Solving the Fractional Rosenau-Hyman Equation via Variational Iteration Method and Homotopy Perturbation Method

Abstract

In this study, fractional Rosenau-Hynam equations is considered. We implement relatively new analytical techniques, the variational iteration method and the homotopy perturbation method, for solving this equation. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for fractional Rosenau-Hynam equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity.

1. Introduction

Recent advances of fractional differential equations are stimulated by new examples of applications in fluid mechanics, viscoelasticity, mathematical biology, electrochemistry, and physics. For example, the nonlinear oscillation of earthquake can be modeled with fractional derivatives [1], and the fluid-dynamic traffic model with fractional derivatives [2] can eliminate the deficiency arising from the assumption of continuum traffic flow. Based on experimental data fractional partial differential equations for seepage flow in porous media are suggested in [3], and differential equations with fractional order have recently proved to be valuable tools to the modeling of many physical phenomena [4]. Fractional partial differential equations also have studied and successfully solved such as the space-time fractional diffusion-wave equation [5–7], the fractional advection-dispersion equation [8, 9], the fractional KdV equation [10], and the linear inhomogeneous fractional partial differential equations [11].

Most nonlinear differential equations are usually arising from mathematical modeling of many physical systems. In most cases, it is very difficult to achieve analytic solutions of these equations. Perturbation techniques are widely used in science and engineering to handle nonlinear problems and do great contribution to help us understand many nonlinear phenomena. However, perturbation techniques are based on the existence of small/large parameter. Therefore, these techniques are not valid for strongly nonlinear problems.

The homotopy perturbation method (HPM) is the new approach for finding the approximate analytical solution of linear and nonlinear problems. The method was first proposed by He [12, 13] and was successfully applied to solve nonlinear wave equation by He [14–16]. The convergence of Homotopy perturbation series to the exact solution is considered in [17]. Similarly, applying the variational iteration method, created by He [3, 18, 19], consists in constructing the appropriate correction functional connected with the considered equation. The correction functional contains a Lagrange multiplier, the determination of which leads to a recurrence formula. Convergence of the VIM method is discussed by Tatari and Dehghan in [20]. Both of the methods examined have found application in determining the approximate solutions of different technical problems [21]. Adaptation of the VIM method for solving fractional heat-wave-like equation and fractional Zakharov-Kuznetsov equation were discussed by Yulita and colleagues in [22, 23]. Whereas, Chun [24] obtained the numerical solution of heat conduction problem by VIM. Recently, the application of the VIM method for solving kuramoto and Sivashinsky equations was presented by Porshokouhi and Ghanbari in [25]. For the application of HPM, this method used for solving fractional vibration equation [26] and partial differential equations of fractional order in finite domains [27].

In the present paper, VIM and HPM will be applied for solving fractional Rosenau-Hay nam equation which written as 𝐷^𝛼_𝑡𝑢=𝑢𝐷_𝑥𝑥𝑥(𝑢)+𝑢𝐷_𝑥(𝑢)+3𝐷_𝑥(𝑢)𝐷_𝑥𝑥(𝑢),𝑡>0, (1.1) subject to the initial condition 𝑢()8𝑥,0=−3𝑐cos²𝑥4, (1.2) where 𝑢=𝑢(𝑥,𝑡), 𝛼 is a parameter describing the order of the fractional derivative (0<𝛼≤1), 𝑡 is the time, and 𝑥 is the spatial coordinate. Fractional RH equation when 𝛼=1.0 has appeared in the study of the formation of patterns in liquid drops [28].

2. Basic Definitions

Fractional calculus unifies and generalizes the notions of integer-order differentiation and 𝑛-fold integration [4, 29]. We give some basic definitions and properties of fractional calculus theory which will be used in this paper:

Definition 2.1. A real function 𝑓(𝑥),𝑥>0 is said to be in the space 𝒞_𝜇,  𝜇∈𝐑 if there exists a real number 𝑝(>𝜇), such that 𝑓(𝑥)=𝑥^𝑝𝑓₁(𝑥), where 𝑓₁(𝑥)∈𝒞[0,∞), and it is said to be in the space 𝒞^𝑚_𝜇 if and only if 𝑓^(𝑚)∈𝒞_𝑚,  𝑚∈𝐍.

The Riemann-Liouville fractional integral operator is defined as follows.

Definition 2.2. The Riemann-Liouville fractional integral operator of order 𝛼≥0, of a function 𝑓∈𝐶_𝜇,  𝜇≥−1, is defined as 𝐽^𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥₀(𝑥−𝑡)^𝛼−1𝑓(𝑡)𝐽d𝑡,𝛼>0,𝑥>0,₀𝑓(𝑥)=𝑓(𝑥) (2.1)

In this paper only real and positive values of 𝛼 will be considered.

Properties of the operator 𝐽^𝛼 can be found in [29] and we mention only the following: for 𝑓∈𝐶_𝜇, 𝜇≥−1,  𝛼,𝛽≥0, and 𝛾≥−1: (1)𝐽^𝛼𝐽^𝛽𝑓(𝑥)=𝐽^𝛼+𝛽𝑓(𝑥), (2)𝐽^𝛼𝐽^𝛽𝑓(𝑥)=𝐽^𝛽𝐽^𝛼𝑓(𝑥), (3)𝐽^𝛼𝑥^𝛾=(Γ(𝛾+1)/Γ(𝛼+𝛾+1))𝑥^𝛼+𝛾. The Reimann-Liouville derivative has certain disadvantages when trying to model real-world phenomena with FDEs. Therefore, we will introduce a modified fractional differential operator 𝐷^𝛼_∗ proposed by Caputo in his work on the theory of viscoelasticity [30].

Definition 2.3. The fractional derivative of 𝑓(𝑥) in Caputo sense is defined as 𝐷^𝛼_∗𝑓(𝑥)=𝐽^𝑚−𝛼𝐷^𝑚_∗𝑓(𝑥)=1Γ(𝑚)^𝑥₀(𝑥−𝑠)^{𝑚−𝛼−1}𝑓^(𝑚)(𝑠)d𝑠,for𝑚−1<𝛼≤𝑚,𝑚∈𝒩,𝑥>0,𝑓∈𝐶^𝑚₋₁ (2.2) In addition, we also need the following property.

Lemma 2.4. If 𝑚−1<𝛼≤𝑚,  𝑚∈𝒩 and 𝑓∈𝐶^𝑚_𝜇,  𝜇≥−1, then 𝐷^𝛼_∗𝐽^𝛼𝑓(𝑥)(𝑥),𝐽=𝑓_𝛼𝐷^𝛼_∗𝑓(𝑥)=𝑓(𝑥)−^𝑚−1_𝑖=0𝑓^(𝑖)0⁺𝑥^𝑖𝑖!,𝑥>0. (2.3)The Caputo differential derivative is considered here because the initial and boundary conditions can be included in the formulation of the problems [4]. The fractional derivative is taken in the Caputo sense as follows.

Definition 2.5. For 𝑚 to be the smallest integer that exceeds 𝛼, the Caputo fractional derivative operator of order 𝛼>0 is defined as 𝐷^𝛼_𝑡𝑢()=1𝑥,𝑡Γ()∫𝑚−𝛼^𝑡0(𝑡−𝑠)^{𝑚−𝛼−1(𝜕𝑚𝑢(𝑥,𝑠)/𝜕𝑠𝑚)}𝜕d𝑠,for𝑚−1<𝛼≤𝑚,_𝑚𝑢(𝑥,𝑡)𝜕𝑡^𝑚,for𝛼=𝑚∈𝐍. (2.4)For more information on the mathematical properties of fractional derivatives and integrals, one can consult [4, 29].

3. Basic Idea of He’s Variational Iteration Method

To clarify the basic ideas of VIM, we consider the following differential equation: 𝐷^𝛼_∗𝑡(𝑢)=𝑓𝑢,𝑢_𝑥,𝑢_𝑥𝑥+𝑔(𝑥,𝑡),𝑚−1<𝛼<𝑚, (3.1) where 𝑢=𝑢(𝑥,𝑡),  𝐷^𝛼_∗𝑡=𝜕^𝛼/𝜕𝑡^𝛼 is the Caputo fractional derivative of order 𝛼, 𝑚∈𝑁, 𝑓 is a nonlinear function, and 𝑔 is the source function. According to VIM, we can write down a correction functional as follows: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡₀𝜆(𝑠𝜕)^𝑚𝜕𝑠^𝑚𝑢_𝑛−𝑓𝑢_𝑛,𝑢_𝑛,𝑥,𝑢_{𝑛,𝑥𝑥}−𝑔(𝑥,𝑠)d𝑠, (3.2) where 𝜆(𝑠) is a general Lagrangian multiplier which can be optimally identified via the variational theory [31]. 𝑢_𝑛 is considered as a restricted variation [32], that is, 𝛿𝑢_𝑛=0 and the subscript 𝑛 indicates the 𝑛th approximation. We have 𝛿𝑢_𝑛+1=𝛿𝑢_𝑛+𝛿^𝑡₀𝜆(𝑠𝜕)^𝑚𝜕𝑠^𝑚𝑢_𝑛−𝑓𝑢_𝑛,𝑢_𝑛,𝑥,𝑢_{𝑛,𝑥𝑥}−𝑔(𝑥,𝑠)d𝑠, (3.3) where 𝑢_𝑛 is considered as restricted variations, that is, 𝛿𝑢_𝑛=0. For 𝑚=1, we have 𝛿𝑢_𝑛+1=𝛿𝑢_𝑛+𝛿^𝑡₀𝜆(𝑠𝜕)𝑢𝜕𝑠^𝑛d𝑠,𝛿𝑢_𝑛+1=1−𝜆^′(𝑡)𝛿𝑢_𝑛(𝑡)𝜕+𝛿𝜆𝑢𝜕𝑡^𝑛+¹₀𝛿𝜆^′′(𝑠)𝑢_𝑛d𝑠. (3.4) Thus, we obtain the following stationary conditions: 𝛿𝑢_𝑛(𝑡)∶1−𝜆^′(𝑡)|_𝑠=𝑡=0,𝛿𝑢^′_𝑛(𝑡)∶𝜆(𝑡)|_𝑠=𝑡=0,𝛿𝑢_𝑛(𝑠)∶𝜆^′′(𝑡)=0. (3.5) Solving this system of equations yields 𝜆(𝑠)=−1. (3.6) Furthermore, substituting (3.6) to (3.2), the iteration formula of VIM can be written as follows: 𝑢_𝑛+1=𝑢_𝑛−^𝑡₀𝜕^𝑚𝜕𝑠^𝑚𝑢_𝑛−𝑓𝑢_𝑛,𝑢_𝑛,𝑥,𝑢_{𝑛,𝑥𝑥}−𝑔(𝑥,𝑠)d𝑠. (3.7) In this case, we begin with the initial approximation: 𝑢₀(𝑥,𝑡)=ℎ(𝑥). (3.8) The correction functional (3.8) will give several approximations, and therefore the exact solution is obtained as 𝑢(𝑥,𝑡)=lim_𝑛→∞𝑢_𝑛(𝑥,𝑡). (3.9)

4. Basic Idea of Homotopy Perturbation Method

Consider the following nonlinear differential equation: 𝐴(𝑢)−𝑓(𝑟)=0, (4.1) with boundary conditions 𝐵𝑢,𝜕𝑢𝜕𝑛=0,𝑟∈Γ, (4.2) where 𝐴 is a general differential operator, 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, generally speaking, be divided into two parts, 𝐿 and 𝑁, where 𝐿 is linear and 𝑁 is nonlinear, therefore (4.1) can be written as 𝐿(𝑢)+𝑁(𝑢)−𝑓(𝑟)=0. (4.3) By using homotopy technique, one can construct a homotopy 𝑣(𝑟,𝑝)∶Ω×[0,1]→ℜ which satisfies 𝐻𝑣,𝑝=1−𝑝𝐿(𝑣)−𝐿𝑢₀+𝑝𝐴(𝑣)−𝑓(𝑟)=0, (4.4) or 𝐻𝑣,𝑝=𝐿(𝑣)−𝐿𝑢₀+𝑝𝐿𝑢₀+𝑝𝑁(𝑣)−𝑓(𝑟)=0, (4.5) where 𝑝∈[0,1] is an embedding parameter, and 𝑢₀ is the initial approximation of (4.1) which satisfies the boundary conditions. Clearly, we have 𝐻(𝑣,0)=𝐿(𝑣)−𝐿(𝑢0)=0, (4.6) or 𝐻(𝑣,1)=𝐴(𝑣)−𝑓(𝑟)=0, (4.7) the changing process of 𝑝 from zero to unity is just that of 𝑣(𝑟,𝑝) changing from 𝑢₀(𝑟) to 𝑢(𝑟). This is called deformation, and also, 𝐿(𝑣)−𝐿(𝑢₀) and 𝐴(𝑣)−𝑓(𝑟) are called homotopic in topology. If the embedding parameter 𝑝(0≤𝑝≤1) is considered as a small parameter, applying the classical perturbation technique, we can assume that the solution of (4.3) and (4.4) can be given as a power series in 𝑝, that is, 𝑣=𝑣₀+𝑝𝑣₁+𝑝²𝑣₂+⋯, (4.8) and setting 𝑝=1 results in the approximate solution of (4.1) as 𝑢=lim_𝑝→1𝑣=𝑣₀+𝑣₁+𝑣₂+⋯. (4.9)

5. Application of HPM and VIM Methods

We consider the application of VIM to fractional Rosenau-Hynam equations which is rewritten as follows: 𝐷^𝛼_𝑡𝑢=𝑢𝐷_𝑥𝑥𝑥(𝑢)+𝑢𝐷_𝑥(𝑢)+3𝐷_𝑥(𝑢)𝐷_𝑥𝑥(𝑢), (5.1) where 𝑢=𝑢(𝑥,𝑡) with the initial conditions of 𝑢()8𝑥,0=−3𝑐cos²𝑥4. (5.2)

5.1. VIM Implement for Fractional Rosenau-Hynam Equation

According to the formula (3.8), the iteration formula for (4.9) is given by 𝑢_𝑛+1=𝑢_𝑛−^𝑡₀𝐷^𝛼_𝑠𝑢_𝑛−𝑢_𝑛𝐷_𝑥𝑥𝑥𝑢_𝑛−𝑢_𝑛𝐷_𝑥𝑢_𝑛−3𝐷_𝑥𝑢_𝑛𝐷_𝑥𝑥𝑢_𝑛d𝑠. (5.3) The iteration starts with an initial approximation which is initial condition in (5.2). Furthermore, using the iteration formula in (5.3), we can directly obtain other components as 𝑢₁()2𝑥,𝑡=−3𝑐²𝑥sin24𝑡−3𝑐𝑥1+cos2,𝑢₂()4𝑥,𝑡=−3𝑐²𝑥sin2+16𝑐²+43𝑐𝑥1+cos2−16𝑐³𝑡²+23𝑐²𝑥sin2𝑡^(2−𝛼)Γ(),𝑢3−𝛼₃()=7𝑥,𝑡6𝑐³𝑡³𝑥cos2−9636𝑐cos²𝑥4−48𝑐36²𝑥𝑡sin2−23𝑥cos2𝑡^3−𝛼Γ()+()𝑐4−𝛼6−2𝛼²𝑥sin2𝑡^2−𝛼Γ()−24−𝛼3𝑐²𝑥sin2𝑡^3−2𝛼Γ(),4−2𝛼 (5.4) and so on.

The exact solution of this equation is given by [33] 𝑢()8𝑥,𝑡=−3𝑐cos²14(𝑥−𝑐𝑡),|𝑥−𝑐𝑡|≤2𝜋, (5.5) where 𝑐 is arbitrary constant [28].

6. Analysis of the Homotopy Perturbation Method (HPM)

Now applying the classical perturbation technique for solving (5.1) with initial condition in (5.2). To solve (5.1) by the homotopy perturbation method, we construct the following homotopy: 1−𝑝𝐷^𝛼_𝑡𝑢(𝑥,𝑡)+𝑝𝐷^𝛼_𝑡𝑢(𝑥,𝑡)−𝑢(𝑥,𝑡)𝐷_𝑥𝑥𝑥𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡)𝐷_𝑥𝑢(𝑥,𝑡)+3𝐷_𝑥𝑢(𝑥,𝑡)𝐷_𝑥𝑥𝑢(𝑥,𝑡)=0, (6.1) or 𝐷^𝛼_𝑡𝑢(𝑥,𝑡)=𝑝𝑢(𝑥,𝑡)𝐷_𝑥𝑥𝑥𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡)𝐷_𝑥𝑢(𝑥,𝑡)+3𝐷_𝑥𝑢(𝑥,𝑡)𝐷_𝑥𝑥𝑢(𝑥,𝑡), (6.2) where 𝑝∈[0,1] is an embedding parameter. If 𝑝=0, then (6.2) becomes a linear equation, 𝐷^𝛼_𝑡𝑢(𝑥,𝑡)=0. (6.3) And when 𝑝=1, then (6.2) turns out to be (5.1). Assume the solution of (6.2) to be in the form 𝑢(𝑥,𝑡)=𝑣₀+𝑝𝑣₁+𝑝²𝑣₂+𝑝³𝑣₃+⋯. (6.4) Substituting (6.4) into (6.2) and equating the terms with identical powers of 𝑝, we obtain the following set of linear differential equations: 𝑝⁰∶𝐷^𝛼_𝑡𝑢₀𝑝=0,₁∶𝐷^𝛼_𝑡𝑢₁=𝑣₀𝜕𝑣₀𝜕𝑥+3𝜕𝑣₀𝜕𝜕𝑥²𝑣₀𝜕𝑥²+𝑣₀𝜕³𝑣₀𝜕𝑥³,𝑝²∶𝐷^𝛼_𝑡𝑢₂=𝑣₁𝜕𝑣₀𝜕𝑥+𝑣₀𝜕𝑣₁𝜕𝑥+3𝜕𝑣₁𝜕𝜕𝑥²𝑣₀𝜕𝑥²+3𝜕𝑣₀𝜕𝜕𝑥²𝑣₁𝜕𝑥²+𝑣₁𝜕³𝑣₀𝜕𝑥³+𝑣₀𝜕³𝑣₁𝜕𝑥³, (6.5)(6.6)(6.7) and so on. Equations (6.6) and (6.7) can be solved by applying the operator 𝐽^𝛼, which is the inverse of the operator 𝐷^𝛼 and then by simple computation, Thus, the solution reads as follows 𝑣₁(𝑥,𝑡)=−2𝑐²𝑡^𝛼sin(𝑥/4)(),𝑣3Γ1+𝛼₂()𝑐𝑥,𝑡=−³𝑡3𝜋^2𝛼𝑥cos2Γ()𝑣−2𝛼sin1𝜋𝛼,₃()𝑐𝑥,𝑡=−⁴𝜋𝑡^3𝛼csc𝜋𝛼sin(𝑥/2)()Γ(𝛼)Γ(),𝑣6Γ1−𝛼1+3𝛼₄()𝑐𝑥,𝑡=−⁵𝜋𝑡^4𝛼cos(𝑥/2)csc𝛼𝜋()Γ(𝛼)Γ(),12Γ1−𝛼1+4𝛼 (6.8) and so on.

In this manner, the rest of the components of the homotopy perturbation series can be obtained. Finally, we approximate the analytical solutions of 𝑢(𝑥,𝑡) by the truncated series 𝑢(𝑥,𝑡)=lim_𝑁→∞𝑢₅(𝑥,𝑡), (6.9) where 𝑢₅(𝑥,𝑡)=∑⁴_𝑛=0𝑣_𝑘(𝑥,𝑡).

7. Numerical Results and Discussion

Tables 1 and 2 show the approximate solutions for (5.1) obtained for different values of 𝛼 using the decomposition method and the variational iteration method in different values of 𝑐, that is, 𝑐=1 and 𝑐=0.5, respectively. Tables 3 and 4 show the absolute error of (5.1) when 𝛼=1.0 in different value of 𝑐, that is, 𝑐=1.0 and 𝑐=0.5, respectively. Figures 1 and 2 show the approximate solutions for (5.1) in different values of 𝑐 using the fifth iterates of VIM, the fifth terms of HPM when 𝛼=1, and exact solution, respectively. From Tables 3 and 4 show that the approximate solution using the VIM is more accurate than the approximate solution obtained using the HPM. It is to be noted that only the fifth iterates of the variational iteration solution and only fifth terms of the homotopy perturbation series were used in evaluating.

Table 1 

The approximate solution of fractional Rosenau-Hynam equation using fifth iterates of VIM and fifth terms of HPM in different values of 𝛼, that is, 𝛼=0.7,0.8, and 0.9 and 𝛼=1.0 when 𝑐=0.5.

Table 2 

The approximate solution of fractional Rosenau-Hynam using fifth iterate of VIM and fifth terms of HPM in different values of 𝛼, that is, 𝛼=0.7,0.8, and 0.9 and 𝛼=1.0 when 𝑐=1.0.

Table 3 

Errors of the approximate solution of fractional Rosenau-Hynam equation using using fifth term of HPM and fifth iterate of VIM when 𝛼=1.0 and 𝑐=0.5.

Table 4 

Errors of the approximate solution of fractional Rosenau-Hynam equation using fifth term of HPM and fifth iterate of VIM when 𝛼=1.0 and 𝑐=0.5.

(a)

(b)

(c)

Figure 1 

The approximate solution of fractional Rosenau-Hynam equation when 𝛼=1.0 and 𝑐=0.5 using such method: (a) HPM, (b) VIM and (c) Exact.

(a)

(b)

(c)

Figure 2 

The approximate solution of fractional Rosenau-Hynam equation when 𝛼=1.0 and 𝑐=1.0 using such method: (a) HPM, (b) VIM, and (c) Exact.

8. Conclusions

The fundamental goal of this work has been to construct an approximate solution of nonlinear partial differential equations of fractional order. For computations and plots, the Mathematica and Maple packages were used. The goal has been achieved by using the variational iteration method (VIM) and the homotopy perturbation method (HPM). The methods were used in a direct way without using linearization or restrictive assumptions. There are four important points that were gotten. First, the VIM and the HPM provide the solutions in terms of convergent series with easily computable components. Second, the approximate solution in (5.1) using the VIM converges faster than the approximate solution using the HPM. Third, the variational iteration method handles nonlinear equations without any need for the so-called He’s polynomials. Finally, the recent appearance of fractional differential equations as models in some fields of applied mathematics makes it necessary to investigate methods of solution for such equations (analytical and numerical) and we hope that this work is a step in this direction.

Acknowledgment

The financial support received from UKM Grant ERGS/1/2011/STG/UKM/01/13 is gratefully acknowledged.