Henry E. Heatherly & Jason P. Huffman
Abstract:
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 | |
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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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