Adomian s Decomposition Method for Solving Singular System of Transistor Circuits

Transcription

1 Applied Mathematical Sciences, Vol. 6, 2012, no. 37, Adomian s Decomposition Method for Solving Singular System of Transistor Circuits K. Krishnaveni veni.fairy@gmail.com S. Raja Balachandar srbala09@gmail.com S. K. Ayyaswamy School of Humanities and Sciences SASTRA University, Thanjavur, India sjcayya@yahoo.co.in Abstract In this paper Adomian s Decomposition Method (ADM) is employed to solve observer design in generalized state space or singular system of transistor circuits. From the example, the solution of singular system of transistor circuits shows that the ADM is more simple and easy to use than Haar Wavelet and Single Term Walsh Series (STWS) method. Mathematics Subject Classification: 34A12, 34A30, 34K07 Keywords: Adomian s Decomposition Method; transistor circuits; Singular matrix 1 Introduction Adomian s Decomposition method (ADM) is an useful and powerful method for solving linear and nonlinear differential equations [5-10]. The Adomian goal is to find the solution of linear and nonlinear, ordinary or partial differential equation without dependence on any small parameter like perturbation method. In this method the solution is considered as a sum of an infinite series which rapidly converges to an accurate solution. A great deal of interest has been focused on the application of ADM to solve a wide variety of stochastic and terministic problems [10]. The algorithm for ADM is presented in [15].

2 1820 K. Krishnaveni, S. Raja Balachandar and S. K. Ayyaswamy The efficiency of the ADM has been compared with other techniques and is presented in the literature. Bellomo and Monaco[13] have compared the ADM and the perturbation technique when they are used in solving random nonlinear differential equations. Rach[30] has showed the advantages of ADM over the Picard s method. He showed that the two methods are not the same and the Picard s method works only if the equation satisfies Lipschitz condition. The comparison of ADM and Runge Kutta methods are studied by Edwards [21] for solving some predator prey model equations. Wazwaz[32] introduced a comparison between the ADM and the Taylor series method. Also he showed that the ADM minimizes the computational difficulties of the Taylor series in that the components of the solution are determined elegantly by using simple integrals. Moreover, he showed that the ADM is fast convergent and it attacks nonlinear problems in a similar manner as linear problems. Recently, the ADM has been used also to solve Higher-order ordinary differential equations [22], Hyperbolic equations with an integral condition [14], Modified regularized long wave equation [26], Coupled Burgers equations [20] and Three-dimensional electrodes [31]. The theoretical treatment of the convergence of Adomian s method is considered in [1-4, 16-19, 23, 27, 29]. In the present paper, we employ ADM to solve singular system of transistor circuits. Kang[25] and Lewis[28] discussed the simplest model for a transistor in the circuit. Balachandran [11,12] and Kalpana[24] have applied the Single Term Walsh Series(STWS) technique and Haar Wavelet method respectively to solve the model for a transistor presented in [25,28]. This paper is organized as follows: - In section 2, we review the ADM to be used to solve the singular system of Transistor circuits. Section 3 deals with an example and the concluding remarks are given in the last section. 2 Analysis of singular systems using ADM Consider the linear singular system K Ẋ(t) =AX(t)+BU(t) (1) Y (t) =CX(t),X(0) = X 0 (2) With X R n,u R m,y R p,k generally a singular matrix. Y (t) is a p output vector. A, B and C are to be chosen as constant matrices with appropriate dimensions. X(t) isan n state vector, U(t) isan m input vector. Next we discuss the ADM algorithm to solve (2) Let L(.) = d (.) then dt L 1 (.) = t 0 (.)dt, equation (1) can be written as KLX(t) =AX(t)+BU(t) (3)

3 Adomian s decomposition method 1821 By applying L 1 to both sides of (3), we have KX(t) = K X(0) + AL 1 (N(x 1,...,x n )) + BL 1 (U(t)) (4) The Adomian decomposition method gives the solution X (t) and the linear function N(x 1,...,x n ) by infinite series X i (t) = X im (t) and N(x 1,...,x n )= A im (5) m=0 m=0 where X im s are the components of the solution X i which are determined recurrently. To formulate Adomian polynomials aim we use the following relation: A im (λ) = 1 [ ] d m m! dλ N i(x m 1 (λ),...,x n (λ)) (6) where x i (λ) = j=0 λ j x ij A i0 (t) =x i0 (t) i =1,...,n. x i0 (t) = K X(0) + BL 1 (U(t)) x im+1 (t) =AL 1 (A im (λ)), n 0, where A is the coefficient matrix of X(t). λ=0 3 Analysis of transistor circuits Consider the simplest model for a transistor in the circuit shown in Figure 1 discussed by Kang and Lewis [25, 28]. The circuit equations are given by u 1 + x 1 + r 1 x 2 =0 u 2 + x 3 + r 2 (x 4 α 1 x 2 )=0 x 2 = c 1 ẋ 1 x 4 = c 2 ẋ 3 y 1 = r 2 (α 1 x 2 x 4 ) y 2 = r L α 2 x 4 (7) Here x 1,x 2,x 3 and x 4 are the capacity voltages. Assuming r 1 = r 2 = r L = α 1 = α 2 = c 1 = c 2 =1, we get a singular system. KẊ(t) =AX + BU Y = CX (8)

4 1822 K. Krishnaveni, S. Raja Balachandar and S. K. Ayyaswamy Figure 1: Model for a transistor circuit where is a singular matrix, K = A = C = B = [ U =[1 0] T and X(0) = [ ] T From the above, we get the relations ] x 1 + x = 0 (9) x 3 + x 1 + x = 0 (10)

5 Adomian s decomposition method 1823 and x 1 + x 2 +1=0; x 2 + x 3 + x 4 =0 In an operator form, Eq. (9) and (10) becomes Lx 1 = 1 x 1 (11) Lx 3 = 1 x 1 x 3 (12) By applying L 1 to both sides of (11) and (12), we have x 1 n+1 = L 1 (A 1 n); x 1 = x 10 + x 11 + x 12 + x and x 10 = t x 3 n+1 = L 1 (A 1 n) L 1 (A 3 n); x 3 = x 30 + x 31 + x 32 + x and x 30 =1 t Proceeding as before we obtained the recursive relationship A 1n = 1 n! [ d n dλ n x 1 (λ) ] λ=0; x 1 (λ) =x 10 (λ)+x 11 (λ 2 )+x 12 (λ 3 )+x 13 (λ 4 )+... A 3n = 1 n! [ d n dλ n x 3 (λ) ] λ=0; x 3 (λ) =x 30 (λ)+x 31 (λ 2 )+x 32 (λ 3 )+x 33 (λ 4 )+... The Adomian polynomials can be determined in the form x 10 = t x 11 = t 0 A 10 dt = t2 2 x 12 = t 0 A 11 dt = t3 6 x 13 = t 0 A 12 dt = t4 24 ;... (13) x 30 =1 t x 31 = t 0 A 10 dt t 0 A 30 dt = t 2 t x 32 = t 0 A 11 dt t 0 A 31 dt = t2 t3 2 2 x 33 = t 0 A 12 dt t 0 A 32 dt = t4 6 t3 6 ;... From Eqs.(13), the solution in a series form is given by (14) x 1 = t + t2 2! t3 3! + t4 4! +... = e t 1 (15)

6 1824 K. Krishnaveni, S. Raja Balachandar and S. K. Ayyaswamy From Eqs.(14), the solution in a series form is given by x 3 =1 t + t 2 t + t2 2 t3 2 + t4 6 t = e t (1 t) (16) In view of Eqs.(15) and (16), we obtain the solution x 2 = e t, x 4 = e t (t 2), y 1 = e t (1 t) and y 2 = e t (t 2) Hence ADM provides exact solutions to singular system of transistor circuits. But Haar Wavelet [24] and Single Term Walsh Series(STWS) [11,12] have taken more computational time to calculate the connection coefficients. 4 Conclusion In this paper, we employed the ADM techniques to solve the singular system of transistor circuits. The solutions can be obtained in explicit form by using ADM which confirms our belief that solving singular system of transistor circuits can be reached by sufficiently simple calculations. The study shows that the ADM is more simple and easy to use than Haar Wavelet and Single Term Walsh Series(STWS) method. References [1] K. Abbaoui,Y. Cherruault, Convergence of Adomians method applied to nonlinear equations,math Comput Modell,20(9) (1994), [2] K. Abbaoui,Y. Cherruault, Convergence of adomians method applied to differential equations. Comput Math Appl,28(5) (1994), [3] K. Abbaoui,Y. Cherruault, New ideas for proving convergence of decomposition methods, Comput Math Appl,29(7) (1995), [4] K. Abbaoui,MJ. Pujol,Y. Cherruault,N. Himoun,P. Grimalt, A new formulation of Adomian method convergence results, Kybernetes,30 (2001), [5] G.Adomian, New approach to nonlinear partial differential equations,j Math Anal Appl,102 (1984), [6] G. Adomian, A review of the decomposition method in applied mathematics,j Math Anal Appl,135 (1988), [7] G.Adomian, Nonlinear stochastic systems theory and applications to physics,dordrecht: Kluwer, (1989).

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