On the Heat and Wave Equations with the Sturm-Liouville Operator in Quantum Calculus
In this paper, we explore a generalised solution of the Cauchy problems for the q-heat and q-wave equations which are generated by Jackson’s and the q-Sturm-Liouville operators with respect to t and x, respectively. For this, we use a new method, where a crucial tool is used to represent functions i...
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| Format: | Article |
| Language: | English |
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Wiley
2023-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2023/2488165 |
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| author | Serikbol Shaimardan Lars-Erik Persson Nariman Tokmagambetov |
| author_facet | Serikbol Shaimardan Lars-Erik Persson Nariman Tokmagambetov |
| author_sort | Serikbol Shaimardan |
| collection | DOAJ |
| description | In this paper, we explore a generalised solution of the Cauchy problems for the q-heat and q-wave equations which are generated by Jackson’s and the q-Sturm-Liouville operators with respect to t and x, respectively. For this, we use a new method, where a crucial tool is used to represent functions in the Fourier series expansions in a Hilbert space on quantum calculus. We show that these solutions can be represented by explicit formulas generated by the q-Mittag-Leffler function. Moreover, we prove the unique existence and stability of the weak solutions. |
| format | Article |
| id | doaj-art-91a1bb82623147c1815fa262e16ebc32 |
| institution | OA Journals |
| issn | 1687-0409 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-91a1bb82623147c1815fa262e16ebc322025-08-20T02:10:10ZengWileyAbstract and Applied Analysis1687-04092023-01-01202310.1155/2023/2488165On the Heat and Wave Equations with the Sturm-Liouville Operator in Quantum CalculusSerikbol Shaimardan0Lars-Erik Persson1Nariman Tokmagambetov2L. N. Gumilyov Eurasian National UniversityDepartment of Mathematics and Computer ScienceKaragandy University of the Name of Academician E.A. BuketovIn this paper, we explore a generalised solution of the Cauchy problems for the q-heat and q-wave equations which are generated by Jackson’s and the q-Sturm-Liouville operators with respect to t and x, respectively. For this, we use a new method, where a crucial tool is used to represent functions in the Fourier series expansions in a Hilbert space on quantum calculus. We show that these solutions can be represented by explicit formulas generated by the q-Mittag-Leffler function. Moreover, we prove the unique existence and stability of the weak solutions.http://dx.doi.org/10.1155/2023/2488165 |
| spellingShingle | Serikbol Shaimardan Lars-Erik Persson Nariman Tokmagambetov On the Heat and Wave Equations with the Sturm-Liouville Operator in Quantum Calculus Abstract and Applied Analysis |
| title | On the Heat and Wave Equations with the Sturm-Liouville Operator in Quantum Calculus |
| title_full | On the Heat and Wave Equations with the Sturm-Liouville Operator in Quantum Calculus |
| title_fullStr | On the Heat and Wave Equations with the Sturm-Liouville Operator in Quantum Calculus |
| title_full_unstemmed | On the Heat and Wave Equations with the Sturm-Liouville Operator in Quantum Calculus |
| title_short | On the Heat and Wave Equations with the Sturm-Liouville Operator in Quantum Calculus |
| title_sort | on the heat and wave equations with the sturm liouville operator in quantum calculus |
| url | http://dx.doi.org/10.1155/2023/2488165 |
| work_keys_str_mv | AT serikbolshaimardan ontheheatandwaveequationswiththesturmliouvilleoperatorinquantumcalculus AT larserikpersson ontheheatandwaveequationswiththesturmliouvilleoperatorinquantumcalculus AT narimantokmagambetov ontheheatandwaveequationswiththesturmliouvilleoperatorinquantumcalculus |