Comparison of Euler's- and Taylor's Expansion Methods for Numerical Solution of Non-linear System of Differential Equation

The paper deals with Euler'sand Taylor's expansion methods for next numerical solution in Matlab environment. There are many applications in technical practise described and modelled by linear or non-linear differential equation (DE) systems. A fictitious exciting functions method makes possible numerical solution of this DE system with non-stationary matrices. The solution of examples with non-linear inductance is presented as well in the paper. 1 ODE System in Matrix State-Space Form There are many applications in technical practise modelled linear or non-linear differential equation (DE) systems. Let’s have system of two first order ODEs (which can be given/rewritten as one ODE of the 2 order) d d − − = , d d − − = . The system can be also presented in matrix state-space form d̅ d = ̅ + , where (1) =  , = are systemand transition matrices, ̅ =  and = are state (state-variables)and exciting vectors, respectively. Such a linear system of ODE can be solved analytically and/or also numerically (e.g. by Euler explicit method) [1], [2], [4]. If the matrix elements are non-stationary (e.g. time dependent ones) then system of equations cannot be solved by the methods using the matrix operation as e.g. Euler implicit or Taylor expansion methods. 2 Principle of Fictitious Exciting Functions Method If and elements of matrix are non-stationary (e.g. time dependent ones) and , , and = 0 then system of Eq. (1) can be rearranged into following form [3] d̅ d = 0 0 ̅ + 1 1 0 0 0 , where = ; ; is fictitious exciting vector and ; are fictitious exciting functions, = 0 0 is modified (fictitious) state matrix, = 1 1 0 0 0 is modified (fictitious) transition matrix of the system. Let’s consider Euler’sand Taylor’s expansion methods for numerical solution of Eq. (.). We obtain a) Euler explicit method yields ̅ ! = "# + h %̅ + h where h is integration step; # is unity matrix. That method is sensitive on integration step. Stability condition is that h should be smaller than 2/|Re{λi}|max [4]. b) Euler implicit method yields ̅ ! = "# − h %& '̅ + h ( where ) = "# − h %& is fundamental matrix of the system. Contrary to above this method is for negative real part of eigenvalues absolutely stable (A-stabile) for any positive step h [4]. c) Taylor expansion yields [4] ) = exp h = h .! 0 12 and similarly ) ! = ! h ! . + 1 ! 0 12 ; 3 = & )4!5 So, choosing appropriated number of series member . one can obtain ̅ ! = )̅ + 3 The method is similarly to Euler implicit above also A-stabile one. All discrete equations carried-out by Euler explicit-, implicitand Taylor expansion methods are easily solvable by numerical computing because their modified (fictitious) matrices are stationary ones. 3 Application of all three method for 2 order electric circuit solving 2 order electric circuit with non-linear element 6 7 is depicted in Fig. 1 Figure 1: 2 order electric circuit with non-linear element 6 7 = f 9: The circuit can be described by two first order ODEs system as follow d9: d − 9: − ; = d ; d − 9: − ; = 0 where: = − <= :>?> = − @A = − = − :>?> = ;, = − <B + C ; = − @D The dynamical inductance 6 7 is non-linear function of the inductor current 6 7 = f 9: The functional dependency can be obtained directly from B-H characteristic of magnetic core of the inductor [6], [7], by measurement or using various functional linearized substitutions. For SIFERRIT U60 material [6] is that dependency shown in Fig. 2 with other functional linearized dependencies. Figure 2: Non-linear dependency 6 7 = f 9: for U 60 material [6] (a) and other functional linearized dependencies (b) in p.u. where x: 9: and y: 6 7 So, 6 7 = f 9: can be expressed by some different models: Linearized model I.: if 9: < F: 7G then 6 7 = 6HI else 6 7 = 6HI − 60 exp'−1 J K . 9: − F:HI ( + 60, where 60 = 0.5 6HI as given in Fig. 2. This model has been used for simulation experiments shown in the F Fig. 3a,b,c. Other models are referred in [5].