Multi-dimensional space fractional systems of nonlinear Schrödinger equations:
Modeling and Computation
Middle Tennessee State University, USA
The fractional nonlinear Schrödinger equation (FNLSE) fractionally generalizes the classical Schrödinger equation, which is a canonical model describing various physical phenomena such as the hydrodynamics, the nonlinear optics and the Bose–Einstein condensate. The FNLSE was first introduced by Laskin [1, 2] replacing Brownian trajectories in Feynman path integrals (corresponding to the classical Schrödinger equation) by the Lévy flights. Correspondingly, this equation includes a space fractional derivative of order α (1 < α < 2) instead of the Laplacian in the classical Schrödinger equation. A rigorous derivation of the space FNLSE can also be found in . There are very few work on space FNLS equations in one dimension , what to say of multi-dimensions .
In this talk we introduce space fractional nonlinear Schrodinger equation. Discuss a fourth order numerical scheme for one dimensional systems of coupled space fractional non-linear Schrodinger equations. A local extrapolation of exponential operator splitting scheme is introduced to solve multi-dimensional space-fractional nonlinear Schrodinger equations. The reliability and adaptability of the proposed methods are tested by implementing on systems of space-fractional nonlinear Schrodinger equations including space-fractional Gross-Pitaevskii equation, which is used to model optical solitons in graded-index fibers. Numerical experiments are shown to demonstrate the efficiency, accuracy and reliability of the methods.
 N. Laskin, Fractional quantum mechanics, Phys. Rev. E 62 (2000) 3135–3145.
 N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000) 298–305.
 K. Kirkpatrick, E. Lenzmann, G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys. 317 (2013), 563-591
 A.Q. M. Khaliq, X. Liang and K.M. Furati, A fourth order Implicit-Explicit scheme for the space fractional coupled nonlinear Schrödinger equations, Numerical Algorithms, 2016.
 X. Zhao, Z. Sun, Z. Hao, A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrodinger equation, SIAM J. SCI. Comput. 36 (2014) A2865-A2886.
Enhancement of Styrene Production in a Coupled Membrane Fixed Bed Catalytic Reactor
Fixed bed catalytic chemical reactors are characterized as flow reactors in which stationary beds of solid catalyst are present. They are operated at superficial velocities below the minimum fluidization velocity identified at the moment of fluidization of the solid beds. Their importance in industry and scientific research comes from the fact that all main catalyzed processes, except the catalytic cracking of gasoline, such as ammonia synthesis, steam reforming, and styrene production are carried out fixed bed catalytic chemical reactors.
Mathematical modeling of fixed bed catalytic chemical reactors is required as a tool to predict reactor responses for various changes which may happen, enhance the design aspects of reactors, and control the quality of products and production rates. Due to the presence of the solid catalysts in the fixed bed catalytic chemical reactors, different phenomena such as mass and heat transport processes, degree of mixing and catalyst deactivation, play significant roles in predicting the reactor performance as a result of varying reactor operating conditions, and interaction between the solid catalyst and the reacting fluid.
Styrene is an important monomer used in the production of the polystyrene, acrylonitrile-butadiene-styrene resins (ABS), and a variety of other polymers. Commercially, it is mostly produced by the dehydrogenation reaction of ethylbenzene in catalytic fixed bed reactors. The dehydrogenation reaction of ethylbenzene is endothermic and reversible which hence limits the styrene production. In this talk, a mathematical modeling for a proposed membrane reactor used to couple dehydrogenation of ethylbenze with hydrogenation of nitrobenzene will be presented. The simulation results of proposed membrane coupled reactor based on the homogeneous and heterogeneous reactor models will be shown and compared.
On Anti-Periodic Boundary Value Problems of Fractional-Order
King Abdulaziz University
Some recent work on anti-periodic boundary value problems of fractional-order will be reviewed. A new problem consisting of nonlinear coupled Liouville-Caputo type fractional differential equations supplemented with coupled anti-periodic boundary conditions will also be discussed.
Numerical Solutions to the Anomalous Flow of Single-Phase Fluid In Porous Media
Abeeb A. Awotunde
Simulation of fluid flow in porous media is an indispensable part of oil and gas reservoir management. Accurate prediction of reservoir performance and profitability of investment rely on our ability to model the flow behavior of reservoir fluids. Over the years, numerical reservoir simulation models have been based mainly on solutions to the normal diffusion of fluids in the porous reservoir. Recently, however, it has been documented that fluid flow in porous media does not always follow strictly the normal diffusion process. Small deviations from normal diffusion, called anomalous diffusion, have been reported in some experimental studies. Such deviations can be caused by different factors such as the viscous state of the fluid, the fractal nature of the porous media and the pressure pulse in the system.
In this work, we present explicit and implicit numerical solutions to the anomalous diffusion of single-phase fluids in heterogeneous reservoirs. An analytical solution is used to validate the numerical solution to the simple homogeneous case. The conventional wellbore flow model is modified to account for anomalous behavior. Example applications are used to show the behavior of wellbore and wellblock pressures during the single-phase anomalous flow of fluids in the reservoirs considered.
Adaptive control of fractional-order nonlinear systems in feedforward form
In this talk, a control design methodology is proposed for a class of fractional-order nonlinear systems in feed-forward form. The systems under consideration may be subject to parameter uncertainty of well-known upper bounds. The design of the stabilizing feedbacks is based on Lyapunov theory and convex-optimization tools. By adaptation of only one parameter in the control law, it will be shown that the design is not rooted to special form of nonlinearities and therefore, broader classes of systems could be considered.
Fractional Convection-diffusion equations, numerical solutions
A brief background describing the fractional convection-diffusion models will be presented. Following which, I will focus on the computational numerical solutions of these models. The numerical schemes will be based on finite elements for the spatial discretization, combined with finite-differences or discontinuous Galerkin methods for the time approximation. Some theoretical results will be discussed. Finally, various numerical results will be delivered.
Fractional damping: Theory and Applications
In this talk I will survey
the damping and dampers (integer order and fractional) from 1495 up to today. I
- the different types of damping and dampers
- the different models describing the appearance of vibrations in dynamical systems physics, chemistry and biology as well as in in mechanical engineering, electrical engineering, civil engineering, electromagnetism, fluid dynamics, electronics, electrochemistry, acoustics and other fields.
Some historical facts, theoretical results and applications will be given.