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WSEAS TRANSACTIONS on MATHEMATICSC. Pestano-Gabino, C. Gonz_Lez-Concepcion, M.C. Gil-Farinaparticular solution, given the initial conditions Y 0 andiY 1 , is Yi+ 2= ( ∑ AjYi − j+ bi) /((i + 2)(i + 1)) i≥0.j=0It is interesting to comment that, from computationalpoint of view, if n is large h is very small and Y iincreases considerably when i increases. To avoiderrors, we consider the formal power series∞* i∑ Yit ,i=0* 2iwith Yi= h Yi. Note that Y(t) is rational if and only ifY * (t) is rational.Let us study some particular examples.Example 52xu tt = u2t(1 − t)xx0≤x≤1, t>0, t≠0, t≠1u(0,t)=0 u(1,t)= t1− t; u(x,0.5)=x2 u t (x,0.5)=4x 2Considering, for instance, n=10 and t 0 =0.5 (theequation has not sense in t=0) the initial conditionsare:22⎛ x ⎞⎛1x ⎞1⎜ 2 ⎟⎜ 2 ⎟x2Y(0.5)= ⎜ ⎟ , Y'(0.5)=4 ⎜ x2⎟⎜ ⋮ ⎟⎜ ⋮ ⎟⎜2x ⎟⎜2⎝ 9 ⎠x ⎟⎝ 9 ⎠⎛ −2x x 0 ⋯ 02 21 1⎜ 2 2 22 2 2a⎜ −i 2 2i= ⎜23−3x 2x x ⋯ 0A 0 x 2x ⋯ 0h ⎜⎜ ⋯ ⋯ ⋯ ⋯ ⋯⎜⎝ 0 0 ⋯ x −2xwhere a 2i =2 2i+3 (i+1), a 2i+1 =2 2i+4 (i+1)⎛ 0 ⎞b i = i +⎜ ⎟2 2 (i + 2)(i + 1) ⎜⋮ ⎟ , i≥0.2h ⎜ 0 ⎟⎜ 2 ⎟⎝ x9⎠2 29 9⎞⎟⎟⎟⎟⎟⎟⎠Fig. 11: Table 1 for Y(t) is:0 1 2 3 4 50 0 0 0 0 0 01 1 1 1 1 1 12 2 2 1 1 1 13 3 3 2 1 1 14 4 4 3 2 1 15 5 5 4 3 2 1It indicates that Y(t)=[1/1] Y . Solving the systemRS(1,1), we obtain⎛ 0.01+ 0.02(t − 0.5) ⎞⎜⎟Y(t)= R 0.04 0.08(t 0.5) 1[1/1] Y = ⎜+ −⎟⎜ ⋮ ⎟1− 2(t − 0.5)⎜⎟⎝ 0.81+ 1.62(t − 0.5) ⎠⎛ 0.01t ⎞⎜ ⎟0.04t 1which can be simplified as Y(t)= ⎜ ⎟⎜ ⋮ ⎟1−t⎜ ⎟⎝ 0.81t ⎠xNote that u(x,t)= 2 tis the solution and that Y(t)1−tcoincides with u(x,t) discretised in x.Example 6Let us see now an example where left approximant ismore suitable (left m.d. are smaller than right m.d.).2xu tt = u2 xxt>0, 0≤x≤1(1 + t)u(0,t)=1, u(1,t)= 12 + t; u(x,0)= 11+ x, u xt(x,0)= −2(1 + x)Considering different values of n (n=5, n=10, n=50) inthis example Table 1 is highly sensitive to thethreshold that we choose to decide whether andelement is to be considered zero or not (Due to thelimitations of finite arithmetic, certain theoretically zeroelements will not be exactly zeros, which is why wehave chosen a threshold so that any number with anabsolute value of below it will be considered as beingzero). However Table 2 for left approximant, in mostcases, shows that (0,1) is a pair of left m.d.1Note that u(x,t)= is the solution and that1+ x(t + 1)Table 2 indicates that Y(t)∈{ L [0/1] Y }. For instance, ifn=5:Y(t) =⎛⎛1 0 0 0⎞ ⎛0.2 /1.2 0 0 0 ⎞ ⎞ ⎛1/1.2⎞⎜⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟⎜⎜ 0 1 0 0⎟0 0.4/1.4 0 0 1/1.4+ ⎜ ⎟ t⎟ ⎜ ⎟⎜⎜0 0 1 0⎟ ⎜ 0 0 0.6/1.6 0 ⎟ ⎟ ⎜1/1.6⎟⎜⎜ ⎟ ⎜ ⎟ ⎜ ⎟0 0 0 1 0 0 0 0.8/1.8 ⎟⎝⎝ ⎠ ⎝ ⎠ ⎠ ⎝1/1.8⎠i.e.; Y(t)=⎛ 1 ⎞⎜1 0.2(t 1)⎟⎜+ +⎟⎜ 1 ⎟⎜⎟⎜1+ 0.4(t + 1) ⎟⎜ 1 ⎟⎜⎟⎜1+ 0.6(t + 1) ⎟⎜ 1 ⎟⎜1+ 0.8(t + 1) ⎟⎝⎠is u(x,t) discretised in x.5 ConclusionsIn this paper we have extended results and practice of[3]. Our aim has been to highlight methods of lookingfor rational fundamental matrices for systems of linearm-order differential equation and particular rationalsolutions for these systems and for partial differentialequations.−1ISSN: 1109-2769 351 Issue 6, Volume 7, June 2008
WSEAS TRANSACTIONS on MATHEMATICSC. Pestano-Gabino, C. Gonz_Lez-Concepcion, M.C. Gil-FarinaThe balanced use of numerical methods, together withapproximation theory is a very interesting alternativeto improve the calculation of these solutions (or,alternatively, approximated solutions).In this paper we combine Galerkin and FiniteDifference methods with MPA.Other applications were considered in [7] inconnection with time series analysis and economicmodels.References[1] Burden, R.L. and Faires, J.D. Análisis Numérico.Grupo Editorial Iberoamιrica. México, 1996.[2] A. Draux, On the Non-Normal Padé Table in aNon-Conmutative Algebra, Journal of Computationaland Applied Mathematics 21, 1998, pp. 271-288.[3] González-Concepción, C. and Pestano-Gabino, C.(1999), Approximated solutions in rational form forsystems of differential equations, NumericalAlgorithms 21, 185-203.[4] Kincaid, D. and Cheney, W. Análisis Numérico.Addison-Wesley Iberoamericana. Argentina, 1994.[5] Marcellαn, F., Casasϊs, L. and Zarzo, A. EcuacionesDiferenciales. Problemas lineales y aplicaciones.McGraw-Hill. Madrid, 1990.[6] Pestano C. and González C. Matrix PadéApproximation of Rational Functions, NumericalAlgorithms, 15 (1997), 1-26.[7] Pestano-Gabino, C. and González-Concepción, C.,Rationality, minimality and uniqueness ofrepresentation of matrix formal power series, Journalof Computational an Applied Mathematics, 94 (1998),23-38.ISSN: 1109-2769 352 Issue 6, Volume 7, June 2008
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