The GDTM-Padé Technique for the Nonlinear Lattice Equations

Abstract

The GDTM-Padé technique is a combination of the generalized differential transform method and the Padé approximation. We apply this technique to solve the two nonlinear lattice equations, which results in the high accuracy of the GDTM-Padé solutions. Numerical results are presented to show its efficiency by comparing the GDTM-Padé solutions, the solutions obtained by the generalized differential transform method, and the exact solutions.

1. Introduction

The nonlinear differential difference equations (NDDEs) have wide applications in various branches of science, including the mechanical engineering, condensed matter physics, biophysics, mathematical statistics, control theory and so on [1–11]. During the past decades, a large number of solution methods such as the Adomian decomposition method [12, 13], the Jacobian elliptic function method [14], the Exp-function method [15], the (𝐺′/𝐺)-expansion method [16], and the variable-coefficient discrete tanh method [17] were proposed to solve the NDDEs. Recently, the generalized differential transform method [18–20] combined with the Padé technique (named as GDTM-Padé technique) was presented in [21] to construct the numerical or exact solutions of the differential difference equations. Due to the Padé approximation, the convergence and the accuracy of the original series solutions can be improved.

In this paper, we focus on solving two nonlinear lattice equations by applying the GDTM-Padé technique. The first nonlinear equation is the hybrid lattice equation [5] defined by𝜕𝑢_𝑛=𝜕𝑡1+𝛼𝑢_𝑛+𝛽𝑢²_𝑛𝑢_𝑛−1−𝑢_𝑛+1, (1.1) which was related with the discretization of the KDV equations or the modified KDV equations. The second equation arose in the study of continuum two-boson KP systems [3, 22], which was called as the Volterra lattice equation𝜕𝑢_𝑛𝜕𝑡=𝑢_𝑛𝑣_𝑛−𝑣_𝑛−1,𝜕𝑣_𝑛𝜕𝑡=𝑣_𝑛𝑢_𝑛+1−𝑢_𝑛. (1.2)

The rest of this paper is organized as follows. In Section 2, we introduce the idea of the GDTM-Padé technique for the NDDEs. The hybrid lattice equation and the Volterra lattice equation are studied in Section 3. Numerical results are presented to verify the efficiency. Finally, some conclusions are given.

2. The GDTM-Padé Technique

To illustrate the basic idea of the GDTM-Padé technique, we consider the general nonlinear difference differential equation𝑁𝑢_𝑛(𝑡),𝑢_𝑛+1(𝑡),𝑢_𝑛−1(𝑡),𝑢_𝑛+2(𝑡),…=0, (2.1) where 𝑁 is a nonlinear differential operator, 𝑢_𝑛(𝑡) is the unknown function with respect to the discrete spatial variable 𝑛 and the temporal variable 𝑡. Applying the one-dimensional differential transform method (GDTM), the differential transform of the 𝑘th derivative of the function 𝑢_𝑛(𝑡) is defined by𝑈_𝑛(𝑘)()=1=𝑈𝑛,𝑘𝑑𝑘!^𝑘𝑢_𝑛(𝑡)𝑑𝑡^𝑘|_𝑡=𝑡0. (2.2) The differential inverse transform of 𝑈_𝑛(𝑘) is read as𝑢_𝑛(𝑡)=^∞_𝑘=0𝑈_𝑛(𝑘)𝑡−𝑡₀^𝑘=^∞_𝑘=0𝑈(𝑛,𝑘)𝑡−𝑡₀^𝑘. (2.3) Particularly, the function 𝑢_𝑛(𝑡) can be formulated as a series when 𝑡₀=0, that is,𝑢_𝑛(𝑡)=^∞_𝑘=0𝑈_𝑛(𝑘)𝑡^𝑘=^∞_𝑘=0𝑈(𝑛,𝑘)𝑡^𝑘. (2.4) In the real applications, we can determine the coefficients 𝑈(𝑛,𝑘)(𝑘=1,…,𝑚) and obtain the 𝑚th-order approximation of the function 𝑢_𝑛(𝑡) given by𝑢_𝑛,𝑚(𝑡)=^𝑚_𝑘=0𝑈_𝑛(𝑘)𝑡^𝑘=^𝑚_𝑘=0𝑈(𝑛,𝑘)𝑡^𝑘. (2.5) The transformed operations for the GDTM are listed in Table 1 [21].

Table 1 

The operations for generalized differential transform method.

To improve the accuracy and convergence of the GDTM solution (2.5), the Padé approximation [23, 24] is used. For simplicity, we denote the [𝐿,𝑀] Padé approximation to 𝑓(𝑥)=∑^∞_𝑘=0𝑎_𝑘𝑡^𝑘 by𝑓[]=𝑃𝐿,𝑀^𝐿(𝑥)𝑄_𝑀(𝑥), (2.6) where𝑃_𝐿(𝑥)=𝑝₀+𝑝₁𝑥+𝑝₂𝑥²+𝑝₃𝑥³+⋅⋅⋅+𝑝_𝐿𝑥^𝐿,𝑄_𝑀(𝑥)=1+𝑞₁𝑥+𝑞₂𝑥²+𝑞₃𝑥³+⋅⋅⋅+𝑞_𝑀𝑥^𝑀, (2.7) with the normalization condition 𝑄_𝑀(0)=1. The coefficients of 𝑃_𝐿(𝑥) and 𝑄_𝑀(𝑥) can be uniquely determined by comparing the first (𝐿+𝑀+1) terms of the functions 𝑓[𝐿,𝑀] and 𝑓(𝑥). In practice, the construction of the [𝐿,𝑀] Padé approximation involves only algebra equations, which are solved by means of the Mathematica or Maple package. We call the solution obtained by the GDTM and the Padé approximation as the GDTM-Padé solution.

3. Numerical Examples

In this section, we will illustrate the validity and advantages of the GDTM-Padé technique for the nonlinear differential difference equations. Two nonlinear lattice equations will be studied, where one is a hybrid lattice and the other is a Volterra lattice.

3.1. The Hybrid Lattice Equation

Consider the hybrid lattice equation (1.1) with the initial condition𝑢_𝑛(0)=−𝛼+𝛼²−4𝛽tanh(𝑑)()2𝛽tanh𝑑𝑛. (3.1) The exact solution to (1.1) [2] is of the form𝑢_𝑛(𝑡)=−𝛼+𝛼²−4𝛽tanh(𝑑)𝛼2𝛽tanh𝑑𝑛+²−4𝛽(𝑑)𝑡2𝛽tanh. (3.2)

Using the GDTM technique, the transformed problem of (1.1) can be expressed in the following recurrence formula:(𝑘+1)𝑈(𝑛,𝑘+1)=𝑈(𝑛−1,𝑘)−𝑈(𝑛+1,𝑘)+𝛼^𝑘_𝑠=0𝑈(𝑛,𝑠)(𝑈(𝑛−1,𝑘−𝑠)−𝑈(𝑛+1,𝑘−𝑠))+𝛽^𝑘𝑠_𝑠=0𝑡=0𝑈(𝑛,𝑡)𝑈(𝑛,𝑠−𝑡)(𝑈(𝑛−1,𝑘−𝑠)−𝑈(𝑛+1,𝑘−𝑠)). (3.3) The transformed initial condition is𝑈(𝑛,0)=−𝛼+𝛼²−4𝛽tanh(𝑑)()2𝛽tanh𝑑𝑛. (3.4) One can also easily construct the implicit initial conditions as follows:𝑈(𝑛−1,0)=−𝛼+𝛼²−4𝛽tanh(𝑑)(𝑑(,𝑈()=2𝛽tanh𝑛−1))𝑛+1,0−𝛼+𝛼²−4𝛽tanh(𝑑)(𝑑(.2𝛽tanh𝑛+1)) (3.5)

Based on the above initial conditions and the recursive formula (3.3), we can derive the coefficients 𝑈(𝑛,𝑘) one by one and obtain the approximate solution 𝑢_𝑛,𝑚(𝑡)=∑^𝑚_𝑘=0𝑈(𝑛,𝑘)𝑡^𝑘. In this example, we set 𝛼=3, 𝛽=2 and 𝑑=0.5. The 5th-order approximate solution at 𝑛=5 is given by𝑢_𝑛,5(𝑡)=−0.6259778749−0.0018551343𝑡+0.0001686607𝑡²−2.3937386136×10⁻⁶𝑡³−9.7827591261×10⁻⁷𝑡⁴−4.0803184434×10⁻⁸𝑡⁵. (3.6)

Applying the GDTM-Padé technique to the solution (3.6), we get the [2,2] GDTM Padé approximation:𝑢[2,2]=−0.6259778749−0.059706823𝑡−0.00445469𝑡²1+0.0924181046𝑡+0.0071119169𝑡². (3.7)

For comparison, we plot the GDTM solutions 𝑢_𝑛,5(𝑡), the GDTM-Padé solutions 𝑢[2,2], and the exact solutions of (1.1) in Figure 1. Figure 2 shows the absolute error of the GDTM solutions and the GDTM-Padé solutions. The GDTM solutions are in good agreement with the exact solutions in the small interval (−5≤𝑡≤5), and high errors appear when 𝑡>5. By the GDTM-Padé technique, the accuracy of the approximation is improved largely.

Figure 1 

The compared results for the GDTM solutions (black), the GDTM-Padé solutions (red), and the exact solutions (blue) of (1.1).

Figure 2 

The error curves for the GDTM solutions (blue) and the GDTM-Padé solutions (red).

3.2. The Volterra Lattice Equations

We further consider the two component Volterra lattice equations (1.2) with the initial conditions𝑢_𝑛(0)=−𝑐coth(𝑑)+𝑐tanh(𝑑𝑛),𝑣_𝑛(0)=−𝑐coth(𝑑)−𝑐tanh(𝑑𝑛). (3.8) We remark that the exact solutions to (1.2) [2] are given by 𝑢_𝑛(𝑡)=−𝑐coth(𝑑)+𝑐tanh(𝑑𝑛+𝑐𝑡) and 𝑣_𝑛(𝑡)=−𝑐coth(𝑑)−𝑐tanh(𝑑𝑛+𝑐𝑡), respectively.

Similarly, using the GDTM-Padé technique, we obtain the following transformed problems:(𝑘+1)𝑈(𝑛,𝑘+1)=^𝑘_𝑠=0𝑈(𝑛,𝑠)(𝑉(𝑛,𝑘−𝑠)−𝑉(𝑛−1,𝑘−𝑠)), (3.9)(𝑘+1)𝑉(𝑛,𝑘+1)=^𝑘_𝑠=0𝑉(𝑛,𝑠)(𝑈(𝑛+1,𝑘−𝑠)−𝑈(𝑛,𝑘−𝑠)), (3.10) with the initial conditions𝑈()(𝑑)()()(𝑑)(𝑑(,𝑉()(𝑑)()()(𝑑)(𝑑(.𝑛,0=−𝑐coth+𝑐tanh𝑑𝑛,𝑈𝑛+1,0=−𝑐coth+𝑐tanh𝑛+1))𝑛,0=−𝑐coth−𝑐tanh𝑑𝑛,𝑉𝑛−1,0=−𝑐coth−𝑐tanh𝑛−1)) (3.11)

We first obtain the coefficients 𝑈(𝑛,𝑘+1) of 𝑢_𝑛(𝑡) by the above initial conditions and (3.9), then derive the coefficients 𝑉(𝑛,𝑘+1) of 𝑣_𝑛(𝑡), which results in the GDTM solutions 𝑢_𝑛,𝑚(𝑡)=∑^𝑚_𝑘=0𝑈(𝑛,𝑘)𝑡^𝑘 and 𝑣_𝑛,𝑚(𝑡)=∑^𝑚_𝑘=0𝑉(𝑛,𝑘)𝑡^𝑘.

We set 𝑐=−0.5, 𝑑=−2 at 𝑛=1, and obtain the 6th-order approximations as follows:𝑢_𝑛,6(𝑡)=−0.0366435703+0.0176627062𝑡−0.008513668𝑡²+0.0026318132𝑡³+0.0005448212𝑡⁴−0.0000317142𝑡⁵+0.00004837𝑡⁶,𝑣_𝑛,6(𝑡)=−1.0006711504−0.0176627062𝑡+0.008513668𝑡²−0.0026318132𝑡³+0.000559098𝑡⁴+0.0001457863𝑡⁵−0.0000126128𝑡⁶. (3.12)

The [2,2] GDTM-Padé solutions to the approximations 𝑢_𝑛,6(𝑡) and 𝑣_𝑛,6(𝑡) can be expressed as𝑢[2,2]=−0.0366435703−0.0274826777𝑡−0.0030536309𝑡²1+1.23201379𝑡+0.4448436759𝑡²,𝑣[2,2]=−1.0006711504−0.5𝑡−0.0833892625𝑡²1+0.48201379𝑡+0.0833333333𝑡². (3.13)

We plot in Figure 3 the curves of the GDTM solutions, the GDTM-Padé solutions, and the exact solutions of (1.2). Figure 4 shows the absolute errors of the GDTM solutions and the GDTM-Padé solutions. The GDTM-Padé method performs better than the GDTM method for this example. We show the absolute errors of |𝑢_𝑛(𝑡)−𝑢_𝑛,6(𝑡)| and |𝑢_𝑛(𝑡)−𝑢[2,2]| in the left column of Table 2. The absolute errors of |𝑣_𝑛(𝑡)−𝑣_𝑛,6(𝑡)| and |𝑣_𝑛(𝑡)−𝑣[2,2]| are shown in the right column. Obviously, the errors of 𝑢[2,2] are reduced significantly, comparing with the approximation 𝑢_𝑛,6(𝑡) when 𝑡>1. This phenomenon also appears in the errors of 𝑣[2,2].

Table 2 

Comparisons of the absolute errors between the GDTM solutions and GDTM-Padé solutions for (1.2) with 𝑐=−0.5, 𝑑=−2 and 𝑛=1.

(a)

(b)

Figure 3 

The compared results for the GDTM solutions (black), the GDTM-Padé solutions (red), and the exact solutions (blue) of (1.2) when 𝑐=−0.5, 𝑑=−2 and 𝑛=1.

(a)

(b)

Figure 4 

The absolute error curves for the the GDTM solutions (blue) and the GDTM-Padé solutions (red) of (1.2) when 𝑐=−0.5, 𝑑=−2 and 𝑛=1.

If 𝑐=1, 𝑑=0.5 at 𝑛=5, the 6th-order approximations are given by𝑢_𝑛,6(𝑡)=−1.1773391156+0.0265922267𝑡−0.0262362711𝑡²+0.0170210046𝑡³−0.0071252574𝑡⁴+0.0028531211𝑡⁵−0.0001774315𝑡⁶,𝑣_𝑛,6(𝑡)=−3.1505677119−0.0265922267𝑡+0.0262362711𝑡²−0.0170210046𝑡³+0.0080477428𝑡⁴−0.0023581228𝑡⁵+0.0007216293𝑡⁶. (3.14)

Similarly, the [2,2] GDTM-Padé solutions to (3.14) are𝑢[2,2]=−1.1773391156−1.2575132091𝑡−0.5105646518𝑡²1+1.0906844246𝑡+0.4360104081𝑡²,𝑣[2,2]=−3.1505677119−3.1349873785𝑡−1.0501892373𝑡²1+0.9866142982𝑡+0.3333333333𝑡². (3.15)

Figure 5 shows the compared results for the solutions including 𝑢_𝑛,6(𝑡), 𝑢[2,2], 𝑢_𝑛(𝑡) and 𝑣_𝑛,6(𝑡), 𝑣[2,2], 𝑣_𝑛(𝑡). The error curves are plotted in Figure 6. In Table 3, we compare the absolute errors of GDTM solutions and GDTM-Padé solutions. Similar to the previous case, the GDTM-Padé method also outperforms the GDTM method.

Table 3 

Numerical results of the absolute errors between the GDTM solutions and the GDTM-Padé solutions for (1.2) with 𝑐=1, 𝑑=0.5 and 𝑛=5.

(a)

(b)

Figure 5 

The compared results for the GDTM solutions (black), the GDTM-Padé solutions (red), and the exact solutions (blue) of (1.2) when 𝑐=1, 𝑑=0.5 and 𝑛=5.

(a)

(b)

Figure 6 

The absolute errors for the the GDTM solutions (blue) and the GDTM-Padé (red) of (1.2) when 𝑐=1, 𝑑=0.5 and 𝑛=5.

4. Conclusions

This paper focused on solving the nonlinear lattice equations by using the GDTM-Padé technique. The numerical results confirmed the effectiveness of this method. In the future work, we will further extend this method to other nonlinear differential difference equations.

Acknowledgments

The work of this author was supported by the Natural Science Foundation of Zhejiang Province (Y6110639) and the project Y201016192 of Education Department of Zhejiang Province.