Electron. J. Diff. Eqns., Vol. 2004(2004), No. 01, pp. 1-18.

First order linear ordinary differential equations in associative algebras

Gordon Erlebacher & Garrret E. Sobczyk

Abstract:
In this paper, we study the linear differential equation
$$ \frac{dx}{dt}=\sum_{i=1}^n a_i(t) x b_i(t) + f(t) $$
in an associative but non-commutative algebra $\mathcal{A}$, where the $b_i(t)$ form a set of commuting $\mathcal{A}$-valued functions expressed in a time-independent spectral basis consisting of mutually annihilating idempotents and nilpotents. Explicit new closed solutions are derived, and examples are presented to illustrate the theory.

Submitted September 6, 2003. Published January 2, 2004.
Math Subject Classifications: 15A33, 15A66, 34G10, 39B12.
Key Words: Associative algebra, factor ring, idempotent, differential equation, nilpotent, spectral basis, Toeplitz matrix.

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Gordon Erlebacher
Department of Mathematics
Florida State University
Tallahassee, FL 32306, USA
email: erlebach@math.fsu.edu
Garret E. Sobczyk
Universidad de las Americas
Departamento de Fisico-Matematicas
Apartado Postal #100, Santa Catarina Martir
72820 Cholula, Pue., Mexico
email: sobczyk@mail.udlap.mx

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