Mild Solutions for the Time-Fractional Navier-Stokes Equations with MHD Effects
Recently, various techniques and methods have been employed by mathematicians to solve specific types of fractional differential equations (FDEs) with symmetric properties. The study focuses on Navier-Stokes equations (NSEs) that involve MHD effects with time-fractional derivatives (FDs). The (NSEs)...
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MDPI AG
2023-01-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/2/280 |
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| author | Kinda Abuasbeh Ramsha Shafqat Azmat Ullah Khan Niazi Muath Awadalla |
| author_facet | Kinda Abuasbeh Ramsha Shafqat Azmat Ullah Khan Niazi Muath Awadalla |
| author_sort | Kinda Abuasbeh |
| collection | DOAJ |
| description | Recently, various techniques and methods have been employed by mathematicians to solve specific types of fractional differential equations (FDEs) with symmetric properties. The study focuses on Navier-Stokes equations (NSEs) that involve MHD effects with time-fractional derivatives (FDs). The (NSEs) with time-FDs of order <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> are investigated. To facilitate anomalous diffusion in fractal media, mild solutions and Mittag-Leffler functions are used. In <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mrow><mi>δ</mi><mo>,</mo><mi>r</mi></mrow></msup></semantics></math></inline-formula>, the existence, and uniqueness of local and global mild solutions are proved, as well as the symmetric structure created. Moderate local solutions are provided in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>J</mi><mi>r</mi></msub></semantics></math></inline-formula>. Moreover, the regularity and existence of classical solutions to the equations in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>J</mi><mi>r</mi></msub></semantics></math></inline-formula>. are established and presented. |
| first_indexed | 2024-03-11T08:04:44Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T08:04:44Z |
| publishDate | 2023-01-01 |
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| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-472e59ff3fd64f60bee4ba2c0e6a520e2023-11-16T23:31:12ZengMDPI AGSymmetry2073-89942023-01-0115228010.3390/sym15020280Mild Solutions for the Time-Fractional Navier-Stokes Equations with MHD EffectsKinda Abuasbeh0Ramsha Shafqat1Azmat Ullah Khan Niazi2Muath Awadalla3Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf 31982, Al Ahsa, Saudi ArabiaDepartment of Mathematics and Statistics, The University of Lahore, Sargodha 40100, PakistanDepartment of Mathematics and Statistics, The University of Lahore, Sargodha 40100, PakistanDepartment of Mathematics and Statistics, College of Science, King Faisal University, Hafuf 31982, Al Ahsa, Saudi ArabiaRecently, various techniques and methods have been employed by mathematicians to solve specific types of fractional differential equations (FDEs) with symmetric properties. The study focuses on Navier-Stokes equations (NSEs) that involve MHD effects with time-fractional derivatives (FDs). The (NSEs) with time-FDs of order <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> are investigated. To facilitate anomalous diffusion in fractal media, mild solutions and Mittag-Leffler functions are used. In <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mrow><mi>δ</mi><mo>,</mo><mi>r</mi></mrow></msup></semantics></math></inline-formula>, the existence, and uniqueness of local and global mild solutions are proved, as well as the symmetric structure created. Moderate local solutions are provided in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>J</mi><mi>r</mi></msub></semantics></math></inline-formula>. Moreover, the regularity and existence of classical solutions to the equations in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>J</mi><mi>r</mi></msub></semantics></math></inline-formula>. are established and presented.https://www.mdpi.com/2073-8994/15/2/280Navier-Stokes equationsCaputo fractional derivativeMittag-Leffler functionsmild solutionsregularity |
| spellingShingle | Kinda Abuasbeh Ramsha Shafqat Azmat Ullah Khan Niazi Muath Awadalla Mild Solutions for the Time-Fractional Navier-Stokes Equations with MHD Effects Symmetry Navier-Stokes equations Caputo fractional derivative Mittag-Leffler functions mild solutions regularity |
| title | Mild Solutions for the Time-Fractional Navier-Stokes Equations with MHD Effects |
| title_full | Mild Solutions for the Time-Fractional Navier-Stokes Equations with MHD Effects |
| title_fullStr | Mild Solutions for the Time-Fractional Navier-Stokes Equations with MHD Effects |
| title_full_unstemmed | Mild Solutions for the Time-Fractional Navier-Stokes Equations with MHD Effects |
| title_short | Mild Solutions for the Time-Fractional Navier-Stokes Equations with MHD Effects |
| title_sort | mild solutions for the time fractional navier stokes equations with mhd effects |
| topic | Navier-Stokes equations Caputo fractional derivative Mittag-Leffler functions mild solutions regularity |
| url | https://www.mdpi.com/2073-8994/15/2/280 |
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