Electronic Journal of Differential Equations 15th annual Conference of Applied Mathematics, Univ. of Central Oklahoma,
Electron. J. Diff. Eqns., Conf. 02, 1999, pp. 11-18.

Noncommutative operational calculus

Henry E. Heatherly & Jason P. Huffman

Oliver Heaviside's operational calculus was placed on a rigorous mathematical basis by Jan Mikusinski, who constructed an algebraic setting for the operational methods. In this paper, we generalize Mikusinski's methods to solve linear ordinary differential equations in which the unknown is a matrix- or linear operator-valued function. Because these functions can be zero-divisors and do not necessarily commute, Mikusinski's one-dimensional calculus cannot be used. The noncommuative operational calculus developed here, however, is used to solve a wide class of such equations. In addition, we provide new proofs of existence and uniqueness theorems for certain matrix- and operator valued Volterra integral and integro-differential equations. Several examples are given which demonstrate these new methods.

Published Nobember 24, 1999.
Subject lassfications: 44A40, 45D05, 34A12, 16S60.
Key words: convolution, Mikusinski, Volterra integral equations, operational calculus, linear operators.

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Henry E. Heatherly
Department of Mathematics
University of Louisiana, Lafayette
Lafayette, LA 70504, USA
e-mail: heh5820@usl.edu
Jason P. Huffman
Department of Mathematical, Computing, and Information Sciences
Jacksonville State University
Jacksonville, AL 36265, USA
e-mail: jhuffman@jsucc.jsu.edu

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