The GDTM-Padé Technique for the Nonlinear Lattice Equations

and Applied Analysis 3 Table 1: The operations for generalized differential transform method. Original function Transformed function f n, t g n, t h n, t F n, k G n, k H n, k f n, t αg n, t F n, k αG n, k f n, t ∂g n, t /∂t F n, k k 1 G n, k 1 f n, t g n, t h n, t F n, k ∑k r 0 G n, r H n, k − r f n, t ∂g n, t /∂t F n, k k m G n, k m f n, t g n s, t F n, k G n s, k 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 0 akt k by f L,M PL x QM x , 2.6 where PL x p0 p1x p2x p3x · · · pLx, QM x 1 q1x q2x q3x · · · qMx, 2.7 with the normalization condition QM 0 1. The coefficients of PL x and QM 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 condition un 0 −α √ α2 − 4β tanh d 2β tanh dn . 3.1 The exact solution to 1.1 2 is of the form un t −α √ α2 − 4β tanh d 2β tanh [ dn α2 − 4β 2β tanh d t ] . 3.2 4 Abstract and Applied Analysis 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 0 U n, s U n − 1, k − s −U n 1, k − s β k ∑ s 0 s ∑ t 0 U n, t U n, s − t U n − 1, k − s −U n 1, k − s . 3.3 The transformed initial condition is U n, 0 −α √ α2 − 4β tanh d 2β tanh dn . 3.4 One can also easily construct the implicit initial conditions as follows: U n − 1, 0 −α √ α2 − 4β tanh d 2β tanh d n − 1 , U n 1, 0 −α √ α2 − 4β tanh d 2β tanh d n 1 . 3.5 Based on the above initial conditions and the recursive formula 3.3 , we can derive the coefficientsU n, k one by one and obtain the approximate solution un,m t ∑m k 0 U n, k t . In this example, we set α 3, β 2 and d 0.5. The 5th-order approximate solution at n 5 is given by un,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 − 0.00445469t2 1 0.0924181046t 0.0071119169t2 . 3.7 For comparison, we plot the GDTM solutions un,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. Abstract and Applied Analysis 5and Applied Analysis 5