Operational Calculus of the Quantum Statistical Fermi–Dirac and Bose–Einstein Functions Leading to the Novel Fractional Kinetic Equations
The sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun is changing. Taking motivation from these facts, we develop and solve a novel fractional kinetic equation con...
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2024-12-01
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| Series: | Fractal and Fractional |
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| author | Asifa Tassaddiq Carlo Cattani Rabab Alharbi Ruhaila Md Kasmani Sania Qureshi |
| author_facet | Asifa Tassaddiq Carlo Cattani Rabab Alharbi Ruhaila Md Kasmani Sania Qureshi |
| author_sort | Asifa Tassaddiq |
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| description | The sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun is changing. Taking motivation from these facts, we develop and solve a novel fractional kinetic equation containing Fermi–Dirac (FD) and Bose–Einstein (BE) functions. Several distributional properties of these functions and their proposed new generalizations are investigated in this article. In fact, it is proved that these functions belong to distribution space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">D</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula> while their Fourier transforms belong to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi mathvariant="script">Z</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>.</mo></mrow></semantics></math></inline-formula> Fourier and Laplace transforms of these functions are computed by using their distributional representation. Thanks to them, we can compute various new fractional calculus formulae and a new relation involving the Fox–Wright function. Some fractional kinetic equations containing the FD and BE functions are also formulated and solved. |
| format | Article |
| id | doaj-art-ec42ffb8529e45e9960f9bfd3d2c1925 |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
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| series | Fractal and Fractional |
| spelling | doaj-art-ec42ffb8529e45e9960f9bfd3d2c19252024-12-27T14:27:11ZengMDPI AGFractal and Fractional2504-31102024-12-0181274910.3390/fractalfract8120749Operational Calculus of the Quantum Statistical Fermi–Dirac and Bose–Einstein Functions Leading to the Novel Fractional Kinetic EquationsAsifa Tassaddiq0Carlo Cattani1Rabab Alharbi2Ruhaila Md Kasmani3Sania Qureshi4Department of Computer Science, College of Computer and Information Sciences Majmaah University, Al Majmaah 11952, Saudi ArabiaDepartment of Mathematics and Informatics, Azerbaijan University, Baku 1007, AzerbaijanDepartment of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi ArabiaInstitute of Mathematical Sciences, Universiti Malaya, Kuala Lumpur 50603, MalaysiaDepartment of Basic Sciences and Related Studies, Mehran University of Engineering & Technology, Jamshoro 76062, PakistanThe sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun is changing. Taking motivation from these facts, we develop and solve a novel fractional kinetic equation containing Fermi–Dirac (FD) and Bose–Einstein (BE) functions. Several distributional properties of these functions and their proposed new generalizations are investigated in this article. In fact, it is proved that these functions belong to distribution space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">D</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula> while their Fourier transforms belong to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi mathvariant="script">Z</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>.</mo></mrow></semantics></math></inline-formula> Fourier and Laplace transforms of these functions are computed by using their distributional representation. Thanks to them, we can compute various new fractional calculus formulae and a new relation involving the Fox–Wright function. Some fractional kinetic equations containing the FD and BE functions are also formulated and solved.https://www.mdpi.com/2504-3110/8/12/749Fermi–Dirac and Bose–Einstein functionsmathematical operatorsFox–Wright functionkinetic equation |
| spellingShingle | Asifa Tassaddiq Carlo Cattani Rabab Alharbi Ruhaila Md Kasmani Sania Qureshi Operational Calculus of the Quantum Statistical Fermi–Dirac and Bose–Einstein Functions Leading to the Novel Fractional Kinetic Equations Fractal and Fractional Fermi–Dirac and Bose–Einstein functions mathematical operators Fox–Wright function kinetic equation |
| title | Operational Calculus of the Quantum Statistical Fermi–Dirac and Bose–Einstein Functions Leading to the Novel Fractional Kinetic Equations |
| title_full | Operational Calculus of the Quantum Statistical Fermi–Dirac and Bose–Einstein Functions Leading to the Novel Fractional Kinetic Equations |
| title_fullStr | Operational Calculus of the Quantum Statistical Fermi–Dirac and Bose–Einstein Functions Leading to the Novel Fractional Kinetic Equations |
| title_full_unstemmed | Operational Calculus of the Quantum Statistical Fermi–Dirac and Bose–Einstein Functions Leading to the Novel Fractional Kinetic Equations |
| title_short | Operational Calculus of the Quantum Statistical Fermi–Dirac and Bose–Einstein Functions Leading to the Novel Fractional Kinetic Equations |
| title_sort | operational calculus of the quantum statistical fermi dirac and bose einstein functions leading to the novel fractional kinetic equations |
| topic | Fermi–Dirac and Bose–Einstein functions mathematical operators Fox–Wright function kinetic equation |
| url | https://www.mdpi.com/2504-3110/8/12/749 |
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