Monotone iterations method for fractional diffusion equations
In recent years, there has been a growing interest on non-local models because of their relevance in many practical applications. A widely studied class of non-local models involves fractional order operators. They usually describe anomalous diffusion. In particular, these equations provide a more f...
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| Format: | Article |
| Language: | German |
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Ivan Franko National University of Lviv
2022-06-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/324 |
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| author | M. Krasnoshchok |
| author_facet | M. Krasnoshchok |
| author_sort | M. Krasnoshchok |
| collection | DOAJ |
| description | In recent years, there has been a growing interest on non-local
models because of their relevance in many practical applications. A
widely studied class of non-local models involves fractional order
operators. They usually describe anomalous diffusion. In
particular, these equations provide a more faithful representation
of the long-memory and nonlocal dependence of diffusion in fractal
and porous media, heat flow in media with memory, dynamics of
protein in cells etc.
For $a\in (0, 1)$, we investigate the nonautonomous fractional
diffusion equation:
$D^a_{*,t} u - Au = f(x, t,u),$
where
$D^a_{*,t}$ is the Caputo fractional derivative and $A$ is a
uniformly elliptic operator with smooth coefficients depending on
space and time. We consider these equations together with initial
and quasilinear boundary conditions.
The solvability of such problems in H\"older spaces presupposes
rigid restrictions on the given initial data. These compatibility
conditions have no physical meaning and, therefore, they can be
avoided, if the solution is sought in larger spaces, for instance in
weighted H\"older spaces.
We give general existence and uniqueness result and
provide some examples of applications of the main theorem. The main
tool is the monotone iterations method. Preliminary we developed the
linear theory with existence and comparison results. The principle
use of the positivity lemma is the construction of a monotone
sequences for our problem. Initial iteration may be taken as either
an upper solution or a lower solution. We provide some examples of
upper and lower solution for the case of linear equations and
quasilinear boundary conditions. We notice that this approach can
also be extended to other problems and systems of fractional
equations as soon as we will be able to construct appropriate upper
and lower solutions. |
| format | Article |
| id | doaj-art-4f4fca1625a541b9b459e4b3ee0ec67d |
| institution | Matheson Library |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2022-06-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-4f4fca1625a541b9b459e4b3ee0ec67d2025-07-08T09:08:39ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202022-06-0157212213610.30970/ms.57.2.122-136324Monotone iterations method for fractional diffusion equationsM. Krasnoshchok0Institute of applied mathematics and mechanics of NAS of Ukraine Sloviansk, UkraineIn recent years, there has been a growing interest on non-local models because of their relevance in many practical applications. A widely studied class of non-local models involves fractional order operators. They usually describe anomalous diffusion. In particular, these equations provide a more faithful representation of the long-memory and nonlocal dependence of diffusion in fractal and porous media, heat flow in media with memory, dynamics of protein in cells etc. For $a\in (0, 1)$, we investigate the nonautonomous fractional diffusion equation: $D^a_{*,t} u - Au = f(x, t,u),$ where $D^a_{*,t}$ is the Caputo fractional derivative and $A$ is a uniformly elliptic operator with smooth coefficients depending on space and time. We consider these equations together with initial and quasilinear boundary conditions. The solvability of such problems in H\"older spaces presupposes rigid restrictions on the given initial data. These compatibility conditions have no physical meaning and, therefore, they can be avoided, if the solution is sought in larger spaces, for instance in weighted H\"older spaces. We give general existence and uniqueness result and provide some examples of applications of the main theorem. The main tool is the monotone iterations method. Preliminary we developed the linear theory with existence and comparison results. The principle use of the positivity lemma is the construction of a monotone sequences for our problem. Initial iteration may be taken as either an upper solution or a lower solution. We provide some examples of upper and lower solution for the case of linear equations and quasilinear boundary conditions. We notice that this approach can also be extended to other problems and systems of fractional equations as soon as we will be able to construct appropriate upper and lower solutions.http://matstud.org.ua/ojs/index.php/matstud/article/view/324fractional; holder spaces; large-time behaviour; comparison principle |
| spellingShingle | M. Krasnoshchok Monotone iterations method for fractional diffusion equations Математичні Студії fractional; holder spaces; large-time behaviour; comparison principle |
| title | Monotone iterations method for fractional diffusion equations |
| title_full | Monotone iterations method for fractional diffusion equations |
| title_fullStr | Monotone iterations method for fractional diffusion equations |
| title_full_unstemmed | Monotone iterations method for fractional diffusion equations |
| title_short | Monotone iterations method for fractional diffusion equations |
| title_sort | monotone iterations method for fractional diffusion equations |
| topic | fractional; holder spaces; large-time behaviour; comparison principle |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/324 |
| work_keys_str_mv | AT mkrasnoshchok monotoneiterationsmethodforfractionaldiffusionequations |