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

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 form𝐷^𝛼𝑡𝑢(𝑥,𝑡)+𝑎𝐷^𝛽𝑥𝑢(𝑥,𝑡)+𝑏𝑢(𝑥,𝑡)+𝑐𝑢^𝛾(𝑥,𝑡)=𝑓(𝑥,𝑡),𝑥∈𝑅,𝑡>0,𝛼,𝛽∈(1,2],(1.1) subjected to the initial condition𝑢^(𝑘)(𝑥,0)=𝑔_𝑘(𝑥),𝑥∈𝑅,𝑘=0,1,(1.2) where 𝐷^𝛼𝑡 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 𝐶_𝜇, 𝜇∈ℝ, if there exists a real number 𝑝>𝜇, such that 𝑓(𝑡)=𝑡^𝑝𝑓₁(𝑡), where 𝑓₁(𝑡)∈𝐶(0,∞), and it is said to be in the space 𝐶^𝑚𝜇 if 𝑓^(𝑚)∈𝐶_𝜇,𝑚∈ℕ.

Definition 2.2. The Riemann-Liouville fractional integral operator of order 𝛼≥0 of a function 𝑓(𝑡)∈𝐶_𝜇,𝜇≥−1 is defined as [28] 𝐽^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡0𝐽(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏,𝛼>0,𝑡>0,₀𝑓(𝑡)=𝑓(𝑡).(2.1) The operator 𝐽^𝛼 satisfy the following properties, for 𝑓∈𝐶_𝜇, 𝜇≥−1, 𝛼,𝛽≥0, and 𝛾>−1:(1)𝐽^𝛼𝐽^𝛽𝑓(𝑡)=𝐽^𝛼+𝛽𝑓(𝑡), (2)𝐽^𝛼𝐽^𝛽𝑓(𝑡)=𝐽^𝛽𝐽^𝛼𝑓(𝑡), (3)𝐽^𝛼𝑡^𝛾=(Γ(𝛾+1)/Γ(𝛾+𝛼+1))𝑡^𝛼+𝛾.

Definition 2.3. The fractional derivative in Caputo sense of 𝑓(𝑡)∈𝐶^𝑚−1𝑚, 𝑚∈𝑁,𝑡>0 is defined as 𝐷^𝛽𝑡𝑓(𝑡)=⎧⎪⎨⎪⎩𝐽^𝑚−𝛽𝑑^𝑚𝑑𝑡^𝑚𝑑𝑓(𝑡),𝑚−1<𝛽<𝑚,_𝑚𝑑𝑡^𝑚𝑓(𝑡),𝛽=𝑚.(2.2) The operator 𝐷^𝛽 satisfy the following properties, for 𝑓∈𝐶^𝑚𝜇, 𝜇≥−1, and 𝛾,𝛽≥0:(1)𝐷^𝛽𝑡[𝐽^𝛽𝑓(𝑡)]=𝑓(𝑡), (2)𝐽^𝛽[𝐷^𝛽𝑡𝑓(𝑡)]=𝑓(𝑡)−∑^{𝑚−1𝑘=0}𝑓^(𝑘)(0)(𝑡^𝑘/𝑘!),𝑡>0, (3)𝐷^𝛽𝑡𝑡^𝛾=(Γ(𝛾+1)/Γ(𝛾−𝛽+1))𝑡^𝛾−𝛽.

3. The Homotopy Perturbation Method (HPM)

Consider the following equation:𝐴(𝑢(𝑥,𝑡))−𝑓(𝑟)=0,𝑟∈Ω,(3.1) with boundary conditions𝐵𝑢,𝜕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 satisfies𝐻(𝑣,𝑝)=(1−𝑝)𝐿(𝑣)−𝐿𝑢₀+𝑝[𝐿(𝑣)+𝑁(𝑣)−𝑓(𝑟)]=0,𝑟∈Ω,(3.4) or𝐻(𝑣,𝑝)=𝐿(𝑣)−𝐿𝑢₀+𝑝𝐿𝑢₀+𝑝[𝑁(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 has𝐻𝐻((𝑣𝑣,,01))==𝐿𝐿((𝑣𝑢))−+𝐿𝑁(𝑢𝑢₀)−,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=lim_𝑝→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=0𝑎_𝑖𝑢^𝑖.(3.9) According to [26], the following homotopy is constructed for (1.1):𝐷^𝛼𝑡𝑢=𝑝(𝐿(𝑢)−𝑓(𝑟))+^∞_𝑖=0𝑝^𝑖𝑎_𝑖𝑢^𝑖,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,𝑢_1,𝑢₂,…, which can be solved by symbolic computation software. Finally, we approximate the solution 𝑢(𝑥,𝑡)=∑^∞𝑛=0𝑢_𝑛(𝑥,𝑡) by the truncated series𝑈_𝑛(𝑥,𝑡)=^𝑛−1_𝑖=1𝑢_𝑖(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 𝐷^𝛼𝑡𝑢−𝐷^𝛽𝑥𝑢+𝑢𝑢³(=𝑥,𝑓0)(𝑥=,𝑡0),,𝑥𝑢≥_𝑡0,𝑡>0,𝛼,𝛽∈(1,2],𝑓(𝑥,𝑡)=Γ(𝛼+1)𝑥_𝛽(𝑥,0)=0,−Γ(𝛽+1)𝑡^𝛼+𝑥^3𝛽𝑡^3𝛼,(4.1) with the exact solution 𝑢(𝑥,𝑡)=𝑥^𝛽𝑡^𝛼.

According to the homotopy (3.10), we obtain the following set of linear partial differential equations of fractional order:𝑝𝑝⁰∶𝐷^𝛼𝑡𝑢₀=0,𝑢₀(𝑥,0)=0,𝑢_0𝑡(𝑥,0)=0,₁𝑝∶𝐷^𝛼𝑡𝑢₁=𝐷^𝛽𝑥𝑢₀+𝑓(𝑥,𝑡),𝑢₁(𝑥,0)=0,𝑢_1𝑡(𝑥,0)=0,₂𝑝∶𝐷^𝛼𝑡𝑢₂=𝐷^𝛽𝑥𝑢₁,𝑢₂(𝑥,0)=0,𝑢_2𝑡(𝑥,0)=0,₃𝑝∶𝐷^𝛼𝑡𝑢₃=𝐷^𝛽𝑥𝑢₂−𝑢³⁰,𝑢₃(𝑥,0)=0,𝑢_3𝑡(𝑥,0)=0,₄⋮∶𝐷^𝛼𝑡𝑢₄=𝐷^𝛽𝑥𝑢₃−3𝑢²⁰𝑢₁,𝑢₄(𝑥,0)=0,𝑢_4𝑡(𝑥,0)=0,(4.2)

Case 1 (𝛼∈(1,2] and 𝛽=2). Solving (4.2), we obtain 𝑢𝑢₀=0,₁=𝑡^𝛼𝑥²−2𝑡^2𝛼Γ(1+𝛼)Γ(1+2𝛼)+𝑡^4𝛼𝑥⁶Γ(1+3𝛼)𝑢Γ(1+4𝛼)₂,=30𝑡^5𝛼𝑥⁴Γ(1+3𝛼)Γ(1+5𝛼)+2𝑡^2𝛼Γ(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 𝑢_2,2 at 𝑡=0.5.

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 𝑢𝑢₀=0,₁=𝑡²𝑥^𝛽+156𝑡⁸𝑥^3𝛽−112Γ(1+𝛽)𝑡⁴𝑢₂,=Γ(1+3𝛽)(5040)Γ(1+2𝛽)𝑡¹⁰𝑥^2𝛽+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 𝑢_2,2 at 𝑡=0.5.

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 𝑢𝑢₀=0,₁=𝑡^𝛼𝑥^𝛽+𝑡^4𝛼𝑥^3𝛽Γ(1+3𝛼)Γ(1+4𝛼)−𝑡^2𝛼Γ(1+𝛼)Γ(1+𝛽)𝑢Γ(1+2𝛼)₂,=𝑡^5𝛼𝑥^2𝛽Γ(1+3𝛼)Γ(1+3𝛽)Γ(1+5𝛼)Γ(1+2𝛽)+𝑡^2𝛼Γ(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 𝑢_2,2 at 𝑡=0.5.

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 𝐷^𝛼𝑡𝑢=𝐷^𝛽𝑥𝑢−34𝑢+23𝑢³𝑢(𝑥,0)=−sech(𝑥),,𝑢_𝑡𝑥≥0,𝑡>0,𝛼,𝛽∈(1,2],(𝑥,0)=12sech(𝑥)tanh(𝑥).(4.6) The corresponding integer-order problem has the exact solution 𝑢_2,2=−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∶𝐷^𝛼𝑡𝑢₀=0,𝑢₀(𝑥,0)=−sech(𝑥),𝑢_0𝑡(𝑥,0)=1𝑝2sech(𝑥)tanh(𝑥),₁∶𝐷^𝛼𝑡𝑢₁=𝑢_0𝑥𝑥−34𝑢⁰𝑝,𝑢₁(𝑥,0)=0,𝑢_1𝑡(𝑥,0)=0,₂∶𝐷^𝛼𝑡𝑢₂=𝑢_1𝑥𝑥−34𝑢¹𝑝,𝑢₂(𝑥,0)=0,𝑢_2𝑡(𝑥,0)=0,₃∶𝐷^𝛼𝑡𝑢₃=𝑢_2𝑥𝑥−34𝑢²+32𝑢³⁰⋮,𝑢₃(𝑥,0)=0,𝑢_3𝑡(𝑥,0)=0,(4.7)

Case 1 (𝛼∈(1,2] and 𝛽=2). Solving (4.7), we have 𝑢₀=−sech(𝑥)+1𝑢2sech(𝑥)tanh(𝑥)𝑡,₁=𝑡^𝛼Γ(𝛼+1)sech(𝑥)34+sech²+𝑡(𝑥)−tanh²(𝑥)_𝛼+1sech(𝑥)tanh(𝑥)Γ(𝛼+2)21−4sech²(𝑥)+−sech²(𝑥)+tanh²(𝑥)−3𝑢₂8,=𝑡^2𝛼Γ(2𝛼+1)sech(𝑥)−196−32sech²+𝑡(𝑥)−5sech⁴(𝑥)_2𝛼Γ(2𝛼+1)sech(𝑥)tanh²(𝑥)32+18sech²+𝑡(𝑥)−tanh²(𝑥)_2𝛼+1Γ(2𝛼+2)sech(𝑥)tanh(𝑥)392+145sech²(𝑥)+612sech⁴+𝑡(𝑥)_2𝛼+1Γ(2𝛼+2)sech(𝑥)tanh³(𝑥)−34−29sech²(𝑥)+12tanh²⋮(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 𝑢_2,2 at 𝑡=0.3.

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: 𝑢₀=−1−𝑥²𝑢2+52𝑥4⁴−6712𝑥0⁶+287076𝑥4⁸+21𝑡𝑥−53𝑥³+6112𝑥0⁵−217070𝑥8⁷[phone]𝑥0⁹₁,=𝑡²+𝑥38_−𝛽−1𝑡𝑥6−31𝑥6²+59𝑡𝑥6³+56𝑥4⁴−6119𝑡2𝑥0⁵−16912𝑥0⁶+217671𝑡2𝑥8⁷+2217570𝑥4⁸−5508502610𝑡8𝑥0⁹𝑡²𝑥²2Γ(3−𝛽)−5𝑡³𝑥³12Γ(4−𝛽)−𝑡²𝑥⁴2Γ(5−𝛽)+61𝑡³𝑥⁵12Γ(6−𝛽)+61𝑡²𝑥⁶+𝑥_−𝛽2Γ(7−𝛽)−1385𝑡³𝑥⁷12Γ(8−𝛽)−1385𝑡²𝑥⁸2Γ(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 sech_𝛽(𝑥)=1−𝑥^𝛽Γ(𝛽+1)+Γ(25𝛽𝑥+^2𝛽1)−Γ(631𝛽𝑥+^3𝛽1)+Γ2(47𝛽7𝑥+^4𝛽1)−⋯,(4.11) for which we have lim_𝛽→2sech_𝛽(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=1−𝑥^𝛽+12+52𝑥4^2𝛽−6712𝑥0^3𝛽+287076𝑥4^4𝛽2𝑡𝑥^𝛽−5𝑥^𝛽+1𝑢3+611𝑥20^2𝛽+1−27170𝑥08^3𝛽+1+5035622𝑥880^4𝛽+1₁,=3𝑡²8−𝑡³𝑥16−𝑥16^4𝛽2217570𝑡²4−50521𝑡³𝑥5806080+Γ(𝛽4+1)𝑡²−5Γ(𝛽+2)72𝑡³+𝑥_𝛽𝑥−136𝑡²−5𝑡³𝑥96−61𝑡³𝑥1920+16414Γ0(Γ3𝛽(2+𝛽1+)1𝑡²)−277𝑡³𝑥Γ(3𝛽+2)+𝑥_3𝛽12096Γ(2𝛽+2)−16912𝑡²0+277𝑡³𝑥16128−277𝑡²Γ(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 𝑢_2,2 at 𝑡=0.3.

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=1−𝑥^𝛽+12+52𝑥4^2𝛽−6712𝑥0^3𝛽+287076𝑥4^4𝛽2𝑡𝑥^𝛽−5𝑥^𝛽+1𝑢3+611𝑥20^2𝛽+1−27170𝑥08^3𝛽+1+5035622𝑥880^4𝛽+1₁,=𝑡^𝛼Γ(𝛼+1)34−38𝑥^𝛽+5𝑥^2𝛽+𝑡_𝛼32−6916𝑥0^3𝛽+217077𝑥52^4𝛽+Γ(𝛽2+1)Γ(𝛼+1)−5𝑥^𝛽Γ(2𝛽+1)24Γ(𝛽+1)+61𝑥^2𝛽Γ(3𝛽+1)720Γ(2𝛽+1)−277𝑥^3𝛽Γ(4𝛽+1)−3𝑡_𝛼+18064Γ(3𝛽+1)+8Γ(𝛼+2𝑡)𝑥−5𝑥6^𝛽+1+611𝑥20^2𝛽+1−27170𝑥08^3𝛽+1+5035622𝑥880^4𝛽+1_2𝛼+1+2Γ(𝛼+𝑡1)Γ(𝛼+2)−65𝑥Γ(𝛽+2)+611𝑥20^𝛽+1ΓΓ((2𝛽𝛽++22)_2𝛼+1)2Γ(𝛼+1)Γ(𝛼+2)−27170𝑥08^2𝛽+1ΓΓ((32𝛽𝛽++22))+50521𝑥^3𝛽+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 𝑢_2,2 at 𝑡=0.3.

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.