First order linear ordinary differential equations in associative algebras

First order linear ordinary differential equations in associative algebras

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 consistin...

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Bibliographic Details
Main Authors: Gordon Erlebacher, Garrret E. Sobczyk
Format: Article
Language:English
Published: Texas State University 2004-01-01
Series:Electronic Journal of Differential Equations
Subjects:
Associative algebra
factor ring
idempotent
differential equation
nilpotent
spectral basis
Toeplitz matrix.
Online Access:http://ejde.math.txstate.edu/Volumes/2004/01/abstr.html
Description
Summary: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.
ISSN:1072-6691

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