A Systematic Approach to Delay Functions

Christopher N. Angstmann, Stuart-James M. Burney, Bruce I. Henry (), Byron A. Jacobs and Zhuang Xu
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Christopher N. Angstmann: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
Stuart-James M. Burney: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
Bruce I. Henry: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
Byron A. Jacobs: Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg 2092, South Africa
Zhuang Xu: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

Mathematics, 2023, vol. 11, issue 21, 1-34

Abstract: We present a systematic introduction to a class of functions that provide fundamental solutions for autonomous linear integer-order and fractional-order delay differential equations. These functions, referred to as delay functions, are defined through power series or fractional power series, with delays incorporated into their series representations. Using this approach, we have defined delay exponential functions, delay trigonometric functions and delay fractional Mittag-Leffler functions, among others. We obtained Laplace transforms of the delay functions and demonstrated how they can be employed in finding solutions to delay differential equations. Our results, which extend and unify previous work, offer a consistent framework for defining and using delay functions.

Keywords: special functions; delay differential equations; fractional differential equations; integral transforms (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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