Approximate Analytical Solutions of the Regularized Long Wave Equation Using the Optimal Homotopy Perturbation Method

Abstract

The paper presents the optimal homotopy perturbation method, which is a new method to find approximate analytical solutions for nonlinear partial differential equations. Based on the well-known homotopy perturbation method, the optimal homotopy perturbation method presents an accelerated convergence compared to the regular homotopy perturbation method. The applications presented emphasize the high accuracy of the method by means of a comparison with previous results.

1. Introduction

A significant part of the natural technological processes and phenomena are usually modelled by means of partial differential equations. Thus it is very important to find solutions of these equations. However, as in many cases the computation of exact solutions is not possible; numerical or approximate solutions must be computed.

In the present paper we present a new approximation method named optimal homotopy perturbation method (OHPM). As the name suggests, the method is based on the homotopy perturbation method [1, 2] and its main feature is an accelerated convergence compared to the regular homotopy perturbation method.

The applications presented show that the approximate solutions obtained by using OHPM requires less iterations in comparison with other iterative methods for approximate solutions of partial differential equations.

2. The Optimal Homotopy Perturbation Method

We consider the following problem: 𝔏(𝑢(𝑥,𝑡))+𝔑(𝑢(𝑥,𝑡))−𝑓(𝑥,𝑡)=0,𝐵(𝑢)=0. (1) Here 𝔏 is a linear operator, 𝑢(𝑥,𝑡) is the unknown function, 𝔑 is a nonlinear operator, 𝑓(𝑥,𝑡) is a known, given function, and 𝐵 is a boundary operator.

If 𝑢 is an approximate solution of (1), we evaluate the error obtained by replacing the exact solution 𝑢 with the approximate one 𝑢 as the remainder: 𝑅(𝑥,𝑡,𝑢)=𝔏(𝑢(𝑥,𝑡))+𝔑(𝑢(𝑥,𝑡))−𝑓(𝑥,𝑡). (2)

The first step in applying OHPM is to attach to the problem (1) the family of equations (see [1, 2]): 𝔏Φ(𝔏ΦΦ(1−𝑝𝑥,𝑡,𝑝−𝑓𝑥,𝑡)+𝑝𝑥,𝑡,𝑝+𝔑𝑥,𝑡,𝑝−𝑓𝑥,𝑡)=0, (3) where 𝑝∈[0,1] is an embedding parameter and Φ(𝑥,𝑡,𝑝) is an unknown function.

When 𝑝=0, Φ(𝑥,𝑡,0)=𝑢₀(𝑥,𝑡) and when 𝑝=1, Φ(𝑥,𝑡,1)=𝑢(𝑥,𝑡). Thus, as 𝑝 increases from 0 to 1, the solution Φ(𝑥,𝑡,𝑝) varies from 𝑢₀(𝑥,𝑡) to the solution 𝑢(𝑥,𝑡), where 𝑢₀(𝑥,𝑡) is obtained from the following: 𝔏𝑢₀(𝑥,𝑡)−𝑓(𝑥,𝑡)=0,𝐵𝑢₀=0. (4)

We consider the following expansion of Φ(𝑥,𝑡,𝑝): Φ𝑥,𝑡,𝑝=𝑢₀(𝑥,𝑡)+_𝑚≥1𝑢_𝑚(𝑥,𝑡)𝑝^𝑚. (5)

Substituting the relation (5) into (3), collecting the same powers of 𝑝, and equating each coefficient of the powers of 𝑝 with zero we obtain 𝔏𝑢_𝑚(𝑥,𝑡)=−𝔑_𝑚−1𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡),…,𝑢_𝑚−1()𝑢𝑥,𝑡𝑚≥1,…,𝐵_𝑚=0, (6) where 𝔑_𝑖, 𝑖≥0 are the coefficients of 𝑝^𝑖 in the nonlinear operator 𝔑: 𝔑(𝑢(𝑥,𝑡))=𝔑₀𝑢₀(𝑥,𝑡)+𝑝𝔑₁𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡)+𝑝²𝔑₂𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡),𝑢₂(𝑥,𝑡)+⋯. (7)

We remark that 𝑢_𝑚, 𝑚≥1 are obtained from the linear equations (6), which are easily solved together with the boundary conditions.

We denote 𝑓_𝑚=𝑢₀+𝑢₁+⋯+𝑢_𝑚.

We consider the set 𝑆_𝑚 (𝑚=0,1,2,…) containing the functions 𝜑_𝑚0,𝜑_𝑚1,𝜑_𝑚2,…,𝜑_𝑚𝑛𝑚, chosen as linearly independent functions in the vector space of the continuous functions on the real domain Ω such that 𝑆_𝑚−1⊆𝑆_𝑚 and 𝑢₀+𝑢₁+⋯+𝑢_𝑚 is a real linear combination of these functions.

We remark that such a construction is always possible. For example we can choose 𝑆_𝑚={𝑢₀,𝑢₁,…,𝑢_𝑚}, 𝑚=0,1,2,…. In this case 𝜑_𝑚0=𝑢₀, 𝜑_𝑚1=𝑢₁, 𝜑_𝑚2=𝑢₂,…,𝜑_𝑚𝑛𝑚=𝑢_𝑚.

Definition 1. We call an HP-sequence of the problem (1) a sequence of functions {𝑠_𝑚(𝑥,𝑡)}_𝑚∈ℕ of the form 𝑠_𝑚(𝑥,𝑡)=∑^𝑛𝑚_𝑘=0𝛼^𝑘_𝑚𝜑_𝑚𝑘, where 𝑚∈ℕ, 𝛼^𝑘_𝑚∈ℝ.
A function of the sequence is called an HP-function of the problem (1).
We call the HP-sequence {𝑠_𝑚(𝑥,𝑡)}_𝑚∈ℕ, convergent to the solution of the problem (1) if lim_𝑚→∞𝑅(𝑥,𝑡,𝑠_𝑚(𝑥,𝑡))=0.

Definition 2. We call an 𝜖-approximate HP-solution of the problem (1) on the real domain Ω an HP-function 𝑢 which satisfies the following condition: |𝑅(𝑥,𝑡,𝑢)|<𝜖 (8) together with the boundary conditions from (1).

Definition 3. We call a weak 𝛿-approximate HP-solution of the problem (1) on the real domain Ω an HP-function 𝑢 satisfying the relation ∫_Ω𝑅²(𝑥,𝑡,𝑢)𝑑𝑥𝑑𝑡≤𝛿, together with the boundary conditions from (1).

We will find a weak 𝜖-approximate HP-solution of the type 𝑢=∑^𝑛𝑚_𝑘=0𝑐^𝑘_𝑚𝜑_𝑚𝑘 where 𝑚≥0 and the constants 𝑐^𝑘_𝑚 are calculated using the following steps.(i)We substitute the approximate solution 𝑢 in (1) and obtain the following expression: ℜ𝑥,𝑡,𝑐^𝑘_𝑚=𝑅(𝑥,𝑡,𝑢). (9)(ii)We attach to the problem (1) the following real functional: 𝐽𝑐^𝑘_𝑚=_Ωℜ²𝑥,𝑡,𝑐^𝑘_𝑚𝑑𝑥𝑑𝑡, (10) where, by imposing the boundary conditions we can determine 𝑙∈𝑁, 𝑙≤𝑚 such that 𝑐^𝑚₀,𝑐^𝑚₁,…,𝑐^𝑚_𝑙 are computed as functions of 𝑐^𝑚_𝑙+1,𝑐^𝑚_𝑙+2,…,𝑐^𝑚_𝑛.(iii)We compute the values of 𝑐^𝑚_𝑙+1,𝑐^𝑚_𝑙+2,…,𝑐^𝑚_𝑛 as the values which give the minimum of the functional (10) and the values of 𝑐^𝑚₀,𝑐^𝑚₁,…,𝑐^𝑚_𝑙 again as functions of 𝑐^𝑚_𝑙+1,𝑐^𝑚_𝑙+2,…,𝑐^𝑚_𝑛 by using the boundary conditions.(iv)Using the constants 𝑐^𝑚₀,…,𝑐^𝑚_𝑛 thus determined, we consider the HP-sequence 𝑠_𝑚(𝑥,𝑡)=^𝑛𝑚_𝑘=0𝑐^𝑘_𝑚𝜑_𝑚𝑘. (11)

The following convergence theorem holds.

Theorem 4. The HP-sequence 𝑠_𝑚(𝑥,𝑡) from (11) satisfies the following property:lim_𝑚→∞Ω𝑅²𝑥,𝑡,𝑠_𝑚(𝑥,𝑡)𝑑𝑥𝑑𝑡=0. (12) Moreover, ∀𝜖>0, ∃𝑚₀∈ℕ such that ∀𝑚∈ℕ, 𝑚>𝑚₀ it follows that 𝑠_𝑚(𝑡) is a weak 𝜖-approximate HP-solution of the problem (1).

Proof. Based on the way the HP-function 𝑠_𝑚(𝑥,𝑡) is computed, the following inequality holds: 0≤_Ω𝑅²𝑥,𝑡,𝑠_𝑚()≤𝑥,𝑡𝑑𝑥𝑑𝑡_Ω𝑅²𝑡,𝑓_𝑚(𝑥,𝑡)𝑑𝑥𝑑𝑡,∀𝑚∈ℕ. (13) It follows that 0≤lim_𝑚→∞Ω𝑅²𝑥,𝑡,𝑠_𝑚(𝑥,𝑡)𝑑𝑥𝑑𝑡≤lim_𝑚→∞Ω𝑅²𝑥,𝑡,𝑓_𝑚(𝑥,𝑡)𝑑𝑥𝑑𝑡=0,∀𝑚∈ℕ. (14) We obtain lim_𝑚→∞Ω𝑅²𝑥,𝑡,𝑠_𝑚(𝑥,𝑡)𝑑𝑥𝑑𝑡=0. (15) From this limit we obtain that ∀𝜖>0, ∃𝑚₀∈ℕ such that ∀𝑚∈ℕ, 𝑚>𝑚₀ it follows that 𝑠_𝑚(𝑥,𝑡) is a weak 𝜖-approximate HP-solution of the problem (1).

Remark 5. Any 𝜖-approximate HP-solution of the problem (1) is also a weak approximate HP-solution, but the opposite is not always true. It follows that the set of weak approximate HP-solutions of the problem (1) also contains the approximate HP-solutions of the problem.

Taking into account the above remark, in order to find 𝜖-approximate HP-solutions of the problem (1) by the OHPM method we will first determine weak approximate HP-solutions, 𝑢. If |𝑅(𝑥,𝑡,𝑢)|<𝜖 then 𝑢 is also an 𝜖-approximate HP-solution of the problem.

3. Applications

In this section we apply OHPM to find approximate analytical solutions for the regularized long wave (RLW) equation.

The RLW equation is a nonlinear evolution equation. These kind of equations are frequently used to model a variety of physical phenomena such as ion-acoustic waves in plasma, magnetohydrodynamics waves in plasma, longitudinal dispersive waves in elastic rods, pressure waves in liquid gas bubble mixtures, and rotating flow down a tube.

The RLW equation was introduced in [3] where it was used to describe the behaviour of the undular bore.

For some restricted initial and boundary conditions, exact analytical solutions for the RLW equation were computed (see, e.g., [4]). However, in most cases it is not possible to find such exact analytical solutions and usually numerical methods are used. Among the numerical methods recently employed for RLW-type equations we mention finite difference methods [5–8], multistep mixed finite element methods [9], the method of lines [10], and meshless finite-point methods [11].

Taking into account the usefulness of analytical solutions versus numerical ones, various approximation methods were also employed to find approximate analytical solutions for various RLW-type equations, such as the homotopy perturbation method [12], the variational iteration method [12], the homotopy asymptotic method [13, 14], and the Riccati expansion method [15].

In the following, for two test problems presented in [12], we compare solutions obtained by using OHPM with previous results obtained by using the homotopy perturbation method and the variational iteration method.

3.1. Application 1

Our first application is the following RLW problem [12]: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+𝑢²2_𝑥𝑢()=0,𝑥,0=𝑥. (16)

In [12] approximate solutions of (16) are computed using the homotopy perturbation method (HPM) and the variational iteration method (VIM).

The exact solution of this problem is 𝑢_𝑒(𝑥,𝑡)=𝑥/(𝑡+1).

The fifth order solution computed in [12] by using the variational iteration method is 𝑢_VIM()−𝑡𝑥,𝑡=𝑥⋅³¹+𝑡109876902975³⁰−𝑡3544416225²⁹[phone]𝑡²⁸−3150592202𝑡²⁷+𝑡6751269²⁶−5953505309𝑡²⁵[phone]𝑡²⁴−5401015202447𝑡²³[phone]𝑡²²−1666980207509𝑡²¹[phone]𝑡²⁰−7144200162179𝑡¹⁹[phone]𝑡¹⁸−2296351080013𝑡¹⁷[phone]𝑡¹⁶−105840063283𝑡¹⁵[phone]𝑡¹⁴−882013141𝑡¹³[phone]𝑡¹²−680401477𝑡¹¹[phone]𝑡¹⁰−567003497𝑡⁹+5670943𝑡⁸−126013𝑡⁷+1543𝑡⁶45−𝑡⁵+𝑡⁴−𝑡³+𝑡²−𝑡+1. (17)

The fifth order solution computed in [12] by using the homotopy perturbation method is of the form (5) 𝑢_HPM()−𝑥,𝑡=𝑥⋅1382𝑡¹¹−1559251382𝑡¹⁰−14175326𝑡⁹−567626𝑡⁸−3151303𝑡⁷−315199𝑡⁶45−𝑡⁵+𝑡⁴−𝑡³+𝑡²−𝑡+1. (18)

Using OHPM, the following steps are performed.(i)Choosing the same homotopy (3) as used in [12] we obtain the same solutions: 𝑢₀(𝑥,𝑡)=𝑥⋅(𝑡+1) 𝑢₁(𝑥,𝑡)=−𝑥⋅𝑡⋅(2+𝑡+𝑡²/3) 𝑢₂(𝑥,𝑡)=2⋅𝑥⋅𝑡²⋅(15+15⋅𝑡+5⋅𝑡²+𝑡³)/15. It follows that we obtain the sets 𝑆₀={𝑥,𝑥⋅𝑡}, 𝑆₁={𝑥⋅𝑡,𝑥⋅𝑡²,𝑥⋅𝑡³}, 𝑆₂={𝑥⋅𝑡²,𝑥⋅𝑡³,𝑥⋅𝑡⁴,𝑥⋅𝑡⁵}. We will compute a second order approximate solution, by taking into account the terms from 𝑆₀, 𝑆₁, and 𝑆₂ and we will compare this solution with the fifth order solutions from [12]. Our second order approximate solution will have the expression 𝑢_OHPM(𝑥,𝑡)=𝑐₀⋅𝑥+𝑐₁⋅𝑥⋅𝑡+𝑐₂⋅𝑥⋅𝑡²+𝑐₃⋅𝑥⋅𝑡³+𝑐₄⋅𝑥⋅𝑡⁴+𝑐₅⋅𝑥⋅𝑡⁵.(ii)Imposing the boundary condition 𝑢_OHPM(𝑥,0)=𝑥 we obtain 𝑐₀=1. Replacing this expression of 𝑐₀ in the expression of 𝑢_OHPM we obtain the following: 𝑢_OHPM(𝑥,𝑡)=𝑥+𝑐₁⋅𝑥⋅𝑡+𝑐₂⋅𝑥⋅𝑡²+𝑐₃⋅𝑥⋅𝑡³+𝑐₄⋅𝑥⋅𝑡⁴+𝑐₅⋅𝑥⋅𝑡⁵. We introduce 𝑢_OHPM in the remainder ℜ given by (2) and (9) and we compute the functional 𝐽(𝑐₁,𝑐₂,𝑐₃,𝑐₄,𝑐₅) of (10). We remark that while the expression of the functional is too long to be included here, the computation is simple and straightforward using a dedicated mathematical software (we used the Wolfram Mathematica 9 software). (iii) We compute the minimum of the functional 𝐽 and, by replacing the corresponding values of the parameters 𝑐₁, 𝑐₂, 𝑐₃, 𝑐₄, 𝑐₅, we obtain the following second order approximation: 𝑢_OHPM(𝑥,𝑡)=−0.109895𝑡⁵𝑥+0.434798𝑡⁴𝑥−0.789112𝑡³𝑥+0.961938𝑡²𝑥−0.997729𝑡𝑥+𝑥.

Figure 1 presents the comparison of the absolute errors (computed as the absolute values of the differences between the exact solutions and the approximate solutions) corresponding to the fifth order approximation obtained by using HPM (red surface), to the fifth order approximation obtained by using VIM (blue surface) and to the second order approximation obtained by OHPM (green surface).

Figure 1 

The absolute differences corresponding to the HPM solution (red surface), VIM solution (blue surface), and OHPM solution (green surface) for problem (16).

Table 1 presents the same comparison for several values of 𝑥 and 𝑡.

Table 1 

The absolute differences corresponding to the HPM solution (red surface), VIM solution (blue surface), and OHPM solution (green surface) for problem (16).

It is easy to see that, overall, the approximations obtained by using OHPM are much more accurate than the ones previously computed by using HPM and VIM. Moreover, our approximate solutions are not only more accurate but also, at the same time, present a much simpler expression since they are second order approximate solutions while the previous ones are fifth order approximate solutions.

3.2. Application 2

Our second application is the RLW problem (also from [12]): 𝑢_𝑡−𝑢_{𝑥𝑥𝑥𝑥}𝑢()(𝑥).=0,𝑥,0=sin (19)

Again in [12] approximate solutions of (16) are computed using the homotopy perturbation method (HPM) and the variational iteration method (VIM).

The exact solution of this problem is 𝑢_𝑒(𝑥,𝑡)=𝑒^−𝑡sin(𝑥).

The fourth order solution computed in [12] by using the variational iteration method is 𝑢_VIM(𝑥,𝑡)=(1/24)(𝑡⁴−4𝑡³+12𝑡²−24𝑡+24)sin(𝑥).

The third order solution computed in [12] by using the homotopy perturbation method is of the form (5) 𝑢_HPM(𝑥,𝑡)=−(1/24)(𝑡⁴+4𝑡³−12𝑡²+24𝑡−24)sin(𝑥).

Using OHPM, the following steps are performed.(i)Choosing the same homotopy (3) as used in [12] we obtain the same solutions: 𝑢₀(𝑥,𝑡)=(𝑡+1)⋅sin(𝑥) 𝑢₁(𝑥,𝑡)=(1/2)⋅𝑡⋅(𝑡+4)⋅(−sin(𝑥)) 𝑢₂(𝑥,𝑡)=(1/6)⋅𝑡²⋅(𝑡+6)⋅sin(𝑥). It follows that we obtain the sets 𝑆₀={sin(𝑥),sin(𝑥)⋅𝑡}, 𝑆₁={sin(𝑥)⋅𝑡,sin(𝑥)⋅𝑡²}, 𝑆₂={sin(𝑥)⋅𝑡²,sin(𝑥)⋅𝑡³}. Hence we will compute a second order approximate solution of the following form: 𝑢_OHPM(𝑥,𝑡)=𝑐₀⋅sin(𝑥)+𝑐₁⋅sin(𝑥)⋅𝑡+𝑐₂⋅sin(𝑥)⋅𝑡²+𝑐₃⋅sin(𝑥)⋅𝑡³.(ii)Imposing the boundary condition 𝑢_OHPM(𝑥,0)=𝑥 we obtain 𝑐₀=1. Replacing this expression of 𝑐₀ in the expression of 𝑢_OHPM we obtain the following: 𝑢_OHPM(𝑥,𝑡)=sin(𝑥)+𝑐₁⋅sin(𝑥)⋅𝑡+𝑐₂⋅sin(𝑥)⋅𝑡²+𝑐₃⋅sin(𝑥)⋅𝑡³. We introduce 𝑢_OHPM in the remainder ℜ given by (2) and (9) and we compute the functional 𝐽(𝑐₁,𝑐₂,𝑐₃) of (10).(iii)We compute the minimum of the functional 𝐽 and, by replacing the corresponding values of the parameters 𝑐₁, 𝑐₂, 𝑐₃, we obtain the following second order approximation: 𝑢_OHPM(𝑥,𝑡)=−0.102902𝑡³sin(𝑥)+0.465235𝑡²sin(𝑥)−0.994455𝑡sin(𝑥)+sin(𝑥).

Figure 2 presents the comparison of the absolute errors corresponding to the third order approximation obtained by using HPM (red surface), to the fourth order approximation obtained by using VIM (blue surface), and to the second order approximation obtained by OHPM (green surface).

Figure 2 

The absolute differences corresponding to the HPM solution (red surface), VIM solution (blue surface), and OHPM solution (green surface) for problem (19).

Table 2 presents the same comparison for several values of 𝑥 and 𝑡.

Table 2 

The absolute differences corresponding to the HPM solution (red surface), VIM solution (blue surface), and OHPM solution (green surface) for problem (19).

Again, overall, the approximations obtained by using OHPM are more accurate than the ones previously computed by using HPM and VIM while, at the same time, they present a much simpler expression.

4. Conclusions

In the present paper the new optimal homotopy perturbation method is introduced as a straightforward and efficient method to compute approximate solutions for nonlinear partial differential equations.

The optimal homotopy perturbation method has an accelerated convergence compared to the regular homotopy perturbation method, fact proved by the included applications. The method is a powerful one since not only were we capable to find more accurate approximations, but also the approximations computed consist of fewer terms than the previous solutions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.