Numerical Solution of Hybrid Dynamical System using Mixture of Elzaki Transform and Differential Transform Method

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1 International Conference on Mathematical Computer Engineering - ICMCE Numerical Solution of Hybrid Dynamical System using Mixture of Elzaki Transform and Differential Transform Method M.G.Fajlul Kareem a, P.S. Sehik Uduman b a kareem@bsauniv.ac.in,b.s.abdur Rahman University, Chennai, India b sheikuduman@bsauniv.ac.in,b.s.abdur Rahman University, Chennai, India Abstract In this paper, we have solved the hybrid dynamical system of equations using Differential Transform Method. Also we have discussed all possibilities of the system variables and the results are compared with numerical solution of the system by Elzaki Transform Method. Key words: Hybrid dynamical system, Differential Transform Method, Elzaki Transform Method 1. INTRODUCTION The reality of natural science can be traced back to Isaac Newton in the seventeenth century who formulated equations related to force and motion and also co-invented calculus. This influential work paved way to inaugurate the field of dynamical systems. The origin of Mathematical modelling of dynamical phenomena is from his laws of force and motion, because calculus generally hates discontinuous and instabilities. The most classical studies of dynamical systems and Mathematical modelling are concentrated on solutions that are smooth and stable. Chaos is a Mathematical term describes complex behaviour in deterministic dynamical systems. It has a short-term predictability but is nevertheless, unstable and unpredictable in a long term. Extensive studies on chaos in the 1980s clarified that chaos is ubiquitous not only in Mathematical models but also in real-world systems. Any dynamical systems can be modelled by any of the following systems described. A linear system is a mathematical model based on the use of linear operator. A non-linear system which is not linear and does not satisfy the superposition principle or whose output is not directly proportional to its input. The third effective system is a Hybrid system, which is dynamic that exhibits both continuous and discrete dynamic behaviour that can be described by a differential equation. A hybrid system has the benefit of encompassing a larger class of systems with its structure, allowing for more flexibility in modelling dynamic phenomena. The hybrid dynamical systems a r e u b i q u i t o u s a n d not yet been formulated by a common mathematical description owing to i t s great diversity. In the previous papers, we consider only one direction of the variable and all the three directions have been carried out in order to complete the study by Differential Transform method [2, 3]. In this paper the system of equations is also solved by Elzaki transform in terms of all the three directions of the system variables are discussed, both results are compared.

2 International Conference on Mathematical Computer Engineering - ICMCE SYSTEM ANALYSIS A famous example of hybrid dynamical system [1] which can be described as follows: dx = α y x h x (2.1) dt dy = x y + z (2.2) dt dz = βy (2.3) dt where x, y, and z are continuous state variables, α and β are positive parameters and h (x ) is a piecewise smooth function of x. The cover image of this Theme Issue is composed of some strange attractors obtained from equations (2.1) (2.3) with h (x) that is more complicated t h a n the original. Consider α = 10, β = 16.82, h x = 0.55x x + 1 x DIFFERENTIAL TRNASFORM METHOD The differential transform method is one of the approximate methods which can be easily applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. The differential transformation method is a numerical method based on a Taylor expansion. This method constructs an analytical solution in the form of a polynomial. Unlike the traditional high order Taylor series method which requires a lot of symbolic computations, the differential transform method is an iterative procedure for obtaining Taylor series solutions. This method will not consume too much computer time when applying to nonlinear or parameter varying systems. This method gives an analytical solution in the form of a polynomial. But, it is different from Taylor series method that requires computation of the high order derivatives. The differential transform method is an iterative procedure that is described by the transformed equations of original functions for solution of differential equations [4, 5, 6, and 7]. 4. SOLUTION BY DIFFERENTIAL TRANSFORM METHOD Solve the system of equations (2.1), (2.2) & (2.3) by Differential Transform Method, The power series obtained is given bellow x t = x 0 + x 1 t + x 2 t 2 + x 3 t 3 + x 4 t 4 + (2.4) y t = y 0 + y 1 t + y 2 t 2 + y 3 t 3 + y 4 t 4 + (2.5) z t = z 0 + z 1 t + z 2 t 2 + z 3 t 3 + z 4 t 4 + (2.6) For the initial values x 0 = 0.01, y 0 = 0.01 & z 0 = 0.01, in equations (2.1),(2.2)& (2.3)by using differential transform method, the values obtained for x, y and z are given in the following table

3 International Conference on Mathematical Computer Engineering - ICMCE Iteration X Y Z Table 4.1-Iteration values of x, y, z By taking values for t (0.0001,0.5) and using equations(2.4),(2.5)&(2.6), the sample values obtained out of nearly 5000 values are tabulated (Table 5.1)for reference. 5. OUT PUT OF THE DYNAMICAL SYSTEM BY DTM In the power series analysis difference in the values of x, y and z are of the system variable is calculated as reference and are plotted as shown below s.no X Y Z x i+1 -x i y i+1 -y i z i+1 -z i Table 5.1

4 International Conference on Mathematical Computer Engineering - ICMCE GRAPHICAL RESULTS FOR DIFFERENTIAL TRANSFORM METHOD Fig. 1. X with X i+1 X i Fig. 2. X with Y i+1 Y i Fig. 3. X with Z i+1 Z i Fig Y with X i+1 X i Fig. 5. Y with Y i+1 Y i Fig. 6. Y with Z i+1 Z i Fig. 7. Z with X i+1 X i Fig. 8. Z with Y i+1 Y i Fig. 9. Z with Z i+1 Z i In Figures 1, 2 and 3, the plot obtained for the system graph of variable x with x i+1 -x i, y i+1 -y i and z i+1 -z i show an exponential growth in the system. Figures 3,4 and 5, show the graph of the system variable y with x i+1 -x i, y i+1 -y i and z i+1 -z i Show that the system growth is almost nil or towards the shrinking mode. Figures 7, 8 and 9 the graph of the system variable z with x i+1 -x i, y i+1 -y i and z i+1 -z i show the system behaviour along in the z direction.

5 International Conference on Mathematical Computer Engineering - ICMCE ELZAKI TRANSFORM There are numerous integral transformations methods widely used in physics and astronomy as well as in engineering. The integral transform method is an efficient method for solving the linear and non-linear systems of ordinary differential equations. The Elzaki transform easier than the Laplace transform for the beginners to understand and apply. The former transform can still serve as an auxiliary method to the latter. A lot of work has been done on the theory and applications of transforms such as Laplace-Fourier-Melin- Hankel and Sumudu, to name a few, but very little on the power series transformation or Elzaki transform, probably because it is little known, and not widely used. The Elzaki transform rivals the Laplace transform in problem Solving, its main advantage is the rivals that it may be used to solve problems without resorting to a new frequency domain because it preserve scales and units of properties. The Elzaki transform may be used to solve intricate problems in engineering, mathematics and applied science without resorting to a domain in new frequency. [8, 9, 10]. 8. SOLUTION BY ELZAKI TRANSFORM Solve the system of equations (2.1), (2.2) & (2.3) by Elzaki Transform Method, The power series obtained is given bellow x t = x 0 + x 1 t + x 2 t 2 + x 3 t 3 + x 4 t y t = y 0 + y 1 t + y 2 t 2 + y 3 t 3 + y 4 t 4 + (8.2) z t = z 0 + z 1 t + z 2 t 2 + z 3 t 3 + z 4 t 4 + (8.3) For the initial values x 0 = 0.01, y 0 = 0.01 & z 0 = 0.01, in equations (2.1),(2.2)& (2.3)by using Elzaki transform method, the values obtained for x, y and z are given in the following table Iteration X y z Table 8.1-Iteration values of x, y, z By taking values for t (0.0001,0.5) and using equations(8.1),(8.2)&(8.3), the sample values obtained out of nearly 5000 values are tabulated (Table 8.2)for reference.

6 International Conference on Mathematical Computer Engineering - ICMCE OUT PUT OF THE DYNAMICAL SYSTEM BY ELZAKI In the power series analysis difference in the values of x, y and z are of the system variable is calculated as reference and are plotted as shown below s.no X Y Z x i+1 -x i y i+1 -y i z i+1 -z i Table GRAPHICAL RESULTS FOR ELZAKI TRANSFORM. Fig. 1. X with X i+1 X i Fig. 2. X with Y i+1 Y i Fig. 3. X with Z i+1 Z i

7 International Conference on Mathematical Computer Engineering - ICMCE Fig. 4. Y with X i+1 X i Fig. 5. Y with Y i+1 Y i Fig. 6. Y with Z i+1 Z i Fig. 7. Z with X i+1 X i Fig. 8. Z with Y i+1 Y i Fig. 9. Z with Z i+1 Z i In Figures 1, 2 and 3, the plot obtained for the system graph of variable x with x i+1 -x i, y i+1 -y i and z i+1 -z i show an exponential growth in the system. Figures 3,4 and 5, show the graph of the system variable y with x i+1 -x i, y i+1 -y i and z i+1 -z i Show that the system growth is almost nil or towards the shrinking mode. Figures 7, 8 and 9 the graph of the system variable z with x i+1 -x i, y i+1 -y i and z i+1 -z i show the system behaviour along in the z direction. 10. CONCLUSION The behaviour of the Dynamical Hybrid system is analyzed with the various possibilities of system variables x,y,z,, along there mutually perpendicular directions by Differential Transform Method and Elzaki Transform Method. From the results of both methods are similar. We conclude that the graphs corresponding to the solution of Hybrid dynamical system growth is in only one direction and not in the other directions References [1]. Kazuyuki, Hideyuki Suzuki. Theory of Hybrid dynamical systems and its applications to Biological and medical systems, Philosophical Trancations, The Royal Society A,2010, 368, [2]. M. G. Fajlul Kareem and P. S. Sehik Uduman, Analysis of Hybrid Dynamical Systems Using SCBZ, International Mathematical Forum, Vol. 7, 2012, No. 23, [3]. M. G. Fajlul Kareem and P. S. Sehik Uduman, Analysis of Hybrid Dynamical Systems in space Using SCBZ Property, Applied Mathematical Sciences,Vol.7,2013, No.74, [4]. Vedat Suat Erturk, Differential Transform method for solving differential equations of lane-emden type, mathematical and computational Applications Vol.12, 2007, No.3, [5].J.Biazar and M. Eslami,Differential Transform Method for Nonlinear Parabolic-hyperbolic Partial Differential Equations, Applications and Applied Mathematics(AAM) Vol.5,No10, 2010, [6]. Montri Thongmoon, Sasitorn Pusjuso,The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations,nonlinear Analysis: Hybrid Systems,Vol. 4, _43 [7]. Javed Ali, One Dimensional Differential Transform Method for Some Higher Order Boundary Value Problems in Finite Domain,Int. J. Contemp. Math. Sciences, Vol. 7, 2012, No. 6, [8]. Tarig M. Elzaki.A Solution for Nonlinear Systems of Differential Equations Using a Mixture of Elzaki Transform and Differential Transform Method, International Mathematical Forum, Vol. 7, 2012, No. 13,

On the Connections Between Laplace and ELzaki Transforms

Adances in Theoretical and Applied Mathematics. ISSN 973-4554 Volume 6, Number (), pp. Research India Publications http://www.ripublication.com/atam.htm On the Connections Between Laplace and ELzaki Transforms