Volume 17, Issue 3

The waveform relaxation method has been successfully applied into solving fractional ordinary differential equations and fractional functional differential equations [11, 5]. In this paper, the waveform relaxation method is further used to solve fractional differential-algebraic equations, which often arise in integrated circuits with new memory materials. We give the iteration scheme of the waveform relaxation method and analyze the convergence of the method under linear and nonlinear conditions for the right-hand of the equations. Numerical examples illustrate the feasibility and efficiency of the method.

In this paper, we consider the multiplicity of positive solutions for a class of nonlinear boundary-value problem of fractional differential equations with integral boundary conditions. By means of a fixed point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of multiple positive solutions to the integral boundary value problem.

Numerical schemes for solving fractional differential equations are computationally heavy, due to the floating-point operations needed and, more importantly, the data flow within the entire memory system of a computer. We choose the fractional Adams-Bashforth-Moulton method as a representative numerical scheme, and review various code optimizations that can be applied to its serial and parallel implementations. As the most important contribution of this paper, we propose a simple methodology to analyze the achievable serial and parallel performance, based on quantifying the amount of data flow through various stages of the entire memory system, together with a small set of easily obtainable hardware parameters. This quantitative approach to performance modeling can in most cases pinpoint the real performance bottleneck, while also verifying the actual performance improvements due to various code optimizations. Moreover, the optimization techniques and performance modeling approach can both be applied to other convolution-intensive numerical methods for solving fractional differential equations.

In this paper, we determine the eigenvalue intervals of λ 1, λ 2, ..., λ n for which the iterative system of nonlinear Sturm-Liouville fractional order two-point boundary value problem possesses a positive solution by an application of Guo-Krasnosel’skii fixed point theorem on a cone.

In this paper, we discuss the solutions to nonlocal initial value problems of fractional order functional differential equations in a Banach space. In particular, we prove the existence and uniqueness of mild and classical solutions assuming that −A generates a resolvent operator family and nonlinear part is a Lipschitz continuous function. We also investigate the global existence of the solution. At the end, a fractional order partial differential equation is given to illustrate the obtained abstract results.

This note is dedicated to the centennial jubilee of the Russian Academician Yury Rabotnov, who was a pioneer in the application of fractional operators in Mechanics of Solids. In the present Note, the authors wish to present a contemporary handling of Rabotnov operators introduced in 1948 and to show their connectedness with fractional derivatives, what provides the ageless interest to his ideas and results.

Editorial Note: The original version of the paper is usually cited as Yu.N. Rabotnov, Equilibrium of an elastic medium with after-effect (in Russian). Prikladnaya Matematika i Mekhanika (J. Appl. Math. Mech.) 12, No 1 (1948), 53–62. This is a re-printed version of the paper, translated from Russian into English by Marina Shitikova, under the kind permission by the Editorial Board of the journal “Prikladnaya Matematika i Mekhanika”, Institute of Mechanics — Russian Academy of Sciences, http://pmm.ipmnet.ru/ru/, a journal published with English translation since 1958, as “Journal of Applied Mathematics and Mechanics” by Elsevier http://www.journals.elsevier.com/journal-of-applied-mathematics-and-mechanics/. Note that nowadays, the fractional exponential function ∋α (β, t) introduced by Rabotnov as (2.5), is known also as the Rabotnov function and as a special case of the Mittag-Leffler function widely used in fractional calculus. Rabotnov spoke also about differential equations with fractional derivatives (Sect. 3) but preferred to work with integral equations methods. Yury Nikolaevich Rabotnov (24 February 1914–15 May 1985) was a great Russian scientist in the field of Mechanics. More about his life and contributions, the readers can find in this journal (Fract. Calc. Appl. Anal.), in the articles: D. Valério, J. Tenreiro Machado, V. Kiryakova, 17, No 2 (2014), 552–578; DOI: 10.2478/s13540-014-0185-1; http://link.springer.com/article/10.2478/s13540-014-0185-1 Yu. A. Rossikhin and M. V. Shitikova, 10, No 2 (2007), 111–122; http://www.math.bas.bf/~fcaa Yu.A. Rossikhin, M.V. Shitikova, 17, No 3 (2014) — this same issue, 674–683; DOI: 10.2478/s13540-014-0192-2.

In the past decade, researchers working on fractional-order systems modeling and control have been considering working on the design and development of analog and digital fractional-order differentiators, i.e. circuits that can perform non-integer-order differentiation. It has been one of the major research areas under such field due to proven advantages over its integer-order counterparts. In particular, traditional integer-order proportional-integral-derivative (PID) controllers seem to be outperformed by fractional-order PID (FOPID or PIλDμ) controllers. Many researches have emerged presenting the possibility of designing analog and digital fractional-order differentiators, but only restricted to a fixed order. In this paper, we present the conceptual design of a variable fractional-order differentiator in which the order can be selected from 0 to 1 with an increment of 0.05. The analog conceptual design utilizes operational amplifiers and resistor-capacitor ladders as main components, while a generic microcontroller is introduced for switching purposes. Simulation results through Matlab and LTSpiceIV show that the designed resistor-capacitor ladders can perform as analog fractional-order differentiation.

In this paper we study the existence and the multiplicity of solutions for an impulsive boundary value problem for fractional order differential equations. The notions of classical and weak solutions are introduced. Then, existence results of at least one and three solutions are proved.

In this paper we discuss fractional generalizations of the filtering problem. The ”fractional” nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process.

We review some basic facts about the λ-cosine transforms with odd kernel on the unit sphere S n−1 in ℝn. These transforms are represented by the spherical fractional integrals arising as a result of evaluation of the Fourier transform of homogeneous functions. The related topic is the hemispherical transform which assigns to every finite Borel measure on S n−1 its values for all hemispheres. We revisit the known facts about this transform and obtain new results. In particular, we show that the classical Funk- Radon-Helgason inversion method of spherical means is applicable to the hemispherical transform of L p-functions.

We study the linear heat equation driven by a random noise which admits a covariance measure structure with respect to the time variable and has a spatial covariance given by the Riesz kernel. We focus our attention on the particular case when the noise behaves as a bifractional Brownian motion in time.

The paper is devoted to the study of representation of complex powers of closed linear operators whose negatives generate equicontinuous (g α, C)-regularized resolvent families (0 < α ≤ 2) on sequentially complete locally convex spaces. Several interesting formulas regarding powers and their domains are proved.

Let b > 0. Let 1 < α ≤ 2. The theory of u 0-positive operators with respect to a cone in a Banach space is applied to study the conjugate boundary value problem for Riemann-Liouville fractional linear differential equations D 0+α u + λp(t)u = 0, 0 < t < b, satisfying the conjugate boundary conditions u(0) = u(b) = 0. The first extremal point, or conjugate point, of the conjugate boundary value problem is defined and criteria are established to characterize the conjugate point. As an application, a fixed point theorem is applied to give sufficient conditions for existence of a solution of a related boundary value problem for a nonlinear fractional differential equation.

We apply the theory for u 0-positive operators to obtain eigenvalue comparison results for a fractional boundary value problem with the Caputo derivative.

For the fractional diffusion-wave equation with the Caputo-Djrbashian fractional derivative of order α ∈ (1, 2) with respect to the time variable, we prove an analog of the principle of limiting amplitude (well-known for the wave equation and some other hyperbolic equations) and a pointwise stabilization property of solutions (similar to a well-known property of the heat equation and some other parabolic equations).

Let $$\mathbb{D}$$ stand for the unit disc in the complex plane ℂ. Given 0 < p < ∞, −1 < λ < ∞, the analytic weighted Besov space $$B_p^\lambda \left( \mathbb{D} \right)$$ is defined to consist of analytic in $$\mathbb{D}$$ functions such that $$\int\limits_\mathbb{D} {\left( {1 - \left| z \right|^2 } \right)^{Np - 2} \left| {f^{\left( N \right)} \left( z \right)} \right|^p d\mu _\lambda \left( z \right) < \infty ,}$$ where dμ λ(z) = (λ + 1)(1 − |z|2)λ dμ(z), $$d\mu (z) = \tfrac{1} {\pi }dxdy$$, and N is an arbitrary fixed natural number, satisfying N p > 1 − λ. We provide a characterization of weighted analytic Besov spaces $$B_p^\lambda \left( \mathbb{D} \right)$$, 0 < p < ∞, in terms of certain operators of fractional differentiation R zα,t of order t. These operators are defined in terms of construction known as Hadamard product composition with the function b. The function b is calculated from the condition that R zα,t (uniquely) maps the weighted Bergman kernel function $$\left( {1 - z\bar w} \right)^{ - 2 - \alpha }$$ to the similar (weight parameter shifted) kernel function $$\left( {1 - z\bar w} \right)^{ - 2 - \alpha - t}$$, t > 0. We also show that $$B_p^\lambda \left( \mathbb{D} \right)$$ can be thought as the image of certain weighted Lebesgue space $$L^p \left( {\mathbb{D},d\nu _\lambda } \right)$$ under the action of the weighted Bergman projection $$P_\mathbb{D}^\alpha$$.