A MATRIX METHOD OF FROBENIUS FOR SOLVING IMPLICIT SECOND ORDER DIFFERENTIAL SYSTEMS

In this paper, a matrix method Frobenius is proposed to construct generalized power series solutions of implicit second order differential systems of the type t A(t) X"(t) + t B(t) X'(t) + C(t) X(t) = 0, where A(t), B(t), C(t) are analytic matrix functions in some interval |t|<a. The method is based in two crucial points, the explicit solution of certain algebraic matrix equations appearing in the recurrence relationships of the matrix coefficients of the series solutions and the study of the convergence of the formal series provides by the method. AMS 1991 Subject classification: 34A09, 34A25, 34A30 ,15A24.

[1]  Solving Linear Operator Equations , 1974, Canadian Journal of Mathematics.

[2]  C. D. Meyer,et al.  Generalized inverses of linear transformations , 1979 .

[3]  H. Keller,et al.  On the Nonunique Equilibrium States and Buckling Mechanism of Spherical Shells , 1965 .

[4]  Stephen L. Campbell,et al.  A general form for solvable linear time varying singular systems of differential equations , 1987 .

[5]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[6]  Ewa Weinmüller,et al.  A difference method for a singular boundary value problem of second order , 1984 .

[7]  Seymour V. Parter,et al.  On the Multiplicity of Solutions Of a Differential Equation Arising in Chemical Reactor Theory , 1975 .

[8]  C. R. Rao,et al.  Generalized Inverse of Matrices and its Applications , 1972 .

[9]  L. Jódar,et al.  A closed-form solution of singular regular higher-order difference initial and boundary value problems , 1992 .

[10]  Karel Rektorys,et al.  The method of discretization in time and partial differential equations , 1982 .

[11]  Rectangular co-solutions of polynomial matrix equations and applications , 1991 .

[12]  E. Hille,et al.  Lectures on ordinary differential equations , 1968 .

[13]  Roswitha März,et al.  Numerical methods for differential algebraic equations , 1992, Acta Numerica.

[14]  M. Rosenblum,et al.  On the operator equation $BX-XA=Q$ , 1956 .

[15]  P. Rentrop,et al.  Differential-algebraic Equations in Vehicle System Dynamics , 1991 .

[16]  J. Ortega.  Numerical Analysis: A Second Course , 1974 .

[17]  L. Jódar,et al.  SOLVING NON-MONiC HIGHER ORDER TWO-POINT BOUNDARY VALUE MATRIX PROBLEMS USING QUASI-GREEN'S MATRIX FUNCTIONS , 1992 .