A Systematic Approach to Delay Functions
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 de...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/21/4526 |
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| author | Christopher N. Angstmann Stuart-James M. Burney Bruce I. Henry Byron A. Jacobs Zhuang Xu |
| author_facet | Christopher N. Angstmann Stuart-James M. Burney Bruce I. Henry Byron A. Jacobs Zhuang Xu |
| author_sort | Christopher N. Angstmann |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T11:25:18Z |
| format | Article |
| id | doaj.art-59dee4d0a6c84986b9f0241fe30710d2 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T11:25:18Z |
| publishDate | 2023-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-59dee4d0a6c84986b9f0241fe30710d22023-11-10T15:08:10ZengMDPI AGMathematics2227-73902023-11-011121452610.3390/math11214526A Systematic Approach to Delay FunctionsChristopher N. Angstmann0Stuart-James M. Burney1Bruce I. Henry2Byron A. Jacobs3Zhuang Xu4School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, AustraliaSchool of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, AustraliaSchool of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, AustraliaDepartment of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg 2092, South AfricaSchool of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, AustraliaWe 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.https://www.mdpi.com/2227-7390/11/21/4526special functionsdelay differential equationsfractional differential equationsintegral transforms |
| spellingShingle | Christopher N. Angstmann Stuart-James M. Burney Bruce I. Henry Byron A. Jacobs Zhuang Xu A Systematic Approach to Delay Functions Mathematics special functions delay differential equations fractional differential equations integral transforms |
| title | A Systematic Approach to Delay Functions |
| title_full | A Systematic Approach to Delay Functions |
| title_fullStr | A Systematic Approach to Delay Functions |
| title_full_unstemmed | A Systematic Approach to Delay Functions |
| title_short | A Systematic Approach to Delay Functions |
| title_sort | systematic approach to delay functions |
| topic | special functions delay differential equations fractional differential equations integral transforms |
| url | https://www.mdpi.com/2227-7390/11/21/4526 |
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