@article{Krasnoshchok_2022, title={Monotone iterations method for fractional diffusion equations}, volume={57}, url={http://www.matstud.org.ua/ojs/index.php/matstud/article/view/324}, DOI={10.30970/ms.57.2.122-136}, abstractNote={<p>In recent years, there has been a growing interest on non-local<br>models because of their relevance in many practical applications. A<br>widely studied class of non-local models involves fractional order<br>operators. They usually describe anomalous diffusion. In<br>particular, these equations provide a more faithful representation<br>of the long-memory and nonlocal dependence of diffusion in fractal<br>and porous media, heat flow in media with memory, dynamics of<br>protein in cells etc.</p> <p><br>For $a\in (0, 1)$, we investigate the nonautonomous fractional<br>diffusion equation:</p> <p>$D^a_{*,t} u - Au = f(x, t,u),$</p> <p> where<br>$D^a_{*,t}$ is the Caputo fractional derivative and $A$ is a<br>uniformly elliptic operator with smooth coefficients depending on<br>space and time. We consider these equations together with initial<br>and quasilinear boundary conditions.</p> <p>The solvability of such problems in H\"older spaces presupposes<br>rigid restrictions on the given initial data. These compatibility<br>conditions have no physical meaning and, therefore, they can be<br>avoided, if the solution is sought in larger spaces, for instance in<br>weighted H\"older spaces.</p> <p>We give general existence and uniqueness result and<br>provide some examples of applications of the main theorem. The main<br>tool is the monotone iterations method. Preliminary we developed the<br>linear theory with existence and comparison results. The principle<br>use of the positivity lemma is the construction of a monotone<br>sequences for our problem. Initial iteration may be taken as either<br>an upper solution or a lower solution. We provide some examples of<br>upper and lower solution for the case of linear equations and<br>quasilinear boundary conditions. We notice that this approach can<br>also be extended to other problems and systems of fractional<br>equations as soon as we will be able to construct appropriate upper<br>and lower solutions.</p>}, number={2}, journal={Matematychni Studii}, author={Krasnoshchok, M.}, year={2022}, month={Jun.}, pages={122-136} }