Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition
This paper is devoted to the fundamental problem of investigating the solvability of initial-boundary value problems for a quasi-linear pseudo-parabolic equation of fractional order with a sufficiently smooth boundary. The difference between the studied problems is that the boundary conditions are s...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-09-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/5/4/134 |
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| author | Serik E. Aitzhanov Abdumauvlen S. Berdyshev Kymbat S. Bekenayeva |
| author_facet | Serik E. Aitzhanov Abdumauvlen S. Berdyshev Kymbat S. Bekenayeva |
| author_sort | Serik E. Aitzhanov |
| collection | DOAJ |
| description | This paper is devoted to the fundamental problem of investigating the solvability of initial-boundary value problems for a quasi-linear pseudo-parabolic equation of fractional order with a sufficiently smooth boundary. The difference between the studied problems is that the boundary conditions are set in the form of a nonlinear boundary condition with a fractional differentiation operator. The main result of this work is establishing the local or global solvability of stated problems, depending on the parameters of the equation. The Galerkin method is used to prove the existence of a quasi-linear pseudo-parabolic equation’s weak solution in a bounded domain. Using Sobolev embedding theorems, a priori estimates of the solution are obtained. A priori estimates and the Rellich–Kondrashov theorem are used to prove the existence of the desired solutions to the considered boundary value problems. The uniqueness of the weak generalized solutions of the initial boundary value problems is proved on the basis of the obtained a priori estimates and the application of the generalized Gronwall lemma. The need to consider and study such initial boundary value problems for a quasi-linear pseudo-parabolic equation follows from practical requirements, such as solving fractional differential equations that simulate physical processes that occur during the study of liquid filtration processes, etc. |
| first_indexed | 2024-03-10T04:05:54Z |
| format | Article |
| id | doaj.art-0a446eb45837411bbff3570f497c454f |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T04:05:54Z |
| publishDate | 2021-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-0a446eb45837411bbff3570f497c454f2023-11-23T08:22:43ZengMDPI AGFractal and Fractional2504-31102021-09-015413410.3390/fractalfract5040134Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary ConditionSerik E. Aitzhanov0Abdumauvlen S. Berdyshev1Kymbat S. Bekenayeva2Department of Mathematics, Al-Farabi Kazakh National University, Almaty A15E3B6, KazakhstanDepartment of Mathematics and Mathematical Modeling, Abai Kazakh National Pedagogical University, Almaty 050010, KazakhstanDepartment of Mathematics and Mathematical Modeling, Abai Kazakh National Pedagogical University, Almaty 050010, KazakhstanThis paper is devoted to the fundamental problem of investigating the solvability of initial-boundary value problems for a quasi-linear pseudo-parabolic equation of fractional order with a sufficiently smooth boundary. The difference between the studied problems is that the boundary conditions are set in the form of a nonlinear boundary condition with a fractional differentiation operator. The main result of this work is establishing the local or global solvability of stated problems, depending on the parameters of the equation. The Galerkin method is used to prove the existence of a quasi-linear pseudo-parabolic equation’s weak solution in a bounded domain. Using Sobolev embedding theorems, a priori estimates of the solution are obtained. A priori estimates and the Rellich–Kondrashov theorem are used to prove the existence of the desired solutions to the considered boundary value problems. The uniqueness of the weak generalized solutions of the initial boundary value problems is proved on the basis of the obtained a priori estimates and the application of the generalized Gronwall lemma. The need to consider and study such initial boundary value problems for a quasi-linear pseudo-parabolic equation follows from practical requirements, such as solving fractional differential equations that simulate physical processes that occur during the study of liquid filtration processes, etc.https://www.mdpi.com/2504-3110/5/4/134pseudo-parabolic equationthe Caputo fractional derivativeMittag–Leffler functiona priori estimatesGalerkin approximationsweak solution |
| spellingShingle | Serik E. Aitzhanov Abdumauvlen S. Berdyshev Kymbat S. Bekenayeva Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition Fractal and Fractional pseudo-parabolic equation the Caputo fractional derivative Mittag–Leffler function a priori estimates Galerkin approximations weak solution |
| title | Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition |
| title_full | Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition |
| title_fullStr | Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition |
| title_full_unstemmed | Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition |
| title_short | Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition |
| title_sort | solvability issues of a pseudo parabolic fractional order equation with a nonlinear boundary condition |
| topic | pseudo-parabolic equation the Caputo fractional derivative Mittag–Leffler function a priori estimates Galerkin approximations weak solution |
| url | https://www.mdpi.com/2504-3110/5/4/134 |
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