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Journal of Applied Mathematics
Volume 2012, Article ID 581481, 13 pages
http://dx.doi.org/10.1155/2012/581481
Research Article

A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein-Gordon Equation of Arbitrary (Fractional) Orders

A. M. A. El-Sayed,1 A. Elsaid,2 and D. Hammad2

1Faculty of Science, Alexandria University, Alexandria, Egypt
2Mathematics & Engineering Physics Department, Faculty of Engineering, Mansoura University, P.O. Box 35516, Mansoura, Egypt

Received 16 October 2011; Accepted 8 December 2011

Academic Editor: Livija Cveticanin

Copyright © 2012 A. M. A. El-Sayed et al. 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

The reliable treatment of homotopy perturbation method (HPM) is applied to solve the Klein-Gordon partial differential equation of arbitrary (fractional) orders. This algorithm overcomes the difficulty that arises in calculating complicated integrals when solving nonlinear equations. Some numerical examples are presented to illustrate the efficiency of this technique.

1. Introduction

The Klein-Gordon equation plays a significant role in mathematical physics and many scientific applications such as solid-state physics, nonlinear optics, and quantum field theory [1, 2]. The equation has attracted much attention in studying solitons [3–6] and condensed matter physics, in investigating the interaction of solitons in a collisionless plasma, the recurrence of initial states, and in examining the nonlinear wave equations [7].

The HPM, proposed by He in 1998, has been the subject of extensive studies and was applied to different linear and nonlinear problems [8–13]. This method has the advantage of dealing directly with the problem without transformations, linearization, discretization, or any unrealistic assumption, and usually a few iterations lead to an accurate approximation of the exact solution [13]. The HPM has been used to solve nonlinear partial differential equations of fractional order (see, e.g., [14–16]). Some other methods for series solution that are used to solve nonlinear partial differential equations of fractional order include the Adomian decomposition method [17–19], the variational iteration method [20–22], and the homotopy analysis method [23–25].

Recently, Odibat and Momani [26] suggested a reliable algorithm for the HPM for dealing with nonlinear terms to overcome the difficulty arising in calculating complicated integrals. In [27], this algorithm is utilized to study the behavior of the nonlinear sine-Gordon equation with fractional time derivative. Our aim here is to apply the reliable treatment of HPM to obtain the solution of the initial value problem of the nonlinear fractional-order Klein-Gordon equation of the form𝐷𝛼𝑡𝑢(𝑥,𝑡)+𝑎𝐷𝛽𝑥𝑢(𝑥,𝑡)+𝑏𝑢(𝑥,𝑡)+𝑐𝑢𝛾],(𝑥,𝑡)=𝑓(𝑥,𝑡),𝑥∈𝑅,𝑡>0,𝛼,𝛽∈(1,2(1.1) subjected to the initial condition𝑢(𝑘)(𝑥,0)=𝑔𝑘(𝑥),𝑥∈𝑅,𝑘=0,1,(1.2) where 𝐷𝛼𝑡 denotes the Caputo fractional derivative with respect to 𝑡 of order 𝛼, 𝑢(𝑥,𝑡) is unknown function, and 𝑎,𝑏,𝑐, and 𝛾 are known constants with 𝛾∈𝑅,𝛾≠±1.

2. Basic Definitions

Definition 2.1. A real function 𝑓(𝑡), 𝑡>0, is said to be in the space 𝐶𝜇, 𝜇∈ℝ, if there exists a real number 𝑝>𝜇, such that 𝑓(𝑡)=𝑡𝑝𝑓1(𝑡), where 𝑓1(𝑡)∈𝐶(0,∞), and it is said to be in the space 𝐶𝑚𝜇 if 𝑓(𝑚)∈𝐶𝜇,𝑚∈ℕ.

Definition 2.2. The Riemann-Liouville fractional integral operator of order 𝛼≥0 of a function 𝑓(𝑡)∈𝐶𝜇,𝜇≥−1 is defined as [28] 𝐽𝛼1𝑓(𝑡)=Γ(𝛼)𝑡0(𝑡−𝜏)𝛼−1𝐽𝑓(𝜏)𝑑𝜏,𝛼>0,𝑡>0,0𝑓(𝑡)=𝑓(𝑡).(2.1) The operator 𝐽𝛼 satisfy the following properties, for 𝑓∈𝐶𝜇, 𝜇≥−1, 𝛼,𝛽≥0, and 𝛾>−1:(1)𝐽𝛼𝐽𝛽𝑓(𝑡)=𝐽𝛼+𝛽𝑓(𝑡), (2)𝐽𝛼𝐽𝛽𝑓(𝑡)=𝐽𝛽𝐽𝛼𝑓(𝑡), (3)𝐽𝛼𝑡𝛾=(Γ(𝛾+1)/Γ(𝛾+𝛼+1))𝑡𝛼+𝛾.

Definition 2.3. The fractional derivative in Caputo sense of 𝑓(𝑡)∈𝐶𝑚−1𝑚, 𝑚∈𝑁,𝑡>0 is defined as 𝐷𝛽𝑡⎧⎪⎨⎪⎩𝐽𝑓(𝑡)=𝑚−𝛽𝑑𝑚𝑑𝑡𝑚𝑑𝑓(𝑡),𝑚−1<𝛽<𝑚,𝑚𝑑𝑡𝑚𝑓(𝑡),𝛽=𝑚.(2.2) The operator 𝐷𝛽 satisfy the following properties, for 𝑓∈𝐶𝑚𝜇, 𝜇≥−1, and 𝛾,𝛽≥0:(1)𝐷𝛽𝑡[𝐽𝛽𝑓(𝑡)]=𝑓(𝑡), (2)𝐽𝛽[𝐷𝛽𝑡∑𝑓(𝑡)]=𝑓(𝑡)−𝑚−1𝑘=0𝑓(𝑘)(0)(𝑡𝑘/𝑘!),𝑡>0, (3)𝐷𝛽𝑡𝑡𝛾=(Γ(𝛾+1)/Γ(𝛾−𝛽+1))𝑡𝛾−𝛽.

3. The Homotopy Perturbation Method (HPM)

Consider the following equation:𝐴(𝑢(𝑥,𝑡))−𝑓(𝑟)=0,𝑟∈Ω,(3.1) with boundary conditions𝐵𝑢,𝜕𝑢𝜕𝑛=0,𝑟∈Γ,(3.2) where 𝐴 is a general differential operator, 𝑢(𝑥,𝑡) is the unknown function, and 𝑥 and 𝑡 denote spatial and temporal independent variables, respectively. 𝐵 is a boundary operator, 𝑓(𝑟) is a known analytic function, and Γ is the boundary of the domain Ω. The operator 𝐴 can be generally divided into linear and nonlinear parts, say 𝐿 and 𝑁. Therefore, (3.1) can be written as𝐿(𝑢)+𝑁(𝑢)−𝑓(𝑟)=0.(3.3) In [9], He constructed a homotopy 𝑣(𝑟,𝑝)∶Ω×[0,1]→𝑅 which satisfies𝐻𝐿𝑢(𝑣,𝑝)=(1−𝑝)(𝑣)−𝐿0[𝐿]+𝑝(𝑣)+𝑁(𝑣)−𝑓(𝑟)=0,𝑟∈Ω,(3.4) or𝐻𝑢(𝑣,𝑝)=𝐿(𝑣)−𝐿0𝑢+𝑝𝐿0[𝑁]+𝑝(𝑣)−𝑓(𝑟)=0,𝑟∈Ω,(3.5) where 𝑝∈[0,1]is an embedding parameter,and 𝑢0 is an initial guess of 𝑢(𝑥,𝑡) which satisfies the boundary conditions. Obviously, from (3.4) and (3.5), one has𝑢𝐻(𝑣,0)=𝐿(𝑣)−𝐿0,𝐻(𝑣,1)=𝐿(𝑢)+𝑁(𝑢)−𝑓(𝑟)=0.(3.6) Changing 𝑝 from zero to unity is just that change of 𝑣(𝑟,𝑝) from 𝑢0(𝑟) to 𝑢(𝑟). Expanding 𝑣(𝑟,𝑝) in Taylor series with respect to 𝑝, one has𝑣=𝑣0+𝑝𝑣1+𝑝2𝑣2+⋯.(3.7) Setting 𝑝=1 results in the approximate solution of (3.1) 𝑢=lim𝑝→1𝑣=𝑣0+𝑣1+𝑣2+⋯.(3.8) The reliable treatment of the classical HPM suggested by Odibat and Momani [26] is presented for nonlinear function 𝑁(𝑢) which is assumed to be an analytic function and has the following Taylor series expansion:𝑁(𝑢)=∞𝑖=0𝑎𝑖𝑢𝑖.(3.9) According to [26], the following homotopy is constructed for (1.1):𝐷𝛼𝑡𝑢=𝑝(𝐿(𝑢)−𝑓(𝑟))+∞𝑖=0𝑝𝑖𝑎𝑖𝑢𝑖[],𝑝∈0,1.(3.10) The basic assumption is that the solution of (3.10) can be written as a power series in 𝑝,𝑢=𝑢0+𝑝𝑢1+𝑝2𝑢2+⋯.(3.11) Substituting (3.11) into (3.10) and equating the terms with identical powers of 𝑝, we obtain a series of linear equations in 𝑢0,𝑢1,𝑢2,…, which can be solved by symbolic computation software. Finally, we approximate the solution ∑𝑢(𝑥,𝑡)=∞𝑛=0𝑢𝑛(𝑥,𝑡) by the truncated series𝑈𝑛(𝑥,𝑡)=𝑛−1𝑖=1𝑢𝑖(𝑥,𝑡).(3.12)

4. Numerical Implementation

In this section, some numerical examples are presented to validate the solution scheme. Symbolic computations are carried out using Mathematica.

Example 4.1. Consider the fractional-order cubically nonlinear Klein-Gordon problem 𝐷𝛼𝑡𝑢−𝐷𝛽𝑥𝑢+𝑢3],=𝑓(𝑥,𝑡),𝑥≥0,𝑡>0,𝛼,𝛽∈(1,2𝑢(𝑥,0)=0,𝑢𝑡(𝑥,0)=0,𝑓(𝑥,𝑡)=Γ(𝛼+1)𝑥𝛽−Γ(𝛽+1)𝑡𝛼+𝑥3𝛽𝑡3𝛼,(4.1) with the exact solution 𝑢(𝑥,𝑡)=𝑥𝛽𝑡𝛼.

According to the homotopy (3.10), we obtain the following set of linear partial differential equations of fractional order:𝑝0∶𝐷𝛼𝑡𝑢0=0,𝑢0(𝑥,0)=0,𝑢0𝑡𝑝(𝑥,0)=0,1∶𝐷𝛼𝑡𝑢1=𝐷𝛽𝑥𝑢0+𝑓(𝑥,𝑡),𝑢1(𝑥,0)=0,𝑢1𝑡𝑝(𝑥,0)=0,2∶𝐷𝛼𝑡𝑢2=𝐷𝛽𝑥𝑢1,𝑢2(𝑥,0)=0,𝑢2𝑡(𝑝𝑥,0)=0,3∶𝐷𝛼𝑡𝑢3=𝐷𝛽𝑥𝑢2−𝑢30,𝑢3(𝑥,0)=0,𝑢3𝑡𝑝(𝑥,0)=0,4∶𝐷𝛼𝑡𝑢4=𝐷𝛽𝑥𝑢3−3𝑢20𝑢1,𝑢4(𝑥,0)=0,𝑢4𝑡⋮(𝑥,0)=0,(4.2)

Case 1 (𝛼∈(1,2] and 𝛽=2). Solving (4.2), we obtain 𝑢0𝑢=0,1=𝑡𝛼𝑥2−2𝑡2𝛼Γ(1+𝛼)Γ(1+2𝛼)+𝑡4𝛼𝑥6Γ(1+3𝛼),𝑢Γ(1+4𝛼)2=30𝑡5𝛼𝑥4Γ(1+3𝛼)Γ(1+5𝛼)+2𝑡2𝛼Γ(1+𝛼)Γ,⋮(1+2𝛼)(4.3) Figure 1 gives the comparison between the HPM 6th-order approximate solution of problem (4.1) in Case 1 with 𝛽=2,𝛼=1.99,1.95,1.90, and 1.85 and the solution of corresponding problem of integer order denoted by 𝑢2,2 at 𝑡=0.5.

581481.fig.001
Figure 1: 𝑢(𝑥,0.5) of Example 4.1 Case 1 for 6th-order HPM approximation as parameterized by 𝛼.

Case 2 (𝛼=2 and 𝛽∈(1,2]). Solving (4.2), we have 𝑢0𝑢=0,1=𝑡2𝑥𝛽+1𝑡568𝑥3𝛽−112Γ(1+𝛽)𝑡4,𝑢2=Γ(1+3𝛽)𝑡(5040)Γ(1+2𝛽)10𝑥2𝛽+112Γ(1+𝛽)𝑡4,⋮(4.4) Figure 2 gives the comparison between the HPM 6th-order approximate solution of problem (4.1) in Case 2 with 𝛼=2,𝛽=1.99,  1.95,1.90, and 1.85 and the solution of corresponding problem of integer order denoted by 𝑢2,2 at 𝑡=0.5.

581481.fig.002
Figure 2: 𝑢(𝑥,0.5) of Example 4.1 Case 2 for 6th-order HPM approximation as parameterized by 𝛽.

Case 3 (both 𝛼 and 𝛽∈(1,2]). Solving (4.2), we have 𝑢0𝑢=0,1=𝑡𝛼𝑥𝛽+𝑡4𝛼𝑥3𝛽Γ(1+3𝛼)Γ(1+4𝛼)−𝑡2𝛼Γ(1+𝛼)Γ(1+𝛽),𝑢Γ(1+2𝛼)2=𝑡5𝛼𝑥2𝛽Γ(1+3𝛼)Γ(1+3𝛽)Γ(1+5𝛼)Γ(1+2𝛽)+𝑡2𝛼Γ(1+𝛼)Γ(1+𝛽),⋮Γ(1+2𝛼)(4.5)
Figure 3 gives the comparison between the HPM 6th-order approximate solution of problem (4.1) in Case 3 with 𝛼 and 𝛽 taking the values 1.99,1.95,1.90, and 1.85 and the solution of corresponding problem of integer order denoted by 𝑢2,2 at 𝑡=0.5.

581481.fig.003
Figure 3: 𝑢(𝑥,0.5) of Example 4.1 Case 3 for 6th-order HPM approximation as parameterized by 𝛼 and 𝛽.

Example 4.2. Consider the fractional-order cubically nonlinear Klein-Gordon problem 𝐷𝛼𝑡𝑢=𝐷𝛽𝑥3𝑢−43𝑢+2𝑢3],,𝑥≥0,𝑡>0,𝛼,𝛽∈(1,2𝑢(𝑥,0)=−sech(𝑥),𝑢𝑡(1𝑥,0)=2sech(𝑥)tanh(𝑥).(4.6) The corresponding integer-order problem has the exact solution 𝑢2,2=−sech(𝑥+𝑡/2) [29].

According to the homotopy (3.10), we obtain the following set of linear partial differential equations of fractional order:𝑝0∶𝐷𝛼𝑡𝑢0=0,𝑢0(𝑥,0)=−sech(𝑥),𝑢0𝑡1(𝑥,0)=2𝑝sech(𝑥)tanh(𝑥),1∶𝐷𝛼𝑡𝑢1=𝑢0𝑥𝑥−34𝑢0,𝑢1(𝑥,0)=0,𝑢1𝑡𝑝(𝑥,0)=0,2∶𝐷𝛼𝑡𝑢2=𝑢1𝑥𝑥−34𝑢1,𝑢2(𝑥,0)=0,𝑢2𝑡𝑝(𝑥,0)=0,3∶𝐷𝛼𝑡𝑢3=𝑢2𝑥𝑥−34𝑢2+32𝑢30,𝑢3(𝑥,0)=0,𝑢3𝑡⋮(𝑥,0)=0,(4.7)

Case 1 (𝛼∈(1,2] and 𝛽=2). Solving (4.7), we have 𝑢01=−sech(𝑥)+2𝑢sech(𝑥)tanh(𝑥)𝑡,1=𝑡𝛼3Γ(𝛼+1)sech(𝑥)4+sech2(𝑥)−tanh2+𝑡(𝑥)𝛼+1sech(𝑥)tanh(𝑥)1Γ(𝛼+2)2−4sech2(𝑥)+−sech2(𝑥)+tanh2−3(𝑥)8,𝑢2=𝑡2𝛼−9Γ(2𝛼+1)sech(𝑥)−3162sech2(𝑥)−5sech4+𝑡(𝑥)2𝛼Γ(2𝛼+1)sech(𝑥)tanh23(𝑥)2+18sech2(𝑥)−tanh2+𝑡(𝑥)2𝛼+19Γ(2𝛼+2)sech(𝑥)tanh(𝑥)+32154sech2(𝑥)+612sech4+𝑡(𝑥)2𝛼+1Γ(2𝛼+2)sech(𝑥)tanh3−3(𝑥)4−29sech21(𝑥)+2tanh2,⋮(𝑥)(4.8) and the solution is obtained as 𝑢=𝑢0+𝑢1+𝑢2+⋯.(4.9) Figure 4 gives the comparison between the HPM 4th-order approximate solution of problem (4.6) in Case 1 with 𝛽=2,𝛼=1.99,1.95,1.90, and 1.85 and the solution of corresponding problem of integer order denoted by 𝑢2,2 at 𝑡=0.3.

581481.fig.004
Figure 4: 𝑢(𝑥,0.3)of Example 4.2 Case 1 for 4th-order HPM approximation as parameterized by 𝛼.

Case 2 (𝛼=2 and 𝛽∈(1,2]). As the attempt to evaluate Caputo fractional derivative of the functions sech(𝑥) and tanh(𝑥) yields hypergeometric function, we substitute sech(𝑥) and tanh(𝑥) by some terms of its Taylor series. Substituting the initial conditions and solving (4.7) for 𝑢0,𝑢1,𝑢2,…, the components of the homotopy perturbation solution for (4.6) are derived as follows: 𝑢0𝑥=−1−22+5𝑥4−2461𝑥6+720277𝑥8+180642𝑡𝑥−5𝑥33+61𝑥5−120277𝑥7+10085052𝑥9,𝑢3628801=𝑡238−𝑡𝑥−163𝑥2+165𝑡𝑥3+965𝑥4−6461𝑡𝑥5−192061𝑥6+1920277𝑡𝑥7+16128277𝑥8−2150450521𝑡𝑥95806080+𝑥−𝛽𝑡2𝑥2−2Γ(3−𝛽)5𝑡3𝑥3−𝑡12Γ(4−𝛽)2𝑥4+2Γ(5−𝛽)61𝑡3𝑥5+12Γ(6−𝛽)61𝑡2𝑥62Γ(7−𝛽)+𝑥−𝛽−1385𝑡3𝑥7−12Γ(8−𝛽)1385𝑡2𝑥8+2Γ(9−𝛽)50521𝑡3𝑥9,⋮12Γ(10−𝛽)(4.10) As the Caputo fractional derivative can not be evaluated for negative powers of the variable at hand, and noting that 𝛽∈(1,2], we can only evaluate the first two components of the series as illustrated. Thus, we suggest to generalize not only the derivatives in the integer-order problem to its fractional form, but also to generalize the conditions as well. For example, a generalized expansion of sech(𝑥) in a fractional form can be written as sech𝛽𝑥(𝑥)=1−𝛽+Γ(𝛽+1)5𝑥2𝛽−Γ(2𝛽+1)61𝑥3𝛽+Γ(3𝛽+1)277𝑥4𝛽Γ(4𝛽+1)−⋯,(4.11) for which we have lim𝛽→2sech𝛽(𝑥)=sech(𝑥). Substituting the generalized form of the initial conditions and solving (4.7) for 𝑢0,𝑢1,𝑢2,…, the components of the homotopy perturbation solution for this case are derived as follows: 𝑢0=𝑥1−𝛽2+5𝑥2𝛽−2461𝑥3𝛽+720277𝑥4𝛽+180642𝑡𝑥𝛽−5𝑥𝛽+13+61𝑥2𝛽+1−120277𝑥3𝛽+1+10085052𝑥4𝛽+1,𝑢3628801=3𝑡28−𝑡3𝑥−𝑥164𝛽16277𝑡2−2150450521𝑡3𝑥+5806080Γ(𝛽+1)4𝑡2−5Γ(𝛽+2)𝑡723𝑥+𝑥𝛽−3𝑡2−165𝑡3𝑥−9661𝑡3𝑥+192061Γ(3𝛽+1)𝑡2−1440Γ(2𝛽+1)277𝑡3𝑥Γ(3𝛽+2)12096Γ(2𝛽+2)+𝑥3𝛽−61𝑡2+1920277𝑡3𝑥−16128277𝑡2Γ(4𝛽+1)+16128Γ(3𝛽+1)50521𝑡3𝑥Γ(4𝛽+2),⋮4354560Γ(3𝛽+2)(4.12) and the solution is obtained as 𝑢=𝑢0+𝑢1+𝑢2+⋯.(4.13) Figure 5 gives the comparison between the HPM 4th-order approximate solution of problem (4.6) in Case 2 with 𝛼=2,𝛽=1.99,1.95,1.90, and 1.85 and the solution of corresponding problem of integer order denoted by 𝑢2,2 at 𝑡=0.3.

581481.fig.005
Figure 5: 𝑢(𝑥,0.3) of Example 4.2 Case 2 for 4th-order HPM approximation as parameterized by 𝛽.

Case 3 (both 𝛼 and 𝛽∈(1,2]). Carrying out the same procedure as in Case 2, we get 𝑢0=𝑥1−𝛽2+5𝑥2𝛽−2461𝑥3𝛽+720277𝑥4𝛽+180642𝑡𝑥𝛽−5𝑥𝛽+13+61𝑥2𝛽+1−120277𝑥3𝛽+1+10085052𝑥4𝛽+1,𝑢3628801=𝑡𝛼3Γ(𝛼+1)4−38𝑥𝛽+5𝑥2𝛽−3261𝑥3𝛽+960277𝑥4𝛽+10752Γ(𝛽+1)2+𝑡𝛼−Γ(𝛼+1)5𝑥𝛽Γ(2𝛽+1)+24Γ(𝛽+1)61𝑥2𝛽Γ(3𝛽+1)−720Γ(2𝛽+1)277𝑥3𝛽Γ(4𝛽+1)−8064Γ(3𝛽+1)3𝑡𝛼+18Γ(𝛼+2)𝑥−5𝑥𝛽+16+61𝑥2𝛽+1−120277𝑥3𝛽+1+10085052𝑥4𝛽+1+𝑡3628802𝛼+12Γ(𝛼+1)Γ(𝛼+2)−56𝑥Γ(𝛽+2)+61𝑥𝛽+1120Γ(2𝛽+2)+𝑡Γ(𝛽+2)2𝛼+1−2Γ(𝛼+1)Γ(𝛼+2)277𝑥2𝛽+11008Γ(3𝛽+2)+Γ(2𝛽+2)50521𝑥3𝛽+1Γ(4𝛽+2),⋮362880Γ(3𝛽+2)(4.14) and the solution is thus obtained as 𝑢=𝑢0+𝑢1+𝑢2+⋯.(4.15) Figure 6 gives the comparison between the HPM 4th-order approximate solution of problem (4.6) in Case 3 with 𝛼,𝛽=1.99,1.95,1.90, and 1.85 and the solution of corresponding problem of integer order denoted by 𝑢2,2 at 𝑡=0.3.

581481.fig.006
Figure 6: 𝑢(𝑥,0.3) of Example 4.2 Case 3 for 4th-order HPM approximation as parameterized by 𝛼 and 𝛽.

5. Conclusion

The reliable treatment HPM is applied to obtain the solution of the Klien-Gordon partial differential equation of arbitrary (fractional) orders with spatial and temporal fractional derivatives. The main advantage of this algorithm is the capability to overcome the difficulty arising in calculating complicated integrals when dealing with nonlinear problems. The numerical examples carried out show good results, and their graphs illustrate the continuation of the solution of fractional-order Klien-Gordon equation to the solution of the corresponding second-order problem when the fractional-order parameters approach their integer limits.

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