A Numerical Computation of a System of Linear and Nonlinear Time Dependent Partial Differential Equations Using Reduced Differential Transform Method

Hindawi Publishing Corporation International Journal of Differential Equations Volume 2016, Article ID 4275389, 8 pages http://dx.doi.org/10.1155/2016/4275389 Research Article A Numerical Computation of a System of Linear and Nonlinear Time Dependent Partial Differential Equations Using Reduced Differential Transform Method Brajesh Kumar Singh and Mahendra Department of Applied Mathematics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow, Uttar Pradesh 226025, India Correspondence should be addressed to Brajesh Kumar Singh; Received 20 April 2016; Revised 18 September 2016; Accepted 4 October 2016 Academic Editor: Davood D. Ganji Copyright © 2016 B. K. Singh and Mahendra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper deals with an analytical solution of an initial value system of time dependent linear and nonlinear partial differential equations by implementing reduced differential transform (RDT) method. The effectiveness and the convergence of RDT method are tested by means of five test problems, which indicates the validity and great potential of the reduced differential transform method for solving system of partial differential equations. 1. Introduction The reduced differential transform method has been successfully employed to solve various types of linear and nonlinear, homogeneous or nonhomogeneous, equations appearing in science and engineering. Partial differential equations have also been applied in modeling many physical engineering problems and differential equations in nonlinear dynamics [1–3]. Most of the partial differential equations cannot be solved exactly, and so, developing schemes for getting accurate and efficient numerical solution differential equations have been an active research area. Burgers’ equation [4], a system of nonlinear fractional differential equations [1], and nonlinear Klein–Gordon equation with a quadratic nonlinear term [2] have been solved using Adomian decomposition method. A system of nonlinear fractional partial differential equations has been solved using homotopy analysis method by Jafari and Seifi [3] and Bataineh et al. [5], using variational iteration method by Wazwaz [6]. In [7], Wang and Cheng adopted variational method and finite element approach to solve damped nonlinear Klein–Gordon equations. The coupled Burgers equation has been solved by using various schemes; among them are variational iteration method [8], Adomian–Pade technique [9], fourth-order compact schemes [10], a composite numerical scheme [11], lattice Boltzmann method [12], finite element and finite difference method [13], two algorithms based on cubic spline function technique [14], a robust finite difference scheme [15], and modified extended cubic B-spline differential quadrature method [16]; while using modified cubic B-spline differential quadrature method Burgers’ and Burgers-Huxley equations have been solved in [17–19], respectively. The fractional models of Burgers equation have been solved in [19–22]. The space- and time-fractional coupled Burgers equations have been solved using generalized differential transform method [19] and homotopy perturbation method [21]. Reduced differential transform method is used to solve (1 + 𝑛)-dimensional Burgers’ equation [20]. Recently, Prakash et al. [22] adopted fractional variational iteration method to solve fractional coupled Burgers equations. Keskin and Oturanç [23] have developed reduced differential transform method to solve partial differential equations of integer order [24] as well as fractional order. After Keskin and Oturanç, RDT method has been implemented for the numerical computation of various physical models of engineering and sciences [25–27]. 2 International Journal of Differential Equations The main goal of this paper is to provide an analytical solution of initial value system of time dependent partial differential equations obtained by employing RDT method developed by Keskin and Oturanç [23]. 2. Reduced Differential Transform Method The basic properties of the fractional reduced differential transform method are described in this section. Let 𝜓(𝑥, 𝑡) be a function of two variables such that 𝜓(𝑥, 𝑡) = 𝑓(𝑥)𝑔(𝑡). By using the properties of the one-dimensional differential transform (DT) method 𝜓(𝑥, 𝑡) can be written as ∞ ∞ 𝑖 𝑗 ∞ ∞ 𝑖 𝑗 𝜓 (𝑥, 𝑡) = ∑𝑓 (𝑖) 𝑥 ∑𝑔 (𝑗) 𝑡 = ∑∑Ψ (𝑖, 𝑗) 𝑥 𝑡 , 𝑖=0 𝑗=0 𝑖=0 𝑗=0 (1) where Ψ(𝑖, 𝑗) is referred to as the spectrum of 𝜓(𝑥, 𝑡) and is defined by Ψ (𝑘, ℎ) = 𝜕𝑘+ℎ 𝜓 (𝑥, 𝑡) 1 ) . ( ℎ!𝑘! 𝜕𝑥𝑘 𝜕𝑡ℎ (𝑥 ,𝑡 ) (2) 0 0 For more details on DT method, see [28] and the references therein. Denote the lowercase 𝜓(𝑥, 𝑡) as the original function while its fractional reduced transformed function is denoted by the uppercase Ψ𝑘 (𝑥). Definition 1. If 𝜓(𝑥, 𝑡) is analytic and continuously differentiable with respect to 𝑥 and 𝑡, then RDT of 𝜓 is given by 𝑈𝑘 (𝑥) = 1 𝜕𝑘 𝑢 (𝑥, 𝑡) ] . [ 𝑘! 𝜕𝑡𝑘 𝑡=0 Property 1. If 𝑤(𝑥, 𝑡) = 𝑢(𝑥, 𝑡)±V(𝑥, 𝑡), then 𝑊𝑘 (𝑥) = 𝑈𝐾 (𝑥)± 𝑉𝑘 (𝑥). Property 2. If 𝑤(𝑥, 𝑡) = 𝛼𝑢(𝑥, 𝑡), then 𝑊𝑘 (𝑥) = 𝛼𝑈𝑘 (𝑥). Property 3. If 𝑤(𝑥, 𝑡) = [𝑥𝑚 𝑡𝑛 ], then 𝑊𝑘 (𝑥) = 𝑥𝑚 𝛿(𝑘 − 𝑛), where {1, when 𝑘 = 𝑛, 𝛿 (𝑘 − 𝑛) = { 0, when 𝑘 ≠ 𝑛. { Property 4. If 𝑤(𝑥, 𝑡) 𝑥𝑚 𝑈𝑘−𝑛 (𝑥). = [𝑥𝑚 𝑡𝑛 𝑢(𝑥, 𝑡)], then 𝑊𝑘 (𝑥) (6) = Property 5. If 𝑤(𝑥, 𝑡) = [𝜕𝑟 𝑢(𝑥, 𝑡)/𝜕𝑡𝑟 ], then 𝑊𝑘 (𝑥) = (𝑘 + 1)(𝑘 + 2) ⋅ ⋅ ⋅ (𝑘 + 𝑟)𝑈𝑘+𝑟 (𝑥) = ((𝑘 + 𝑟)!/𝑘!)𝑈𝑘+𝑟 (𝑥). Property 6. If 𝑤(𝑥, 𝑡) = [𝜕𝑢(𝑥, 𝑡)/𝜕𝑥], then 𝑊𝑘 (𝑥) = 𝜕𝑈𝑘 (𝑥)/ 𝜕𝑥. Property 7. If 𝑤(𝑥, 𝑡) ∑𝑘𝑖=0 𝑈𝑖 (𝑥)𝑉𝑘−𝑖 (𝑥). = 𝑢(𝑥, 𝑡)V(𝑥, 𝑡), then 𝑊𝑘 (𝑥) = Property 8. If 𝑤(𝑥, 𝑡) = [𝑢(𝑥, 𝑡)]𝑚 , then 𝑘 = 0, 𝑈0 (𝑥) , { { { 𝑊𝑘 (𝑥) = { 𝑘 (𝑚 + 1) 𝑛 − 𝑘 (7) { {∑ 𝑈𝑛 (𝑥) 𝑊𝑘−𝑛 (𝑥) , 𝑘 ≥ 1. {𝑛=1 𝑘𝑈0 (𝑥) (3) For details on RDT method we refer the readers to [23–25]. The reduced inverse differential transform of 𝑈𝑘 (𝑥) is defined as follows: ∞ 𝑢 (𝑥, 𝑡) = ∑ 𝑈𝑘 (𝑥) 𝑡𝑘 . 2.1. Some Basic Properties and Notation of RDT Method. In this section, the properties of RDT method as in [23–25] have been revisited to complete our study. (4) 3. Results and Discussion In this section, we give five test problems of linear and nonlinear partial differential equations (PDEs) using reduced differential transform (RDT) method. 𝑘=0 Example 2. Consider the initial value system of linear PDEs: Equations (3) and (4) together reduce to 𝜕𝑢 (𝑥, 𝑡) 𝜕V (𝑥, 𝑡) = − V (𝑥, 𝑡) − 𝑢 (𝑥, 𝑡) , 𝜕𝑡 𝜕𝑥 ∞ 1 𝜕𝑘 𝑢 (𝑥, 𝑡) ) 𝑡𝑘 . ( 𝑘 𝑘! 𝜕𝑡 𝑡=0 𝑘=0 𝑢 (𝑥, 𝑡) = ∑ (5) The basic properties of RDT method are found in [1, 4] and can be deduced from (3) and (4), given in the following. 𝜕V (𝑥, 𝑡) 𝜕𝑢 (𝑥, 𝑡) = − V (𝑥, 𝑡) − 𝑢 (𝑥, 𝑡) , 𝜕𝑡 𝜕𝑥 𝑢 (𝑥, 0) = sinh (𝑥) , V (𝑥, 0) = cosh (𝑥) . (8) International Journal of Differential Equations 3 On using RDT method (8) reduces to a set of recurrence relations as follows: (1 + 𝑘) 𝑈𝑘+1 (𝑥) = 𝜕 𝑉 (𝑥) − 𝑉𝑘 (𝑥) − 𝑈𝑘 (𝑥) , 𝜕𝑥 𝑘 (1 + 𝑘) 𝑉𝑘+1 (𝑥) (...truncated)