Further Developments of Bessel Functions via Conformable Calculus with Applications
The theory of Bessel functions is a rich subject due to its essential role in providing solutions for differential equations associated with many applications. As fractional calculus has become an efficient and successful tool for analyzing various mathematical and physical problems, the so-called f...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2021/6069201 |
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| author | Mahmoud Abul-Ez Mohra Zayed Ali Youssef |
| author_facet | Mahmoud Abul-Ez Mohra Zayed Ali Youssef |
| author_sort | Mahmoud Abul-Ez |
| collection | DOAJ |
| description | The theory of Bessel functions is a rich subject due to its essential role in providing solutions for differential equations associated with many applications. As fractional calculus has become an efficient and successful tool for analyzing various mathematical and physical problems, the so-called fractional Bessel functions were introduced and studied from different viewpoints. This paper is primarily devoted to the study of developing two aspects. The starting point is to present a fractional Laplace transform via conformable fractional-order Bessel functions (CFBFs). We establish several important formulas of the fractional Laplace Integral operator acting on the CFBFs of the first kind. With this in hand, we discuss the solutions of a generalized class of fractional kinetic equations associated with the CFBFs in view of our proposed fractional Laplace transform. Next, we derive an orthogonality relation of the CFBFs, which enables us to study an expansion of any analytic functions by means of CFBFs and to propose truncated CFBFs. A new approximate formula of conformable fractional derivative based on CFBFs is provided. Furthermore, we describe a useful scheme involving the collocation method to solve some conformable fractional linear (nonlinear) multiorder differential equations. Accordingly, several practical test problems are treated to illustrate the validity and utility of the proposed techniques and examine their approximate and exact solutions. The obtained solutions of some fractional differential equations improve the analog ones provided by various authors using different techniques. The provided algorithm may be beneficial to enrich the Bessel function theory via fractional calculus. |
| format | Article |
| id | doaj-art-a4f108d514174ca184aeb90a87556063 |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-a4f108d514174ca184aeb90a875560632025-02-03T07:24:25ZengWileyJournal of Function Spaces2314-88962314-88882021-01-01202110.1155/2021/60692016069201Further Developments of Bessel Functions via Conformable Calculus with ApplicationsMahmoud Abul-Ez0Mohra Zayed1Ali Youssef2Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, EgyptMathematics Department, College of Science, King Khalid University, Abha, Saudi ArabiaMathematics Department, Faculty of Science, Sohag University, Sohag 82524, EgyptThe theory of Bessel functions is a rich subject due to its essential role in providing solutions for differential equations associated with many applications. As fractional calculus has become an efficient and successful tool for analyzing various mathematical and physical problems, the so-called fractional Bessel functions were introduced and studied from different viewpoints. This paper is primarily devoted to the study of developing two aspects. The starting point is to present a fractional Laplace transform via conformable fractional-order Bessel functions (CFBFs). We establish several important formulas of the fractional Laplace Integral operator acting on the CFBFs of the first kind. With this in hand, we discuss the solutions of a generalized class of fractional kinetic equations associated with the CFBFs in view of our proposed fractional Laplace transform. Next, we derive an orthogonality relation of the CFBFs, which enables us to study an expansion of any analytic functions by means of CFBFs and to propose truncated CFBFs. A new approximate formula of conformable fractional derivative based on CFBFs is provided. Furthermore, we describe a useful scheme involving the collocation method to solve some conformable fractional linear (nonlinear) multiorder differential equations. Accordingly, several practical test problems are treated to illustrate the validity and utility of the proposed techniques and examine their approximate and exact solutions. The obtained solutions of some fractional differential equations improve the analog ones provided by various authors using different techniques. The provided algorithm may be beneficial to enrich the Bessel function theory via fractional calculus.http://dx.doi.org/10.1155/2021/6069201 |
| spellingShingle | Mahmoud Abul-Ez Mohra Zayed Ali Youssef Further Developments of Bessel Functions via Conformable Calculus with Applications Journal of Function Spaces |
| title | Further Developments of Bessel Functions via Conformable Calculus with Applications |
| title_full | Further Developments of Bessel Functions via Conformable Calculus with Applications |
| title_fullStr | Further Developments of Bessel Functions via Conformable Calculus with Applications |
| title_full_unstemmed | Further Developments of Bessel Functions via Conformable Calculus with Applications |
| title_short | Further Developments of Bessel Functions via Conformable Calculus with Applications |
| title_sort | further developments of bessel functions via conformable calculus with applications |
| url | http://dx.doi.org/10.1155/2021/6069201 |
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