Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach

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1 Commun. Theor. Phys. 57 (2012) 5 9 Vol. 57, No. 1, January 15, 2012 Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach G. Darmani, 1, S. Setayeshi, 2 and H. Ramezanpour 2 1 Department of Electrical Engineering, Sadjad Institute of Higher Education, Mashhad, Iran 2 Department of Nuclear Engineering and Physics, Amirkabir University of Technology, Tehran, Iran (Received June 7, 2011; revised manuscript received September 22, 2011) Abstract In this paper an efficient computational method based on extending the sensitivity approach (SA) is proposed to find an analytic exact solution of nonlinear differential difference equations. In this manner we avoid solving the nonlinear problem directly. By extension of sensitivity approach for differential difference equations (DDEs), the nonlinear original problem is transformed into infinite linear differential difference equations, which should be solved in a recursive manner. Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained. Numerical examples are employed to show the effectiveness of the proposed approach. PACS numbers: x Key words: differential difference equation, sensitivity approach, exact solution 1 Introduction Nonlinear differential-difference equations (NDDEs) play an important role in the study of modern physics and also in numerical simulation of nonlinear partial differential equations, queuing problems, discretization in solid state and quantum physics. Furthermore, modelling of many phenomena in different fields, ranging from condensed matter and biophysics to mechanical engineering is based on NDDEs, e.g. atomic chains [1 2] with onsite cubic nonlinearities, molecular crystals, [3] biophysical systems, [4] electrical lattices, [5] and nonlinear optical wave guides. [6 7] Unlike difference equations, which are fully discretized, DDEs are semi-discretized, with some (or all) of their spatial variables discretized, while the time variable is usually kept continuous. At the same time, finding exact solutions of DDEs is extremely important in mathematical physics. Some methods for solving DDEs are the inverse scattering method, [8] the Hirota bilinear method, [9] the variables separate method, [10] Backlund transformation, [11] Darboux transformation and other methods. [12 17] Wealthy information about integrable differentialdifference equations can be found in papers by Suris, [18 20] and a lot of works were developed to analyze the properties of solutions of DDEs. [21 24] In this paper we extend sensitivity approach which has been presented in [25 27] to solve various kinds of optimal control problems and analysis of time delay systems. In this approach, by introducing a sensitivity parameter, the original nonlinear DDE is transferred into a linear sequence of DDEs. The response of equation consists of an accurate linear term and a nonlinear compensating series. Iterations are required only for nonlinear compensation series, i.e., the solution of a sequence of linear DDEs leads to nonlinear term for compensation. The rest of Paper is organized as follows: In Sec. 2, the problem is formulated. In Sec. 3, the extended SA is presented for solving ND- DEs. In Sec. 4, we choose two nonlinear DDEs, namely, the Volterra equation, mkdv equation, to illustrate the validity and advantage of this method. Finally, the conclusion and discussion are given in Sec Problem Formulation Consider the nonlinear DDEs in the form of: t y n(t) = N D (f(y )), y n (1) in which, f is a nonlinear polynomial function, y is an unknown function, g(n) is the initial condition and t, n are independent variables. Difference operator also is defined as follows: N D = b m D m + b m 1 D m 1 + b m 2 D m b 0, (2) in which D denotes the difference with respect to the n, and b i : i = 0, ±1,..., ±m are known constant coefficients. In the following we assume that a unique solution is existed for (1). Generally, it is difficult to obtain exact solution of DDE (1). In most cases, only approximate solutions (either numerical solutions or analytical solutions) can be expected. In the next section we extend the SA to find an exact solution of nonlinear DDE (1) analytically. Corresponding author, ghazal.darmani@gmail.com c 2011 Chinese Physical Society and IOP Publishing Ltd

2 6 Communications in Theoretical Physics Vol Extended Sensitivity Approach In [25] SA has been used to solve nonlinear two point boundary value problems arisen in optimal control problems. In this approaches a sensitivity parameter ε, which varies between zero and unity, is embedded into nonlinear terms of differential difference equations. When ε = 0 the nonlinear problem deform to a simple problem (and most of the cases linear) which has an analytic solution. Also when ε = 1 the original nonlinear problem is obtained. This deformation leads to solving a sequence of linear DDEs instead of solving a nonlinear DDE. To clear up this let rewrite (1) as follows: where t y n(t) = M D (y n ) + M D (Ψ(y )), y n (3) M D = a m D m + a m 1 D m 1 + a m 2 D m a 0, M D = ā m D m + ā m 1 D m 1 + ā m 2 D m ā 0, (4) in which D denotes the difference with respect to the n, and ā, a i : i = 0, ±1,..., ±m are known constant coefficients. Now we introduce a sensitivity parameter ε in (3) and construct the following sensitized DDE: t y n(t, ε) = M D (y n (t, ε)) + ε M D (Ψ(y n (t, ε))), t 0 t, y n (t, ε) t=t0 = g(n), (5) where 0 ε 1 is a scalar. In the following discussion, we always assume that the solution of (5) is uniquely existed and y n (t, ε) with ε is infinitely differentiable with respect to the ε around ε = 0, and its Maclaurin series is convergent at ε = 1. Obviously when ε = 1 (5) is equivalent to the original problem (1). According to the assumption we can write: ε i y n (t, ε) =, (6) where () (i) = 1 i () i! ε i ε=0. Now by substituting (6) into (5) and equating terms with the same order of ε on each side we have: ε 0 t y(0) = M D (y n (0) (t)), y n (0) (7a) ε 1 t y(1) = M D (y n (1) (t)) + h (0), y n (1) (7b) (t 0 ) = 0, t y(i) = M D (y n (i) (t)) + h (i 1), y (i) (7c) where h (i 1) (t) is the coefficient of ε i 1 in the expanding of g and can be determined in the following manner: h (i 1) (t) = 1 i 1 (M D (Ψ(y n (t, ε)))) ε=0 (i 1)! ε i 1. (8) It should be noted that (7a) yields linear approximate solution to the equation, (7b) yields correction term to linear approximate solution by considering second order nonlinearity and so on. Notice that if the above process caries on, at each step, a system of inhomogeneous linear DDE s is obtained in which inhomogeneous terms are known from the previous step. Thus, solving the presented sequence is a recursive process. After indentifying y (i) (t) for i 0, it is obvious that ε = 1 should be sat in (5) and (6) so that they deform to the exact solution of (1) and so we have: y n (t, 1) = y (i). (9) Thus the original nonlinear DDE has been transformed into a sequence of linear DDEs, which should be solved in a recursive process and this overcomes the difficulty of working with nonlinear DDEs. Remark It is true that by using sensitivity approach, a nonlinear DDE is transformed into a sequence of linear DDEs, but solving this sequence is also very difficult except in a few simple cases. Thus for overcoming this difficulty, each sub-problem of presented sequence in (7), can be solved iteratively, again by using SA. In this case we avoid solving each linear DDE directly and a simple integration with respect to variable t is remedial. For example, in order to solve the first equation in (7a), the following sensitized linear DDE is constructed: By assuming t y(0) n (t, ε) = ε M D (y n (0) (t, ε)), n (t 0, ε) = g(n). (10) n (t, ε) = ε j y (0,j). (11) Now by substituting (11) into (10) and equating terms with the same order of ε on each side we have: ε 0 t y(0,0) = 0, y (0,0) (12a) ε 1 ε j n t y(0,1) = M D (y n (0,0) (t)), y (0,1) t y(0,j) n y (0,j) (t) = M D (y n (0,j 1) (t)), (12b) (12c) Presented sequence, just needs simple integration to be solved:

3 No. 1 Communications in Theoretical Physics 7 ε 0 ε 1 ε j t y(0,0) = 0, y n (0,0) t y(0,1) = M D (y n (0,0) (t)), y (0,1) y n (0,0) (t) = g(n), t y(0,j) = M D (y n (0,j 1) (t)), y 0,(j) y (0,1) = M D (g(n))t, y n (0,j) (t) = M j tj D (g(n)) j! (13) in which M j D () = M D ( M D (... ())). {{ j times Finally we have: y n (0) (t) = y n (0) (t, 1) = y n (0,j) (t) = M j tj D (g(n)) j!. (14) Similarly, for y (i) and i 1, such a sub-problem like (7c) is needed to be solved. After some similar calculation we have: y n (i) (t) = y(i,0) n + y n (i,1) + y n (i,2) + y n (i,3) + (15) In which: y n (i,0) = 0, y n (i,1) = h (i 1) dt, y (i,j) n = MD (y (i,j 1) n )dt, j 2. (16) Although these steps seems to be enormous, but just few iteration of sub-problems and original problem is enough to get an acceptable accurate solution. Most of the times just two iterations of problem and sub-problems yield a high level of accuracy. Remark Since for finding the exact solution of y, infinite terms in series (7) and (15) is needed and it is almost impossible, thus in practiced applications by replacing with finite positive integers Nand K in series (7) and (15) we may obtain an approximate closed-form solution, i.e., y K N = y n (i,j) (t). (17) We can gain a more accurate solution with increasing the numbers N and K. 4 Examples Example 1: Volterra Equation Consider the following Volterra equation: [23] y = y (y n+1 (t) y n 1 (t)), t y n (0) = n, (18) whose the exact solution can be written as y = n/(1 2t). For solving this equation, the following new equation is constructed with sensitivity parameter: y n (t, ε) = ε y n (t, ε)(y n+1 (t, ε) y n 1 (t, ε)), t y n (0, ε) = n. (19) Now assume: y n (t, ε) = y n 1 (t, ε) = y n+1 (t, ε) = ε i y n (i) (t), (20) n 1 (t), (21) n+1 (t). (22) Substituting (20), (21), and (22) into (19) and equating the terms with the same power of ε yields: ε 0 t y(0) = 0, y(0) n (0) = n, (23a) ε 1 t y(1) = y(0) (y(0) n+1 (t) y(0) n 1 {{ (t)), y n (1) (0) = 0, h (0) (23b)

4 8 Communications in Theoretical Physics Vol. 57 ε 2 t y(2) = y(0) (y(1) n+1 (t) y(1) n 1 (t)) + y(1) (y(0) n+1 (t) y(0) n 1 {{ (t)), y n (2) (0) = 0, h (1) It is seen that the nonlinear original DDE was transformed into a set of linear recursive DDEs in which at each step, the imhomogeneous terms are calculated from the previous steps and this procedure can be handled very easily just with a paper and pencil. In this example, because there is no linear difference terms in original nonlinear DDE, i.e., {ā i = 0 : i = 0, ±1,..., ±m, consequently sequence (23) is solved just by simple integration and there is no need to solve a linear DDE sub-problem for each equation of (23). Thus, solving (23) yields: y = n {{ + {{ 2nt + 4nt {{ 2 y (1) which is the exact solution. Example 2: mkdv Lattice Equation Let us consider the hybrid nonlinear DDE: [28] y t y (2) (23c) + = n(1 + 2t + 4t 2 + ) = n 1 2t, (24) = (1 y ) 2 (y n+1 (t) y n 1 (t)), y n (0) = tanh(k)tanh(kn). (25) The hybrid nonlinear difference equation (25), describes the discretization of the Korteweg-de Vries (KdV) equations whose exact solution can be written as: Equation (25) can be written as follows: y t y = tanh(k)tanh(kn + 2 tanh(k)t). (26) = (y n+1 (t) y n 1 (t)) y 2 (y n+1 (t) y n 1 (t)), y n (0) = tanh(k)tanh(kn). (27) {{{{ M D(y n(t)) M D(Ψ(y n(t))) Now by using (7) the resulted DDE sequence is obtained as: ε 0 t y(0) = ( n+1 (t) y(0) n 1 (t)), y n (0) (0) = tanh(k) tanh(kn), ε 1 t y(1) = (y (1) n+1 (t) y(1) n 1 (t)) + h(0), y n (1) (0) = 0, Equation (28a) is a linear DDE which should be solved according to (12) (14). After that h (0) = (y(0) n )2 ( n+1 (t) y(0) n 1 (t)) is determined by using (8) and then (16) is calculated for i = 1. Tables 1 and 2 show the numerical approximate solution by choosing N = 3 and K = 1 with a high degree of accuracy. Figures 1 and 2 are also illustrative the simulated results agree very well to the exact solution. Table 1 For constant k = 0.1, and time t = 0.5. n Exact solution Approximate solution Absolute error (28a) (28b)

5 No. 1 Communications in Theoretical Physics 9 Table 2 For constant k = 0.1, and time t = 1.5. n Exact solution Approximate solution Absolute error Fig. 1 Comparison of the exact solution and approximate solution obtained by SA for k = 0.1, and time t = 0.5. Fig. 2 Comparison of the exact solution and approximate solution obtained by SA for k = 0.1, and time t = Conclusion In this paper, by extension of the SA for nonlinear DDEs, firstly, we obtain the exact solution of Volterra equation. Secondly, we obtain the approximate solution of mkdv lattice equation. This method is a powerful tool that enables one to search for solutions of various linear and nonlinear problems. Also, the method is extremely simple, easy to use, and very accurate for solving nonlinear DDEs. Comparisons are made between the results of the proposed method and exact solutions. The results show that the SA is an attractive method for solving DDEs. References [1] A.C. Scott and L. Macneil, Phys. Lett. A 98 (1983) 87. [2] A.J. Sievers and S. Takeno, Phys. Rev. Lett. 61 (1988) 970. [3] W.P. Su, J.R. Schrieffer, and A.J. Heeger, Phys. Rev. Lett. 42 (1979) [4] A.S. Davydov, J. Theor. Biol. 38 (1973) 599. [5] P. Marquii, J.M. Bilbault, and M. Rernoissnet, Phys. Rev. E 51 (1995) [6] H.S. Eisenberg, Y. Silberberg, R. Moraudotti, A. Boyd, and J. Aitchison, Phys. Rev. Lett. 81 (1998) [7] H.S. Eisenberg, Y. Silberberg, R. Moraudotti, A. Boyd and Y. Silberberg, Phys. Rev. Lett. 83 (1999) [8] T. Tsuchida, H. Ujino, and M. Wadati, J. Phys. A: Math. Gen. 32 (1999) [9] R. Hirota, The Direct Method in Soliton Theory, ed. and transl. A. Nagai, Cambridge University Press, Cambridge (2004). [10] X.M. Qian, S.Y. Lou, and X.B. Hu, J. Phys. A: Math. Gen. 37 (2004) [11] W.X. Ma and X.G. Geng, CRM Proc. Lect. Notes 29 (2001) 313. [12] A. Yildirim, Int. J. Comput. Math. 29 (2008) [13] A. Wang, L. Zou, and H.H. Zhang, Phys. Lett. A 369 (2007) 77. [14] Y. Ahmet, Math. Prob. Eng. doi: /2008/ [15] C. Dai, J. Zhang, Chaos, Solitons & Fractals 27 (2006) [16] A. Yildirim, Int. J. Non. Sci. Numer. Simul. 9 (2008) 111. [17] Z. Wang, Comput. Phys. Commun. 180 (2009) [18] Yu. B. Suris, J. Phys. A: Math. Gen. 30 (1997) [19] Yu. B. Suris, J. Phys. A: Math. Gen. 30 (1997) [20] Yu. B. Suris, Rev. Math. Phys. 11 (1999) 727. [21] D.J. Zhang, Chaos, Solitons & Fractals 23 (2005) [22] K. Narita, Chaos, Solitons & Fractals 3 (1993) 279. [23] Z. Wang and H.Q. Zhang, Chin. Phys. 15 (2006) [24] Z. Wang and H.Q. Zhang, Appl. Math. Comput. 178 (2006) 431. [25] G. Tang, N. Xie, and P. Liu, Proc. of ICSMC 99 5 (1999) 104. [26] M. Malek-Zavarei and M. Jamshidi, Time Delay Systems: Analysis, Optimization and Applications, North Holland, New York (1987). [27] P.P. Khargonkar and K. Zhou, Proc. of the 25th IEEE CDC, Athens, Greece, 3 (1986) [28] M.J. Ablowitz and J.F. Ladic, Stud. Appl. Math. 57 (1977) 1.