Dept. of Mathematical & Computational Sciences
In this talk we consider distributed order time space anomalous diffusion with non-homogenous boundary conditions. Time fractional derivative is defined in Caputo sense and Matrix Transfer Technique (MTT) is utilized for fractional spatial discretization. Midpoint quadrature rule is applied to approximate the spatial integral term. Numerical experiments are presented by employing a recently developed split-step method for space fractional reaction diffusion equations. Computational complexity and efficiency of the method is discussed as well as reliability and stability.
Saudi Aramco Fellow
Gigantic reservoirs contribute significantly to meet the world energy demand.Computer simulation is largely used as an effective tool for managing the existing fields and discovering new resources. Numerical simulators used for this purpose are comprised of large sets of nonlinear and coupled partial differential equations. These equations describe the flow of multi-phase multi component compressible fluids in porous media and in pipes.
The PDEs are discretized both space and time resulting in discrete nonlinear equations which are solved by iterative methods. Because of the large size of the reservoirs, the number of computational elements are generally in the order of millions to billions or more. Solving such systems requires specific numerical methods and evolving computer hardware technology utilizing hundreds of thousands of CPUs and GPUS.
EXPEC ARC, GPT Saudi ARAMCO-KSA
Physical experiments show wave amplitudes attenuation linearly increase with respect to frequencies for certain materials, including many types of rocks studied in geophysics. However, a long standing problem is there is no time domain wave equation to precisely describe this type of attenuation, despite many approximations having been proposed. We present a simple time domain wave equation to precisely simulate such linear attenuation (i.e., both dissipations and dispersions). The key idea is to represent the wavefield (the function to be solved) by complex value functions, rather than by real value functions which all previous time domain wave equations have employed For verification, we have compared CWE numerical solutions with analytical solutions for acoustic waves in constant velocity media, and obtained nearly perfect matching between these two solutions.
The School of Mathematical Sciences
In this talk, we will present mixed finite element methods for space-fractional partial differential equations for flow and transport in fractured porous media. For the flow in fractured porous media, a space-fractional Darcy’s law will be introduced and it motivate us to use mixed finite element method for the flow by coupling the fractional Darcy’s law and mass conservation law. The mass-conservative property of the numerical scheme is naturally obtained which is important for the simulation of transport in fractured porous media. Then an upwind mixed finite element method will be introduced to simulate the transport with a fractional diffusion in fractured porous media. Performance of our schemes including numerical stability and solution accuracy will also be discussed.
Earth Sciences & Engineering
In this talk we will introduce a kind of space-fractional partial differential equations to model and simulate the flow and transport in fractured porous media. Firstly, we will introduce a space-fractional Darcy’s law for the flow in fractured porous media, which is coupled with the mass conservation law to model the porous media flow. The new fractional partial differential equations for flow (and transport) in fractured porous media is more physical than that in literature; in particular, our formulation satisfies the principle of Galilean invariance. We will also compare our new model with the discrete fracture model and with the dual porosity dual permeability model for flow (and transport) in fractured porous media.
Vu Kim Tuan
University of West Georgia-USA
In this talk we consider a fractional diffusion process on a finite length rod modeled by
We are concerned with the recovery of the diffusion coefficient from the measurement of the flux at one end of the rod only. This problem is close in spirit with the boundary control, where is a given input on the boundary and we can observe its response. However the boundary control method is usually applied to wave equations because of their finite wave speed propagation. We prove that we can uniquely recover from a single boundary measurement and provide a constructive procedure for its recovery. The algorithm is based on the well known Gelfand-Levitan-Gasymov inverse spectral theory of Sturm-Liouville operators and the Laplace transform method. More precisely, we have.
MESA Lab of University of California-USA
This talk tries to connect
stochasticity and fractional calculus in a generic sense. It has
been widely experienced that randomness, when properly
introduced, could enhance performance in optimization process,
modeling and control etc. When asking what is the “optimal”
randomness, in many cases, heavy-tailness (HT, or algebraic
tail) emerges. A widely known example is the Levy flights used
in population-based random search (e.g. Cuckoo search).
Fractional calculus is shown to play an important role in
characterizing HT processes. Thus the fractional order can be
regarded as a tuning knob to achieve “optimal stochasticity”. In
turn, we can say “optimal stochasticity” entails fractional
calculus. In machine learning problems, optimal stochasticity
may lead us to better than the best optimization performance. An
illustrative example is given. For big data research, how to
best quantify the variability leads us to the so-called
“fractional-order data analytics (FODA)” using fractional
calculus based methods. In summary, this talk attempts to
convince the audience that whenever there is randomness, there
is a chance to ask what is the optimal randomness, and in turn a
chance to use fractional calculus.