First order linear ordinary differential equations in associative algebras. (English) Zbl 1042.34001

Erlebacher, Gordon; Sobczyk, Garret E.

Electron. J. Differ. Equ. 2004, Paper No. 01, 18 p. (2004).

Summary: Here, 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 noncommutative 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.

MSC:

34A05	Explicit solutions, first integrals of ordinary differential equations
15B33	Matrices over special rings (quaternions, finite fields, etc.)
15A66	Clifford algebras, spinors
34A99	General theory for ordinary differential equations

Keywords:

Associative algebra; factor ring; idempotent; differential equation; nilpotent; spectral basis; Toeplitz matrix