We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.

Fractional Calculus: Integral and Differential Equations of Fractional Order

Consider the function defined for $\alpha _{1},\ \alpha _{2} \in \mathbb{R}$ (α 1 2 +α 2 2 ≠ 0) and $\beta _{1},\beta _{2} \in \mathbb{C}$ by the series 

$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$

(6.1.1)

Such a function with positive α 1 > 0, α 2 > 0 and real $\beta _{1},\beta _{2} \in \mathbb{R}$ was introduced by Dzherbashian [Dzh60].

Multi-index Mittag-Leffler Functions

In some of the earliest work on quantum computing, Feynman showed how to implement universal quantum computation with a time-independent Hamiltonian. I show that this remains possible even if the Hamiltonian is restricted to be the adjacency matrix of a low-degree graph. Thus quantum walk can be regarded as a universal computational primitive, with any quantum computation encoded in some graph. The main idea is to implement quantum gates by scattering processes.

Universal Computation by Quantum Walk

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete-time quantum walks.

Quantum walks: a comprehensive review

The problem of deriving approximations for multinormal integrals is examined using results of asymptotic analysis. The boundary of the integration domain given by g(x¯)=0 is simplified by replacing g(x¯) by its Taylor expansion at the points on the boundary with minimal distance to the origin. Two approximations which are obtained by using a linear or quadratic Taylor expansion are compared. It is shown that, applying a quadratic Taylor expansion, an asymptotic approximation for multinormal integrals can be obtained, whereas using linear approximations large relative errors may occur.

Asymptotic approximations for multinormal integrals

Asymptotic Expansions of Integrals

The title problem is posed as a linear heat equation in one space dimension $(x > 0)$ and time $(t > 0)$, with a nonlinear radiative-type boundary condition on the surface $(x = 0)$. Existence and ...

Diffusion in a Semi-Infinite Region with Nonlinear Surface Dissipation

It is known that the nonlinear Volterra integral equation \[ \varphi (t)\pi ^{( - 1 / 2)} \,\int_0^t (t - s)^{{ - 1 / 2} } [ {f(s) - \varphi ^n (s)} ]ds,\quad t\geqq 0,\geqq n\geqq 1,\] has a continuous solution $\varphi (t) \geqq 0$ which is unique for each bounded and locally lntegrable function $f(t) \geqq 0$ Our prior investigation considered the asymptotic behavior, as $t \to \infty $, of $\varphi (t)$ when $f(t) \sim \gamma _0 t^{ - a_0 } + \cdots ,\gamma _0 > 0,a_0 \geqq 0$. The goal here is to complete this analysis so that the behavior of $\varphi (t)$, as $t \to \infty $, is provided for all $a_0 \geqq 0,n \geqq 1$. In achieving this, we must treat some cases with properties that are markedly different from those previously considered.

Asymptotic Solution to a Class of Nonlinear Volterra Integral Equations

Asymptotic expansion of a class of integral transforms with algebraically dominated kernels

Asymptotic expansions for λ ≫1 of functions defined by integrals of the form 
$$I(\lambda ;\theta ) = \mathop \smallint \limits_\Gamma \exp \{ i\lambda \phi (k;\theta )\} g(k;\theta )dk$$

 are considered in the case where there are two stationary points of φ which approach ±∞ as the second variable θ approaches some critical value, say θ
0. In this limit the results of the classical methods of stationary phase and steepest descents become invalid. This paper is devoted to the development of an asymptotic expansion of I that remains valid even for θ near and equal to θ
0. The motivating physical problem is the propagation of signals in dispersive media. Indeed, the results of the present paper can be used to study the behavior of that portion of a signal called the “precursor” in a neighborhood of its front of propagation. The technique used to obtain the uniform expansion is an adaptation of the method originally developed by Chester, Friedman and Ursell in their treatment of the problem of two nearby stationary points. Here, however, we find that a certain family of Bessel functions play the role of the Airy functions in that problem. We also obtain the interesting result that our expansion remains valid for λ merely bounded away from zero and θ → θ
0. In fact a theorem is proved which establishes the asymptotic nature of our results in the relevant limits. Finally, two examples are considered to illustrate the use of these results.

Richard A. Handelsman

Papers

Asymptotic Expansions of Integrals

Diffusion in a Semi-Infinite Region with Nonlinear Surface Dissipation

Asymptotic Solution to a Class of Nonlinear Volterra Integral Equations

Asymptotic expansion of a class of integral transforms with algebraically dominated kernels

Uniform asymptotic expansions of integrals that arise in the analysis of precursors