**May 28, 2018 at 09:00**

**Speaker(s):**
** M. D'ELIA, Q. DU, V. KOLOKOLTSOV, J. NIETO ROIG, A.J. SALGADO**

**Center(s):**
Sandia National Laboratories, Albuquerque, New Mexico, USA; Columbia University, New York, USA; University of Warwick, Coventry, UK; Universidad de Santiago de Compostela, Santiago de Compostela, Spain; University of Tennessee, Knoxville, Tennessee, USA

**Summer School on Fractional and Other Nonlocal Models**
**May 28-31, 2018**
**BCAM- Basque Center for Applied Mathematics**

Alameda Mazarredo 14, Bilbao, Basque Country, Spain
In many settings, differential equation models do not always provide adequate fidelity to the continuum phenomena being modeled. This occurs in diverse areas such as crack nucleation and propagation in solids, charge transport in semiconductors, subsurface and polymer flows, and animal migration in ecological systems, just to name a few. A common feature in these and other settings is that interactions can occur at a distance and not just in infinitesimal neighborhoods as is the case for differential equation models. The summer school focuses on the continuum modeling, analysis, simulation, and numerical analysis of settings that feature such nonlocal interaction.

Although the universe of continuum nonlocal models is large, two approaches have gained popularity due to their wide applicability. On one hand, we have fractional derivative models that can be used to describe many non-Fickian diffusion processes that are observed in practice. Both spatial and temporal fractional derivative models are used for this purpose. On the other hand, we find integral equation models that can treat problems such as those in solid mechanics, which cannot be modeled using fractional derivatives. These integral equation models can be viewed as generalizations of fractional derivative models and can be treated through the use of a nonlocal vector calculus.

The general aim of the school is for students to become familiar with these two approaches. In both cases, the lectures cover models arising in applications, the mathematical analysis of the models, algorithms for obtaining approximate solutions, and the numerical analysis of those algorithms. In addition, the close connection between some of the models discussed and stochastic processes is also a subject of interest.

In addition to attending the summer school lectures, (junior) attendees who are already working on problems related to the subject matter of the school can present a lecture about their own research. PhD students and postdoctoral researchers attending the school may be financially supported by the organization.

**Registration:**
Attendance at the school is free (there is no registration fee), but participants are requested to fill in the registration form.

Please go to: **http://www.bcamath.org/en/activities/workshops**, click on the "Registration" button corresponding to this Summer School, and fill the form with your information.

If your research is on problems related to the subject matter of the school and you are willing to give a short presentation (20 minutes talk + 5 minutes for questions) please include title and abstract in the "Comments" section when you register for the school.

**Financial support: **

PhD students and postdoctoral researchers attending the school may be financially supported by the host organisation. If you are interested, please say so in the "Comments" section in the Registration form.

Note that priority will be given to PhD students. Moreover, funding availability is limited and hence support is not guaranteed for all applicants

**Courses:**

M. D'ELIA, Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico, USA

**Mathematical and numerical analysis of nonlocal models for diffusion**

In these lectures I will introduce nonlocal diffusion equations characterized by finite interactions and present the mathematical analysis (well-posedness and regularity results) of their variational form by means of the nonlocal vector calculus. I will also describe available numerical methods for their discretization (e.g. meshless methods, Galerkin methods) mainly focusing on the finite element method. I will talk about the computational challenges associated with the numerical solution of nonlocal models and possible avenues to address them (e.g. fast solvers and coupling methods). Finally, I will illustrate how to treat fractional differential equations using the techniques developed for nonlocal models with finite interaction radius and show that the nonlocal operator treated in this set of lectures represents a general definition that includes the fractional Laplacian as a special instance.

Q. DU, Dept. of Applied Physics and Applied Mathematics, Columbia University, New York, USA

**Nonlocal models of mechanics: analysis and computation**

Our lectures will be centered on some nonlocal models of continuum mechanics that, unlike more popularly studied scalar nonlocal models, often deal with vector and tensor fields and hence give rise to more complex systems with richer mathematical structures. To offer a systematic framework for the rigorous analysis and the effective computation of these nonlocal systems (as well as those relevant to diffusion processes), we will introduce some basic elements of the nonlocal vector calculus, nonlocal calculus of variations, and asymptotically compatible discretizations. As illustrative examples, we will present applications to nonlocal peridynamics in solid mechanics and nonlocal Stokes models in fluid mechanics. Both of these nonlocal systems involve nonlocal operators that are characterized by a finite horizon parameter measuring the range of nonlocal interactions. Relations to their traditional local counterparts as the horizon vanishes will be explored. Mesh-based and meshfree discretization will also be discussed.

V. KOLOKOLTSOV, Dept. of Statistics, University of Warwick, Coventry, UK

**The probabilistic methods of solving and analysing the fractional differential equations**

We shall present in detail various approaches to the analysis of fractional ordinary and partial differential equations (fractional diffusions, Schrodinger, general kinetic equations, etc) and their numerous extensions based on the methods of stochastic analysis. This will include the random time-change by the inverse Lévy processes and more general stoppings of Markov processes on the attempt to cross the boundary. Resulting formulas for the solutions of linear equations based on the generalized operator-valued Mittag-Leffler function and the chronological operator-valued Feynman-Kac formulae yield the path integral representations for solutions amenable to numeric simulations and qualitative analysis. This approach would allow us further to represent the related nonlinear fractional partial differential equations, as infinite-dimensional multiplicative integral equations, yielding again a natural setting for their analysis and numeric solution schemes.

J. NIETO ROIG, Dept. of Mathematical Analysis, Universidad de Santiago de Compostela, Santiago de Compostela, Spain

**Differential equations of fractional order**

The purpose is to introduce the main tools to study fractional differential equations: Fractional calculus and some relevant functions such as the classical Mittag-Leffler functions. Then some details on fractional differential equations in one variable and on partial fractional equations such as the fractional Laplacian are presented. Finally some real problems where fractional differential equations appear will be introduced.

Breakdown of course:

Part 1: Fractional calculus. Different types of fractional derivatives. Mittag-Leffler functions.

Part 2: Fractional differential equations in one variable

Part 3: Some mathematical models using fractional derivatives.

Part 4: The fractional Laplacian. An example of analytical solution for a partial differential equation of fractional order.

A.J. SALGADO, Dept. of Mathematics, University of Tennessee, Knoxville, Tennessee, USA

**Numerical methods for space and space-time fractional diffusion**

We present and analyze finite element methods (FEM) for the numerical approximation of the spectral fractional Laplacian. These methods hinge on the extension to an infinite cylinder in one more dimension. We discuss rather delicate numerical issues that arise in the construction of reliable FEMs and in the a priori and a posteriori error analyses of such FEMs for linear and nonlinear problems.

We will also consider a space-time fractional parabolic problem where the spatial operator is the spectral fractional Laplacian and a Caputo fractional derivative of order $gamma in (0,1)$. We overcome the spatial nonlocality by means of the previously presented extension approach and discuss the regularity of the solution to this problem, which allows us to obtain error estimates for fully implicit schemes.

We show illustrative simulations, applications, and mention challenging open questions.

**Organizing committee:**

Elena AKHMATSKAYA (BCAM, Ikerbasque)

Nicole CUSIMANO (BCAM)

Luca GERARDO-GIORDA (BCAM)

Max GUNZBURGER (Florida State University, USA)

Marina MURILLO (Universitat Jaume I, Castelló de la Plana)

Jesús María SANZ-SERNA (U. Carlos III, Madrid)