Solutions for Discrete Toda Equation with Homotopy Analysis Method
In this letter, we apply the homotopy analysis method (HAM) to solving the differential-difference equations, and the approximate solution for the model was obtained. HAM contains the auxiliary parameter h, which provides us with a convenient way to adjus
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Solutions for Discrete Toda Equation with Homotopy Analysis Method Xiu Rong Chen and Jia Shang Yu Abstract In this letter, we apply the homotopy analysis method (HAM) to solving the differential-difference equations, and the approximate solution for the model was obtained. HAM contains the auxiliary parameter h, which provides us with a convenient way to adjust and control convergence region and rate of solution series. The results show that the method is feasible for the discrete Toda equation studies Keywords Differential-difference equations • Homotopy analysis method • Discrete Toda equation
75.1 Introduction The nonlinear differential-difference equations (DDEs) play an important role in modeling complicated physical phenomena. However, it is usually difficult to solve them either theoretically or numerically. Up to now, many powerful methods for constructing exact solutions are proposed, such as tanh-function method [1], Jacobi elliptic function expansion method [2], Backlund transformation [3], Darboux transformation [4] and so on. Homotopy analysis method (HAM), firstly proposed by Liao [5], is a powerful analytic method for nonlinear problems. The HAM contains a certain auxiliary parameter h, which provides us with a simple way to adjust and control the convergence region and rate of convergence of the series solution and has been successfully employed to solve explicit analytic solutions for many types of nonlinear problems [6, 7]. However, HAM is a powerful and easy-to-use analytic tool to solve systems of DDEs. Here, we generalized the method to discrete Toda Equation [8, 9]. X. R. Chen (*) Department of Science and Information, Qingdao Agricultural University, Qingdao 266109, China e-mail: [email protected] J. S. Yu Dean’s Office, Heze College, Heze 274200, China e-mail: [email protected]
Z. Zhong (ed.), Proceedings of the International Conference on Information Engineering and Applications (IEA) 2012, Lecture Notes in Electrical Engineering 217, DOI: 10.1007/978-1-4471-4850-0_75, © Springer-Verlag London 2013
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X. R. Chen and J. S. Yu
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75.2 The Solution by HAM Consider the nonlinear discrete Toda equation [10]
du (t) n = u n (t)(u n−1 − u n+1 + v n − v n+1 ) dt dv n (t) = v (t)(u n n−1 − u n ) dt
with initial conditions u n (0) = −kλ cot h(kd) + kλ tan h(kdn + c) v n (0) = kλ cot h(kd) − kλ tan h(kdn + c)
(75.1)
(75.2)
whose bell wave type solution can be written as
u n (t) = −kλ cot h(kd) + kλ tan h(kdn + kλt + c) v n (t) = kλ cot h(kd) − kλ tan h(kdn + kλt + c)
(75.3)
where λ is amplitude, k, c and d are real constants. Due to the governing Eq. (75.1), we choose
L[ϕn (t, p)] =
∂ϕn (t, p) ∂t
(75.4)
As our auxiliary operator, where p ∈ [0, 1] is an embedding parameter. Note that the linear operator L has the property
L[C1 ] = 0,
L[C2 ] = 0
(75.5)
where C1 and C2 are integral constants. Furthermore, due to (75.1), we define nonlinear operators
N1 [φn (t, p)] =
∂φn − φn (φn−1 − φn+1 + ϕn − ϕn+1 ) ∂t
(75.6)
∂ϕn − ϕn (φn−1 − φn ) ∂t
(75.7)
N2 [
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