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# Archive: Fractional Derivatives

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 Dr. Werner Linde, University of Delaware Dr. Werner Linde, University of Delaware Ewg 336: Title: Gaussian Processes generated by Linear Operators – The Riemann-Liouville Process
Let $$H$$ be a separable Hilbert space and let $(T,\rho)$ be a compact metric space. Each (linear, bounded) operator from $$H$$ to $$C(T)$$, the Banach space of continuous functions on $$T$$, generates in natural way a centered Gaussian process $$X=\{X(t) : t\in T\}$$. One basic example is the Brownian Motion generated in this way by the Volterra integration operator $$V : L_2[0,1]\to C[0,1]$$ acting as $$(V f)(t)=\int_0^t f(x)\text{d}x\,, \quad 0\le t\le 1\,.$$ If $$\alpha>1/2$$, then the Riemann-Liouville operator $$R^\alpha : L_2[0,1]\to C[0,1]$$ defined by $$(R^\alpha f)(t)=\frac{1}{\Gamma(\alpha)}\int_0^\alpha (t-x)^{\alpha-1} f(x)\text{d}x\,,\quad 0\le t\le 1\,,$$ generates a Gaussian process $$\{R_\alpha(t) : t\in[0,1]\}$$, called Riemann-Liouville process of order $$\alpha$$. In the talk we present in detail how Gaussian random processes arise from linear operators on Hilbert spaces. Based on this construction, we introduce the Riemann-Liouville process and state some properties of it. Among them are the construction of optimal series representations for $$R_\alpha$$ and its relation to the fractional Brownian Motion provided that 1/2
11/28/2016 7:30:00 PM 11/28/2016 8:30:00 PM False Ryan Evans, University of Delaware Ryan Evans, University of Delaware EWG 336

Title: Biochemical Reactions: An Application of Fractional Calculus
Many chemical reactions in nature involve a stream of chemical reactants flowing through a fluid-filled volume, over a surface to which other reactants are confined. Examples include blood clotting and drug absorption. These reactions, referred to as surface-volume reactions, are a natural application of fractional calculus. We will discuss a mathematical model for surface-volume reactions, and how this model led to a new result in fractional integral equations.

3/24/2016 5:00:00 PM 3/24/2016 6:00:00 PM False Werner Linde, University of Delaware Werner Linde, University of Delaware EWG 336

Title: Fractional Integration with Varying Smoothness

For $$\alpha >0$$ the fractional integration operator (or Riemann-Liouville operator) $$R^\alpha$$ is defined by $$(R^\alpha f) (t) = \frac{1}{\Gamma (\alpha)} \int_0^t (t-x)^{\alpha-1} f(x) \, \text{d}x.$$ If $$\alpha > \text{max}\{ \frac{1}{p}-\frac{1}{q},0\}$$, then $$R^\alpha$$ is a bounded operator from $$L_p [0,1]$$ to $$L_q[0,1]$$, $$1 \leq p,q \leq \infty$$. The number $$\alpha >0$$ describes the type of smoothness of the image $$g = R^\alpha f$$.

We investigate a problem motivated by probabilistic applications. What happens if the degree of smoothness changes by time $$t \geq 0$$? That is, for some measurable function $$\alpha : [0,1] \to (0,\infty)$$ we investigate boundedness and compactness properties of $$R^{\alpha(\cdot )}$$ defined by $$(R^{\alpha (\cdot )} f) (t) = \frac{1}{\Gamma (\alpha (t))} \int_0^t (t-x)^{\alpha (t)-1} f(x) \, \text{d}x, \quad 0\leq t \leq 1.$$ One of the questions we treat is whether or not $$R^{\alpha( \cdot )}$$ is bounded from $$L_p [0,1]$$ to $$L_q[0,1]$$. It turns out that this is so if $$p>1$$ while it becomes wrong for $$p=1$$.

The presented results are taken from a joint work with M.A. Lifshits from St. Petersburg (Russia).

3/10/2016 6:00:00 PM 3/10/2016 7:00:00 PM False Udita Katugampola Udita Katugampola EWG 336

Title of the talk: What is a fractional derivative?

In this tutorial seminar, we will introduce the concept of fractional derivatives along with some historical backgrounds. We will talk about several approaches to form several different fractional derivatives. Among such derivatives, we will spend some time for the Riemann-Liouville derivative, Caputo derivative, Hadamard derivative, Riesz derivative and finally Katugampola derivative. We will then talk about what we mean by a fractional differential equation and some approaches to such differential equations. One of the main goals of the talk is to motivate students for research work in this beautiful realm of research.

3/4/2016 4:20:00 PM 3/4/2016 5:10:00 PM False