A Non-Local Problem for the Fractional-Order Rayleigh–Stokes Equation
A nonlocal boundary value problem for the fractional version of the Rayleigh–Stokes equation, well-known in fluid dynamics, is studied. Namely, the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow>...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-06-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/7/6/490 |
| _version_ | 1797594721806188544 |
|---|---|
| author | Ravshan Ashurov Oqila Mukhiddinova Sabir Umarov |
| author_facet | Ravshan Ashurov Oqila Mukhiddinova Sabir Umarov |
| author_sort | Ravshan Ashurov |
| collection | DOAJ |
| description | A nonlocal boundary value problem for the fractional version of the Rayleigh–Stokes equation, well-known in fluid dynamics, is studied. Namely, the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>=</mo><mi>β</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>+</mo><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is an arbitrary real number, is proposed instead of the initial condition. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, then we have the inverse problem in time, called the backward problem. It is well-known that the backward problem is ill-posed in the sense of Hadamard. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, then the corresponding non-local problem becomes well-posed in the sense of Hadamard, and moreover, in this case a coercive estimate for the solution can be established. The aim of this work is to find values of the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>, which separates two types of behavior of the semi-backward problem under consideration. We prove the following statements: if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>≥</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>, then the problem is well-posed; if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, then depending on the eigenvalues of the elliptic part of the equation, for the existence of a solution an additional condition on orthogonality of the right-hand side of the equation and the boundary function to some eigenfunctions of the corresponding elliptic operator may emerge. |
| first_indexed | 2024-03-11T02:26:14Z |
| format | Article |
| id | doaj.art-3a2c16724896453bb3a1310e74c35b78 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-11T02:26:14Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-3a2c16724896453bb3a1310e74c35b782023-11-18T10:30:01ZengMDPI AGFractal and Fractional2504-31102023-06-017649010.3390/fractalfract7060490A Non-Local Problem for the Fractional-Order Rayleigh–Stokes EquationRavshan Ashurov0Oqila Mukhiddinova1Sabir Umarov2Institute of Mathematics, Uzbekistan Academy of Science, University Str., 9, Olmazor District, Tashkent 100174, UzbekistanInstitute of Mathematics, Uzbekistan Academy of Science, University Str., 9, Olmazor District, Tashkent 100174, UzbekistanDepartment of Mathematics, University of New Haven, 300 Boston Post Road, West Haven, CT 06516, USAA nonlocal boundary value problem for the fractional version of the Rayleigh–Stokes equation, well-known in fluid dynamics, is studied. Namely, the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>T</mi><mo>)</mo><mo>=</mo><mi>β</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo><mo>+</mo><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is an arbitrary real number, is proposed instead of the initial condition. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, then we have the inverse problem in time, called the backward problem. It is well-known that the backward problem is ill-posed in the sense of Hadamard. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, then the corresponding non-local problem becomes well-posed in the sense of Hadamard, and moreover, in this case a coercive estimate for the solution can be established. The aim of this work is to find values of the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>, which separates two types of behavior of the semi-backward problem under consideration. We prove the following statements: if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>≥</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>, then the problem is well-posed; if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, then depending on the eigenvalues of the elliptic part of the equation, for the existence of a solution an additional condition on orthogonality of the right-hand side of the equation and the boundary function to some eigenfunctions of the corresponding elliptic operator may emerge.https://www.mdpi.com/2504-3110/7/6/490Rayleigh–Stokes problemnon-local problemfractional derivativeMittag–Leffler functionFourier method |
| spellingShingle | Ravshan Ashurov Oqila Mukhiddinova Sabir Umarov A Non-Local Problem for the Fractional-Order Rayleigh–Stokes Equation Fractal and Fractional Rayleigh–Stokes problem non-local problem fractional derivative Mittag–Leffler function Fourier method |
| title | A Non-Local Problem for the Fractional-Order Rayleigh–Stokes Equation |
| title_full | A Non-Local Problem for the Fractional-Order Rayleigh–Stokes Equation |
| title_fullStr | A Non-Local Problem for the Fractional-Order Rayleigh–Stokes Equation |
| title_full_unstemmed | A Non-Local Problem for the Fractional-Order Rayleigh–Stokes Equation |
| title_short | A Non-Local Problem for the Fractional-Order Rayleigh–Stokes Equation |
| title_sort | non local problem for the fractional order rayleigh stokes equation |
| topic | Rayleigh–Stokes problem non-local problem fractional derivative Mittag–Leffler function Fourier method |
| url | https://www.mdpi.com/2504-3110/7/6/490 |
| work_keys_str_mv | AT ravshanashurov anonlocalproblemforthefractionalorderrayleighstokesequation AT oqilamukhiddinova anonlocalproblemforthefractionalorderrayleighstokesequation AT sabirumarov anonlocalproblemforthefractionalorderrayleighstokesequation AT ravshanashurov nonlocalproblemforthefractionalorderrayleighstokesequation AT oqilamukhiddinova nonlocalproblemforthefractionalorderrayleighstokesequation AT sabirumarov nonlocalproblemforthefractionalorderrayleighstokesequation |