 2 þ uu3 ¼ 0, @t a @x
U1 ðx,tÞ ¼
160 sin2 ðxÞ82 sin3 ðxÞ10sinð4xÞÞ,
U2 ðx,tÞ ¼
^ Thus, the approximate solution is
U3 ðx,tÞ ¼
ta ð1 þ 3sinðxÞ uðx,tÞ ¼ 1 þsinðxÞ Gða þ1Þ t 2a þsin2 ðxÞÞ þ ð11sinðxÞ þ 12 sin2 ðxÞ Gð2a þ 1Þ
x
t 2a
3
5
ð3 sechðxÞ34 sech ðxÞ18 sech ðxÞÞ,
t 3a
3
5
ð64 sech ðxÞ288 sech ðxÞ
Gð3a þ 1Þ
ð22Þ
^
u (x, t)
2 1.5 1 0.5 0 -2
0.01 0.008 0.006 0.004
-1
0.002
1
3
ð2 sechðxÞ3 sech ðxÞÞ,
Gð2a þ 1Þ
0 x
2 0
0.002
1 2
2 1.5 1 0.5 0 -2
u (x, t)
u (x, t)
ta
Gða þ 1Þ
7
0.01 0.008 0.006 0.004 t 0
ð21Þ
þ 240 sech ðxÞÞ,
0
-1
t Z 0,
with the initial condition u(x,0)= sech(x). Substituting (a= 1, b= 0 and c= 1) into Eqs. (11), (16) and (17) we get the first few terms
t 3a ð1857sinðxÞ U3 ðx,tÞ ¼ Gð3a þ 1Þ
-200 -2
ð20Þ
Example 3. Consider the following fractional nonlinear Klein–Gordon equation as
ð11sinðxÞ þ 12 sin2 ðxÞ þ 2 sin3 ðxÞÞ,
-100
ð1857sinðxÞ160 sin2 ðxÞ
In Fig. 2 we have shown the graphs of u(x,t) corresponding to the values a ¼ 0:01, 0:5 and 1.
ta ð1 þ 3sinðxÞ þ sin2 ðxÞÞ, U1 ðx,tÞ ¼ Gða þ 1Þ U2 ðx,tÞ ¼
Gð3a þ 1Þ
82 sin3 ðxÞ10sinð4xÞÞ . . . :
We construct a Homotopy substituting (a = 0, b= 1 and c= 0) into Eq. (11). In view of Eqs. (16) and (17) and corresponding Homotopy we obtain
2a
t 3a
þ2 sin3 ðxÞÞ þ
Example 2. Consider fractional nonlinear Klein–Gordon differential equation @a uðx,tÞ @2 u 2 þ u2 ¼ 0, @t a @x
449
0.01 0.008 0.006 0.004
-1 0 x
t
0.002
1 2 0
Fig. 2. Graph of the u(x,t) corresponding to the values a ¼ 0:01,0:5 and 1 from left to right.
0
t
450
A.K. Golmankhaneh et al. / Signal Processing 91 (2011) 446–451
30 20 10 0 -2
u (x, t)
u (x, t)
0.01 0.008 0.006 t
0.004
-1 0 x
-0.4 -0.6 -0.8 -1 -2
0.01 0.008 0.006 0.004 t
-1 0
0.002
0.002
x
1
1 2 0
u (x, t)
2 0
-0.4 -0.6 -0.8 -1 -2
0.01 0.008 0.006 0.004
-1 0 x
t
0.002 1 2 0
Fig. 3. Graph of the u(x,t) corresponding to the values a ¼ 0:01,0:5 and 1 from left to right.
Hence the approximation solution is ta 3 ð2 sechðxÞ3 sech ðxÞÞ uðx,tÞ ¼ sechðxÞ Gða þ1Þ t 2a 3 5 ð3 sechðxÞ34 sech ðxÞ18 sech ðxÞÞ Gð2a þ1Þ t 3a 3 5 ð64 sech ðxÞ288 sech ðxÞ Gð3a þ1Þ 7
þ 240 sech ðxÞÞ . . . :
ð23Þ
In Fig. 3 we have sketched the plots of u(x,t) corresponding to the values a ¼ 0:01, 0:5 and 1. 5. Conclusion In this study, the Homotopy perturbation method with new strategies has employed to obtain approximate analytical solution of nonlinear Klein–Gordon equations. It is quite important to notice that higher number of iteration and higher orders of p are needed to gain more accuracy. The number of iterations and the order of p depend on the nonlinear term of the equation. This algorithm provides accurate numerical solutions without discretization for nonlinear differential equations. In this method, in spite of Adomian decomposition method we do not need to calculate Adomian polynomials or variational iteration method needs to find Lagrange multiplier by making correction functional stationary in the fractional case. The result obtained appears to be
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