Applications of Differential Transform Method for ENSO Model with compared ADM and VIM M. Gübeş

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1 Applications of Differential Transform Method for ENSO Model with compared ADM and VIM M. Gübeş Department of Mathematics, Karamanoğlu Mehmetbey University, Karaman/TÜRKİYE Abstract: We consider some of ENSO nonlinear differential equation models to obtain approximate solutions with differential transform method. Efficiency, convergence, error rates and computational times of CPU of these approximations are compared with analytic solution and some well-known numerical methods. Also, illustrative examples are presented. All calculations in the entire article are computed commonly used MAPLE program. Keywords: Nonlinear equation, Differential Transform Method, Adomian Decomposition Method, Variational Iteration Method. 1. Introduction In the past fifty years, nonlinear problems which appeared physical phenomena, engineering applications and various of scientific areas are modelled and they are investigated by using so many computable algorithms and approximating methods. Some of these numerical methods are Homotopy Perturbation Method (HPM), Homotopy Analysis Method (HAM), Adomian Decomposition Method (ADM), Variational Iteration Method (VIM) and Differential Transformation Method (DTM). Many authors studied nonlinear models to compute approximate solutions and their convergences with these methods [1-8]. One of the nonlinear mathematical models is El Nino/La Nina Southern Oscillation (ENSO) model which is a climatic pattern that appear in the atmosphere and ocean of tropical areas. It occurs with two phases, the warm oceanic phase El Nino and the cold phase La Nina. The Natural ENSO event has a prolonged effect on the global climate. Therefore, the nonlinear ENSO models are very important and interesting subject to investigate ocean climate, atmospheric physics and dynamical systems (see more [9-15]). The Enso delayed oscillator, delay sea-air oscillator model and sea-air coupled dynamical system were solved approximately, simulated graphically with perturbation, ADM, Modified VIM, Homotopy and recently Laplace-Adomian-Pade Technique by authors in [9-15]. In this research paper, we consider some cases of ENSO Models and compute the approximate solutions and error rates of these solutions by using differential transform

2 method (DTM). Unlike the adomian decomposition method (ADM),variational iteration method (VIM) and homotopy perturbation method (HPM), the DTM transforms the nonlinear models into algebraic equations. So, we don't get into difficulties such as linearizations, integral equations, perturbations and calculations of the Adomian polynomials. Due to these advantages and the simple implementation of the DTM, we applied it to some cases of ENSO model. Numerical examples related to these models are shown to verify the efficiency of the DTM compared with the VIM. Furthermore, presented method calculated time of CPU is much shorter than ADM. Finally, these results are shown graphically. 2. Basic definitions of DTM Differential transform method is a numerical method based on Taylor expansion. This method tries to find coefficients of series expansion of unknown function by using the initial data on the problem. The concept of differential transform method was first proposed by Zhou [8] Definition: The one-dimensional differential transform of a function yx ( ) at the point x x 0 is defined [8],[1],[16] k 1 d Y( k) [ y( x)] k xx (1) 0 k! dt where yx ( ) is the original function and Yk ( ) is the transformed function Definition: The differential inverse transform of Yk ( ) is defined as follows [8][1],[16] 0 (2) k0 y( x) Y( k)( x x ) k From (1) and (2) we write down 1 d y( x) [ y( x)] x x k k x x( 0 0) k (3) k0 k! dt Related the above definitions and [1],[3],[8],[16], we present some basic properties of DTM as follow: Original function Transformed form f ( x) g( x) h( x) F( k) G( k) H( k) f ( x) cg( x) F( k) cg( k)

3 Table 1: Some properties of DTM 3. Governing Enso Equation Model and Approximate Solution by DTM The coupled dynamical ENSO system was considered to describe the oscillating physical mechanism as follows [15] (4) where and are physical constants, is temperature of the eastern equatorial Pasific sea surface and is the thermo-cline depth anomaly. Case 1: When is small enough and positive (physical constant), (4) can be rewritten (5) As the shown of [13], equation (5) has an exact solution following type (6) Let, we can applied the differential transform method (DTM) to equation (5). denote the original function of the equation (5) and denote the transformed function. By using (1),(3) and Table 1, we obtain;

4 (7) If we write computed values of (7) instead of the value of the equation (2) and with the suitable initial condition from (6), the approximate solution results (8) With the initial approximation from (6), and transform form, applying (7) in the equation (8), so we get the following approximate solution; (9) It is clearly seen that the solution (9) converges efficiently to exact solution (6) and this solution is the same as the taylor expansion of (6). The following numerical example is shown that the DTM solutions for various values of and are better than ADM and VIM solutions in terms of CPU computation time and convergence respectively. Example 1: By substituting and to equation (5), we obtain the our presented approximate solution of (5) as (10) and the ADM and VIM solutions of equation (5) are obtained respectively as

5 (11) (12) From (10),(11) and (12), we can say that the approximation with ADM and DTM is almost the same convergence, also we can say that the approximation with VIM is worse than ADM and DTM as shown in Table (2-8) and graphics (1-8). Case 2: By making some changes on the equation (4), we consider the following Enso delayed oscillator model (see more [9]); dh dt 3 1 H H (13) where is time delayed and in (4). This equation is the oscillator model which emphasizes the ocean and atmosphere interactions in the equatorial eastern Pacific and anomaly variations only in this coupling region [9]. In (13), and are positive physical variables. Again, we apply the procedures (7)-(8) to equation (13) with the same initial condition. Thus, we obtain the DTM solution of (13) as follows H ( t) t t (14) which converges efficiently to exact solution of (13). Example 2: Let, we consider the numerical form of (13) by substituting, then compute approximate solutions with respect to DTM as bellowing

6 (15) Now, by considering equations (5) and (13), for different values of, we will give some tables and graphics following to compare presented method with VIM and ADM in terms of efficiency, accuracy and calculation speed of CPU. CPU steps/ Computation times of DTM Computation times of ADM 5steps sec sec 10steps sec sec 15steps sec sec 20steps sec sec Table 2: For c=1 and =0.01, compared the Adm and Dtm solutions of Eq. (5) in terms of CPU computation times. Dtm Vim Exact ErrDtm ErrVim Table 3: For c=1 and =0.01,compared the 4th order approximation of Eq.(5) with Dtm and Vim, also given absolute errors. Dtm Vim Exact ErrDtm ErrVim Table 4: For c=1 and =0.01,compared the 13th order approximation of Eq.(5) with Dtm and Vim, also given absolute errors.

7 Dtm Vim Adm Exact Table 5: For =1, =0.5, =0.5 and =0.0001,compared the 13th order approximation of Eq.(13) with Dtm, Adm and Vim. ErrDtm ErrAdm ErrVim Table 6: Absolute error values of Table 5. Graphic 1: For c=1, ε=0.0001, compared the approximate solutions of Eq. (5) with DTM, ADM, VIM and exact solution Graphic 2: For c=1, ε= , compared the approximate solutions of Eq. (5) with DTM, ADM, VIM and exact solution

8 Graphic 3: For c=1, ε=0.0001, Relative errors of approximate solutions of Eq. (5) with DTM, ADM, VIM Graphic 4: For c=1, ε= , Relative errors of approximate solutions of Eq. (5) with DTM, ADM, VIM Graphic 5: For α=1, =0.5, σ=0.5, ε=0.001, compared the approximate solutions of Eq. (13) with DTM, ADM, VIM and exact solution Graphic 6: For α=1,, =0.5, σ=0.5, ε=0.0001, compared the approximate solutions of Eq. (13) with DTM, ADM, VIM and exact solution

9 Graphic 5: For α=1, =0.5, σ=0.5, ε=0.001, Relative errors of approximate solutions of Eq. (13) with DTM, ADM, VIM Graphic 6: For α=1, =0.5, σ=0.5, ε=0.0001, Relative errors of approximate solutions of Eq. (13) with DTM, ADM, VIM 4. Conclusion Differential transform method(dtm) has been successfully applied to the some cases of ENSO nonlinear models. Analytic solutions, DTM, VIM and ADM solutions are compared in terms of accuracy, time elapsed for the calculations and efficiency for various range of time (t). We have been obtained fairly good results comparison to VIM and it has been clearly seen that DTM much faster than ADM. In conclusions, given examples have been shown the effectiveness, accuracy and convergence of the numerical solutions of DTM. 5. References [1] Y Keskin, A Kurnaz, Μ. E Kiris, G Oturanc, "Approximate solutions of generalized pantograph equations by the differential transform method", International Journal of Nonlinear Sciences and Numerical Simulation VOL. 8 (2), [2] J. H. He, "A variational iteration approach to nonlinear problems and its applications", Mech. Appl. 20, 30-31, [3] Haldun Alpaslan Peker, Onur Karaoğlu and Galip Oturanç, "The Differential Transformation Method and Pade Approximant for a Form of Blasius Equation", Mathematical and Computational Applications, Vol. 16, No. 2, pp , 2011.

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