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A matrix method of Frobenius, and application to a generalized Bessel function. (English) Zbl 0764.34004

Numerical mathematics and computing, Proc. 21st Manitoba Conf., Winnipeg/ Can. 1991, Congr. Numerantium 86, 7-17 (1992).
[For the entire collection see Zbl 0745.00024.]
The method of Frobenius can be used for the scalar differential equation \(t^ 2X''(t)+tA(t)X'(t)+B(t)X(t)=0\) when \(A,B\) are analytic for \(| t|<a\). If \(A,B\) are analytic matrix valued functions, then substitution of a Frobenius type series into the matrix differential equation for \(X\) gives algebraic equations on the coefficients. The algebraic structure incorporates Moore-Penrose inverses and, when all the algebraic requirements are met, the formal Frobenius series solution converges. The process is also applied to a matrix generalization of Bessel’s equation.

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34A30 Linear ordinary differential equations and systems
34M99 Ordinary differential equations in the complex domain
15A24 Matrix equations and identities
15A09 Theory of matrix inversion and generalized inverses
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)

Citations:

Zbl 0745.00024
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