The Matrix Padé Approximation in Systems of Differential ... - WSEAS

WSEAS TRANSACTIONS on MATHEMATICSC. Pestano-Gabino, C. Gonz_Lez-Concepcion, M.C. Gil-Farinasome rational elements and others which are not. Nextwe proceed to illustrate different cases.Our aim in this section is to detect rationalfundamental matrices (which are rectangular when theorder of the system is greater than 1). In other case, ifthis is not possible, then to detect particular rationalcolumns or particular rational solutions. If F(t) isrational then any solution of (1) is rational complyingwith the aims. However, when F(t) is not rational andis in analytical form it would be interesting to know ifthere exist any initial conditions, such that theassociated particular solution to be rational. It is noteasy to find the answer to this question in generalcases.We start with an example where the fundamentalmatrix is not rational but the system does have rationalsolutions.Example 1. Let Y''(t)=C(t) Y(t) be a differential2⎛ 1 t + 2 ⎞⎜ 2 3 ⎟(2 t) (2 t)system where C(t)= ⎜ − − ⎟ . Considering⎜24t − 8 6t ⎟⎜2 3 2 2(t + 2) (t + 2)⎟⎝⎠C(t)=∞∑i=0C tii, we obtain recursively the coefficients ofthe series for a fundamental matrix F(t)=follows:∞ ∞ ∞i i i+ +i+2=i ii= 0 i= 0 i=0∞∑i=0F tii∑(i 2)(i 1)F t ( ∑C t )( ∑ F t )(5)⎛Then, taking into account (3), F 0 = 0 0 1 0 ⎞⎜ ⎟⎝0 0 0 1and⎠⎛F 1 = 1 0 0 0 ⎞⎜ ⎟ , and considering (5) we calculate the⎝0 1 0 0⎠ other coefficients using the following expression:F i+2 =( ∑ i CjF i−j) /(i + 2)(i + 1) , i≥0 (6)j=0Fig. 1: Table 1 for F(t)0 1 2 3 4 50 0 0 0 0 0 01 2 2 2 2 2 22 4 4 4 4 4 43 6 6 6 6 6 64 8 8 8 8 8 85 10 10 10 10 10 10Note that F(t) is not rational of orders (p,q) such that0≤p