Electronic Journal of Differential Equations
15th annual Conference of Applied Mathematics, Univ. of Central Oklahoma,
Electron. J. Diff. Eqns., Conf. 02, 1999, pp. 1118.
Noncommutative operational calculus
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 operatorvalued function. Because these functions can be
zerodivisors and do not necessarily commute, Mikusinski's onedimensional
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 integrodifferential 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
email: heh5820@usl.edu 

Jason P. Huffman
Department of Mathematical, Computing, and Information Sciences
Jacksonville State University
Jacksonville, AL 36265, USA
email: jhuffman@jsucc.jsu.edu 
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