# Registered Data

## [00082] Development in fractional diffusion equations: models and methods

**Session Date & Time**:- 00082 (1/3) : 1C (Aug.21, 13:20-15:00)
- 00082 (2/3) : 1D (Aug.21, 15:30-17:10)
- 00082 (3/3) : 1E (Aug.21, 17:40-19:20)

**Type**: Proposal of Minisymposium**Abstract**: The mathematical study of diffusion and its applications has played an important role in modern mathematics. The study of fractional diffusion has become a new trend as a mathematical framework to describe anomalous diffusion. Indeed, in the real world anomalous diffusion is common. We wish to present the last and novel techniques regarding modeling with FDE and its mathematical analysis. In particular we are interested in modeling with the help of FDE, the resulting IBV problems, including free boundary problems. We also pursue the study of the qualitative properties of solutions including self-similar and fundamental solutions.**Organizer(s)**: Sabrina Roscani, Piotr Rybka**Classification**:__35R11__,__26A33__,__80A22__**Speakers Info**:- Vaughan Voller (The University of Minnesota)
- Łukasz Płociniczak (Wrocław University of Science and Technology)
- Gianni Pagnini ( Basque Center for Applied Mathematics)
**Sabrina Roscani**(CONICET - Universidad Austral)- Masahiro Yamamoto (The University of Tokyo)
- Serena Dipierro (The University of Western Australia)
- Katarzyna Ryszewska (Warsaw University of Technology)
- Petra Wittbold (University of Duisburg)
- Marvin Fritz (Technical University of Munich)
- Manuela Rodrigues (University of Aveiro)
- Piotr Rybka (University of Warsaw)
- Nelson Vieira (University of Warsaw)

**Talks in Minisymposium**:**[00119] Weak and entropy solutions of time-fractional porous medium type equations****Author(s)**:**Petra Wittbold**(University of Duisburg-Essen)

**Abstract**: We present results on existence and uniqueness of bounded weak and also unbounded entropy solutions to a degenerate quasilinear subdiffusion problem of porous medium type with bounded measurable diffusion coefficients that may explicitly depend on time. The integro-differential operator in the equation includes, in particular, the time-fractional derivative case. A key ingredient in the proof of existence is a new compactness criterion of Aubin-Lions type which involves the non-local in time operator.

**[00160] Regularity of weak solutions to parabolic-type problems with distributed order time-fractional derivative****Author(s)**:**Katarzyna Ryszewska**(Warsaw University of Technology)

**Abstract**: In this talk we will discuss Holder continuity of weak solutions to evolution equations with distributed order time-fractional derivative. It is a generalization of the result for a single order fractional derivative obtained by Prof. Zacher in 2010. The main difficulty to overcome in this case, is the lack of a natural scaling property of the equation. This is a joint work with Adam Kubica and Prof. Rico Zacher.

**[00161] Atom dislocation in crystals: from partial differential equations to dynamical systems****Author(s)**:**Serena Dipierro**(University of Western Australia)

**Abstract**: We reconsider the classical Peierls-Nabarro model for crystal dislocations in light of nonlocal equations, variational methods and dynamical systems, discussing heteroclinic, homoclinic, and multibump solutions. We also analyze the system in the mesoscopic and macroscopic limit, in which the dislocation function has the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior potential, which can be either repulsive or attractive, depending on the relative orientations of the dislocations. For opposite orientations, collisions occur, after which the system relaxes exponentially fast.

**[00162] Solution of a fractional Stefan problem using a Landau transformation****Author(s)**:**Vaughan Richard Voller**(University of Minnesota)

**Abstract**: The Stefan problem, tracking the motion of a heat conduction driven melt interface, is the classical moving boundary problem. A means of obtaining a solution for such problems is through the use of a variable transformation --- the Landau transformation --- immobilizing the melt interface. Here we apply this technique for the approximate solution of a fractional Stefan problem where the integer time derivative, in the governing diffusion equation, is replaced with a fractional derivative.

**[00163] Numerical methods for nonlocal and nonlinear parabolic equations with applications in hydrology and climatology****Author(s)**:**Lukasz Plociniczak**(Wroclaw University of Science and Technology)

**Abstract**: We present some of our results concerning numerical discretizations of nonlinear and fractional in time parabolic equations. Along with a collection of various methods and statements about their convergence and stability, we stress their motivation and real-world applications.

**[00166] Fractional diffusion as an intermediate asymptotic regime****Author(s)**:**Gianni Pagnini**(BCAM - Basque Center for Applied Mathematics)- Paolo Paradisi (ISTI-CNR, Pisa)
- Silvia Vitali (Eurecat Centre Tecnològic de Catalunya Barcellona )

**Abstract**: A continuous-time random walk driven by two different Markovian hopping-trap mechanisms is investigated and it is shown that paradigmatic features of anomalous diffusion are met. More precisely, anomalous diffusion results from a process that goes through the action of two co-existing Markovian mechanisms acting with different statistical frequency, and the probability of occurrence of this switch between the two Markovian settings originates and fully characterizes the anomalous diffusion. In fact, ensemble and single-particle observables of this model have been studied and they match the main characteristics of anomalous diffusion as they are typically measured in living systems. In particular, the celebrated transition of the walker’s distribution from exponential to stretched-exponential and finally to Gaussian distribution is displayed by including also the Brownian yet non-Gaussian interval. Moreover, the model dynamically provides the power-law exponent of the mean-square displacement as a function of the probability of switching between the two Markovian states, namely the fractional order. Hence, fractional diffusion emerges as an intermediate asymptotic regime. Finally, within the present approach, fractional diffusion can be interpreted as a mathematical method for bridging two co-existing equilibrium states in a disordered medium. This talk is based on: Vitali S, Paradisi P and Pagnini 2022 J. Phys. A: Math. Theor. 55 224012

**[00167] On different formulations for time-fractional Stefan problems****Author(s)**:**Sabrina Roscani**(CONICET - Universidad Austral)

**Abstract**: We present a one-dimensional fractional Stefan problem for a memory flux derived from thermodynamic balance statements and provide a memory enthalpy formulation related to the previous model. The Stefan condition for each problem at the free interface is analyzed and numerical simulations obtained from the enthalpy model are given.

**[02596] Dissipativity of the energy functional in time-fractional gradient flows****Author(s)**:**Marvin Fritz**(Johann Radon Institute for Computational and Applied Mathematics (RICAM))

**Abstract**: In this talk, the monotonicity of the energy functional of time-fractional gradient flows is investigated. It is still unknown whether the energy is dissipating in a timely manner. This characteristic is critical for integer-order gradient flows, and many numerical systems utilize it. We suggest an energy functional that incorporates the solution's history, which is reasonable given that time-fractional partial differential equations are nonlocal in time and feature a natural memory effect. On the basis of this new energy, we demonstrate that a time-fractional gradient flow is equivalent to an integer-order flow. In addition, this connection guarantees the dissipative nature of the augmented energy and permits the development of numerical schemes.

**[02617] Fractional diffusion equation with psi-Hilfer derivative****Author(s)**:**M. Manuela Rodrigues**(University of Aveiro, Portugal)

**Abstract**: We consider the multidimensional time-fractional diffusion equation with $\psi$-Hilfer derivative. An integral representation of the solution to the associated Cauchy problem involving Fox H-functions is obtained. Fractional moments of arbitrary order are computed. Series representations of the first fundamental solution are presented. Some plots of the fundamental solution are presented for particular choices of the function $\psi$ and the fractional parameters. Joint work with N. Vieira (UA - CIDMA), M. Ferreira (IPLeiria~ \& ~ CIDMA)

**[02623] On the $\psi$-Hilfer time-fractional telegraph equation in higher dimensions****Author(s)**:**Nelson Vieira**(CIDMA - University of Aveiro)

**Abstract**: We consider the time-fractional telegraph equation in $\mathbb{R}^n \times \mathbb{R}^+$ with $\psi$-Hilfer fractional derivatives. For the solution, we present an integral representation involving Fox H-functions of two variables, and in terms of double series. For $n=1$, we prove the conditions under which we can interpret the first fundamental solution as a probability density function. Some plots of the fundamental solutions are presented. Joint work with M. Ferreira $($IPLeiria & CIDMA$)$ and M.M. Rodrigues $($CIDMA$)$.

**[02767] The fundamental solution to the space fractional diffusion equation****Author(s)**:**Piotr Rybka**(University of Warsaw)- Tokinaga Namba (Nippon Steel Corporation)
- Shoichi Sato (University of Tokyo)

**Abstract**: We consider a one dimensional diffusion equation equation involving the divergence of the Caputo space derivative of order less than one. We construct a self-similar solution, $\mathcal{E}$, which permits us to derive the representation formulas for boundary value problem on the half line. We also present properties of $\mathcal{E}$, and we show the infinite speed of signal propagation. This is a joint research with T.Namaba and S.Sato.

**[04911] Uniqueness for inverse source problems for time-fractional diffusion-wave equations****Author(s)**:**Masahiro Yamamoto**(The Univ. Tokyo)

**Abstract**: For time-fractional diffusion-wave equations, $\partial_t^{\alpha} u(x,t) = -Au + \mu(t)f(x)$ for $x\in \Omega, 0