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**FMSE17
Titles & Abstracts **

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**Nonlinear
Fractional Models and Computational Algorithms for **

**Pattern
Formation in Biological Science**

A.Q.M. Khaliq

Middle Tennessee State University, USA

Nonlinear space fractional reaction diffusion equations are seen to provide a powerful modeling approach for understanding spatial heterogeneity in pattern formation. However, numerical solutions are particularly challenging when solving large systems of multidimensional space fractional nonlinear reaction diffusion equations. In this talk, we will introduce highly efficient and reliable time stepping methods to meet computational challenges introduced by nonlocality of the fractional Laplacian. Several numerical examples are presented for pattern formation in Biological systems. The talk is geared towards boarder audiences. Although, the talk will have serious research contents, it’s also meant to attract graduate students and early career faculty to opt the fascinating field of fractional modeling and computational algorithms in science and engineering.

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**A Journey
in Anomalous Diffusion:**

**Basics,
Fractional Models, and Numerics**

Khaled M. Furati

KFUPM, Dhahran, Saudi Arabia

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Normal diffusive phenomena are typically modeled by the diffusion equation, which could be obtained through Fickian-type fluxes or as a continuum limit of a stochastic process. For such models, the probability density function is Gaussian and the Mean Square Displacement (MSD) is linear in time. However, it has been observed that this is not the case in many complex systems in physics, chemistry, biology, economy, hydrology, etc. In such systems, the diffusion is slower or faster than the standard case, the probability density function has a long tail, and also the MSD becomes nonlinear in time. This transport phenomenon is referred to as anomalous diffusion. Fractional diffusion models emerged to be appropriate for modelling systems with such anomalous features. In such fractional models, integro-differential operators are employed to account for the memory and global dependence. In this talk, anomalous diffusion and fractional diffusion models will be highlighted. Then, an overview of the numerical treatment of such models will be presented.

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**Expansion
of Fractional Derivatives in Terms of an Integer Derivative Series: Physical
and Numerical Applications to Fractional Solitons**

Usama ALKhawaja

UAE University, Al Ain, UAE

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We use the displacement operator to derive an infinite series of integer order derivatives for the Grünwald-Letnikov fractional derivative and show its correspondence to the Riemann-Liouville and Caputo fractional derivatives. We demonstrate that all three definitions of a fractional derivative lead to the same infinite series of integer order derivatives. We find that functions normally represented by Taylor series with a finite radius of convergence have a corresponding integer derivative expansion with an infinite radius of convergence. Specifically, we demonstrate robust convergence of the integer derivative series for the hyperbolic secant (tangent) function, characterized by a finite radius of convergence of the Taylor series R= π/2, which describes bright (dark) soliton propagation in non-linear media. We also show that for a plane wave, which has a Taylor series with an infinite radius of convergence, as the number of terms in the integer derivative expansion increases, the truncation error decreases. Finally, we illustrate the utility of the truncated integer derivative series by solving two linear fractional differential equations, where the fractional derivative is replaced by an integer derivative series up to the second order derivative. We find that our numerical results closely approximate the exact solutions given by the Mittag-Leffler and Fox-Wright functions. Thus, we demonstrate that the truncated expansion is a powerful method for solving linear fractional differential equations, such as the fractional Schrödinger equation.

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**Realization of Fractional-Order Filters and PID
Controllers**

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Reyad El-Khazali

Khalifa University, Abu Dhabi – UAE

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Realization of fractional-order continuous or discrete-time filters of PID controllers depends heavily on the use of the proper approximation of the fractional-order Laplacian operators. Such approximations are usually represented by rational transfer functions that are not all necessarily stable nor of minimum phase. They can be obtained using Backward-Euler method, Trapezoidal (Tustin) discretization, Al-Alaoui Operator, a Hybrid interpolation of Simpson’s and Trapezoidal discrete-time integrators and El-Khazali Operator. Hence, it is crucial to use the proper approximation that does not jeopardize stability of the overall system. This will be demonstrated by several approximation methods that are popular in the literature. In addition to the previous requirements, some researchers do not pay attention to the accuracy nor to the complexity of the approximation, which may not yield satisfactory frequency response.

To simplify the design of fractional-order filters, El-Khazali operators are used to approximate the Laplacian operators, which provide a method for a systematic circuit and system design. Similarly, fractional-order digital filters are demonstrated by using modular structure of discrete-time fractional-order discrete-time operators, which can be used to synthesize discrete-time filters and PID controllers. These operators are of 1st-, 2nd-, 3rd-, or 4th-orders that only depend on the order of the fractional-order operator. They are stable and of minimum phase. A comparison between such operators and those ones obtained using the continued-fractional expansion method proves that one must be careful into what approximation to use.

The phase-diagram of the approximated models is usually ignored by many researchers. It usually provides sufficient information to define straightforward design methods for both continuous and discrete-time Lag, Lead, Lag-Lead, and PID controllers. It will also be shown that the order of the fractional-order operator plays a significant rule in the design process.

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**Fractional Modeling and Characterization of
Physiological Systems**

Meriem T. Laleg

KAUST, Thuwal, Saudi Arabia

Fractional operators are powerful tools for modeling physical phenomena involving memory effect or delays. Many studies have investigated modeling and analyzing biomedical and biological systems using fractional derivatives. In addition to the physiological insights that these fractional models provide, the differentiation orders offer new parameters that allow capturing more details that the integer order models fail to describe accurately and with fewer equations. I will present in this talk some examples of fractional models proposed to model physiological systems such as the neurovascular coupling and the blood vessel. I will also introduce efficient estimation methods for the calibration and the estimation of fractional models.

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**Particle
and Finite-Difference Solutions of **

**Space-Fractional
Diffusion Equations**

Omar M. Knio

KAUST, Thuwal, Saudi Arabia

This talk will overview recent progress with particle simulation of unsteady space fractional diffusion equations, as well as finite-difference solution of steady fractional diffusion equations with random diffusivity. We will in particular outline the construction of different particle-based approaches to the simulation of one-dimensional fractional sub-diffusion equations in unbounded domains. We rely on smooth particle approximations, and consider five methods for estimating the fractional diffusion term. The first method is based on a direct differentiation of the particle representation; it follows the Riesz definition of the fractional derivative and results in a non-conservative scheme. Three methods follow the particle strength exchange (PSE) methodology and are by construction conservative, in the sense that the total particle strength is time invariant. A fifth method is proposed based on the diffusion-velocity approach, where the diffusion term is turned into a transport term. The performance of all five approaches is assessed for the case of a one-dimensional fractional diffusion equation with constant diffusivity. This enables us to take advantage of known analytical solutions, and consequently conduct a detailed analysis of the performance of the methods. This includes a quantitative study of the various sources of error, namely filtering, quadrature, domain truncation, and time integration, as well as a space-time self-convergence analysis.

We will finally discuss the simulation of steady fractional diffusion equations with random coefficients. To this end, the random diffusivity field is decomposed in terms of a Karhunen-Loeve expansion, which is suitable truncated so as to capture the energy of the fluctuating field. Using a non-intrusive sampling formalism leads us to a set of deterministic equations that are solved using a recently constructed finite-difference scheme. Computational experiments are conducted to assess the performance of the stochastic approach thus constructed. This includes a computational study of the effects of the correlation length and variance of the diffusivity field, and of the order the fractional derivation on the statistics of the solution. We finally illustrate the capability of the present approach in supporting variance-based sensitivity studies, as well as model inference and calibration.

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**The
Finite Difference Method for Fractional Parabolic Equations **

**with
Fractional Laplacian**

Changpin Li

Shanghai University, China

In this talk, we present the finite difference method for fractional parabolic equation with fractional Laplacian, where the time derivative is the Caputo derivative with derivative order in (0,1) and the spatial derivative is the fractional Laplacian. Stability, convergence, and error estimate are displayed. Illustrative examples that support the theoretical analysis are provided.

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**Nonlinear
Nonlocal Behavior of Electrically Actuated Carbon Nanotube Resonators Assuming
Fractional Continuum Mechanics Theory**

Hassen M. Ouakad

KFUPM, Dhahran, Saudi Arabia

In the past few decades, problem formulation based on the Classical Continuum Mechanics (CCM) permitted to develop powerful, robust and reliable simulation tools to solve complex problems in the field of structural mechanics of micro and nano-eletromechanical systems (MEMS and NEMS). However, it is well known that at the molecular level, the mater is somehow discrete and heterogeneous, and therefore the hypotheses of the CCM, recognized as size-independent, are no longer valid in the small-scale. The CCM resulting governing equations lack an internal size dependent length scale, therefore it cannot predict any size effect and may fail when effects like size-dependency and scaling of mechanical phenomena play a crucial role, certainly do in the nano-scale. The above problems could be addressed using discrete models but all of them require a great computational effort. This fact provides a motivation towards developing modified and generalized continuum mechanics theories that are capable to capture the size effects through introducing intrinsic lengths in their respective formulation. Within this category fall the classical couple stress theory and the strain gradient theory, started in 1960s with the works of Mindlin and Tiersten. Another size-dependent continuum theory, which contains only one material length scale parameter, is the nonlocal continuum mechanics initiated by Eringen and coworkers back in 1972, which has been widely used to analyze many localized problems, such as wave propagation, dislocation, and crack singularities.

A different approach to non-local mechanics has been recently introduced in the context of fractional calculus. Fractional calculus (FC) is a branch of mathematical analysis that studies the differential operators of an arbitrary order. The attractiveness of FC application lays in the fact that: (1) fractional differential operators are nonlocal, and (2) there are many definitions of fractional derivatives. In the last decades, fractional differential equations have been used to capture physical phenomena in the nano-scale that cannot be caught by classical differential models. This talk will discuss some of the ongoing theoretical research of electrically actuated carbon nanotubes (CNTs) based NEMS resonators, where the fractional continuum mechanics (FCM) approach will be utilized to modal their respective nonlocal structural behavior. The nano-resonator static, eigenvalue problem (natural frequencies and modal shapes), and dynamic responses are obtained and the effects of the length-scale parameter are discussed and contrasted with those obtained with the solutions derived from the CCM. The presented model provides a basis for the study of the linear and nonlinear structural behaviour of elastic nano-structures showing significant nonlocal length/scale effects.

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