Electronic Journal of Differential Equations,
Conference 02 (1999), pp. 11-18.
Title: Noncommutative operational calculus
Authors: Henry E. Heatherly (Univ. of Louisiana, Lafayette, LA, USA)
Jason P. Huffman (Jacksonville State Univ., AL, USA)
Abstract:
Oliver Heaviside's operational calculus was placed on a rigorous mathematical
basis by Jan Mikusi\'{n}ski, who constructed an algebraic setting for the
operational methods. In this paper, we generalize Mikusi\'{n}ski'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, Mikusi\'{n}ski'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 December 9, 1999.
Math Subject Classifications: 44A40, 45D05, 34A12, 16S60.
Key Words: convolution; Mikusinski; Volterra integral equations;
operational calculus; linear operators.