1. Mathematical and numerical analysis of nonlocal models for diffusion
M. D'ELIA, Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico, USA
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.
2. Nonlocal models of mechanics: analysis and computation
Q. DU , Dept. of Applied Physics and Applied Mathematics, Columbia University, New York, USA
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.
3. The probabilistic methods of solving and analysing the fractional differential equations
V. KOLOKOLTSOV, Dept. of Statistics, University of Warwick, Coventry, UK
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.
4. Differential equations of fractional order
J.J. NIETO-ROIG, Dept. of Mathematical Analysis, Universidad de Santiago de Compostela, Santiago de Compostela, Spain
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.
5. Numerical methods for space and space-time fractional diffusion
A.J. SALGADO Dept. of Mathematics, University of Tennessee, Knoxville, Tennessee, USA
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 in γ ∈ (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.