We establish the existence of smallest eigenvalues for the fractional linear boundary value problems D0+αu+λ1p(t)u=0$D_{0 + }^\alpha u + {\lambda _1}p\left( t \right)u = 0$ and D0+αu+λ2q(t)u=0$D_{0 + }^\alpha u + {\lambda _2}q\left( t \right)u = 0$, 0<t<1 with each satisfying the right focal boundary conditions u(0)=u′(1)=0$u\left( 0 \right) = u'\left( 1 \right) = 0$. A comparison result is then obtained.

Research Paper

Publicly Available March 9, 2016

High-Order Algorithms for Riesz Derivative and their Applications (III)

Hengfei Ding, Changpin Li

Page range: 19-55

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Abstract

Numerical methods for fractional calculus attract increasing interest due to its wide applications in various fields such as physics, mechanics, etc. In this paper, we focus on constructing high-order algorithms for Riesz derivatives, where the convergence orders cover from the second order to the sixth order. Then we apply the established schemes to the Riesz type turbulent diffusion equation (or, Riesz space fractional turbulent diffusion equation). Numerical experiments are displayed which support the theoretical analysis.

Research Paper

Publicly Available March 9, 2016

Existence and Uniqueness for a Class of Stochastic Time Fractional Space Pseudo-Differential Equations

Ke Hu, Niels Jacob, Chenggui Yuan

Page range: 56-68

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Abstract

Some classes of stochastic fractional differential equations with respect to time and a pseudo-differential equation with respect to space are investigated. Using estimates for the Mittag-Leffler functions and a fixed point theorem, existence and uniqueness of mild solutions of the equations under consideration are established.

Research Paper

Publicly Available March 22, 2016

Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data

Bangti Jin, Raytcho Lazarov, Dongwoo Sheen, Zhi Zhou

Page range: 69-93

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Abstract

In this work, we consider the numerical solution of a distributed order subdiffusion model, arising in the modeling of ultra-slow diffusion processes. We develop a space semidiscrete scheme based on the Galerkin finite element method, and establish error estimates optimal with respect to data regularity in L 2 (Ω) and H 1 (Ω) norms for both smooth and nonsmooth initial data. Further, we propose two fully discrete schemes, based on the Laplace transform and convolution quadrature generated by the backward Euler method, respectively, and provide optimal L 2 (Ω) error estimates, which exhibits exponential convergence and first-order convergence in time, respectively. Extensive numerical experiments are provided to verify the error estimates for both smooth and nonsmooth initial data.

Research Paper

March 9, 2016

Bogolyubov-Type Theorem with Constraints Generated by a Fractional Control System

Xiaoyou Liu, Youjun Xu

Page range: 94-115

Abstract

We first study the existence results and properties of the solution set of a control system described by fractional differential equations with nonconvex control constraint. Then a problem of minimizing an integral functional over the solution set of the control system is considered. Along with the original minimizing problem, we also consider the problem of minimizing the integral functional whose integrand is the bipolar (with respect to the control variable) of the original integrand over the solution set of the same system but with the convexified control constraint. We prove that the relaxed problem has an optimal solution and obtain some relationships between these two minimizing problems. Finally, an example is given to illustrate the results.

Research Paper

March 9, 2016

Numerical Solution of Nonstationary Problems for a Space-Fractional Diffusion Equation

Petr N. Vabishchevich

Page range: 116-139

Abstract

An unsteady problem is considered for a space-fractional diffusion equation in abounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary conditions of Robin type. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the fractional power of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The results of numerical experiments are presented for a model two-dimensional problem.

Research Paper

March 9, 2016

Solving 3D Time-Fractional Diffusion Equations by High-Performance Parallel Computing

Wei Zhang, Xing Cai

Page range: 140-160

Abstract

Numerically solving time-fractional diffusion equations, especially in three space dimensions, is a daunting computational task. This is due to the huge requirements of both computation time and memory storage. Compared with solving integer-ordered diffusion equations, the costs for time and storage both increase by a factor that equals the number of time steps involved. Aiming to overcome these two obstacles, we study in this paper three programming techniques: loop unrolling, vectorization and parallelization. For a representative numerical scheme that adopts finite differencing and explicit time integration, the performance-enhancing techniques are indeed shown to dramatically reduce the computation time, while allowing the use of many CPU cores and thereby a large amount of memory storage. Moreover, we have developed simple-to-use performance models that support our empirical findings, which are based on using up to 8192 CPU cores and 12.2 terabytes.

Discussion Survey

March 9, 2016

Physical and Geometrical Interpretation of Grünwald-Letnikov Differintegrals: Measurement of Path and Acceleration

Radosław Cioć

Page range: 161-172

Abstract

A function f ( t ) of the independent variable t changing with every increment dt can be formulated as a functional sequence. If g ( f ( t )) is a derivative or an integral of f ( t ) and the value of dt is interpreted subject to an error Δ T , then g ( f ( t )) is Grünwald-Letnikov differintegral of that sequence with an order closely related to dt and Δ T . This paper illustrates this relationship and proposes a geometrical and physical interpretation of a fractional order Grünwald-Letnikov differintegrals using the example of path and acceleration measurements of a point in motion.

Discussion Survey

March 9, 2016

Some Applications of Fractional Velocities

Dimiter Prodanov

Page range: 173-187

Abstract

Fractional velocity is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. The fractional velocity can be suitable for characterizing singular behavior of derivatives of Hölderian functions and non differentiable functions. Relations to integer-order derivatives and other integral-based definitions are discussed.It is demonstrated that for Hölder functions under certain conditions the product rules deviates from the Leibniz rule. This deviation is expressed by another quantity, fractional co-variation.

Research Paper

March 9, 2016

Maximum Principles for Multi-Term Space-Time Variable-Order Fractional Diffusion Equations and their Applications

Zhenhai Liu, Shengda Zeng, Yunru Bai

Page range: 188-211

Abstract

In this paper, we deal with maximum principles for multi-term space-time variable-order Riesz-Caputo fractional differential equations (MT-STVO-RCFDEs, for short). We firstly derive several important inequalities for variable-order fractional derivatives at extreme points. Based on these inequalities, we obtain the maximum principles. Finally, these principles are employed to show that the uniqueness of solutions of the (MT-STVO-RCFDEs) and continuous dependance of solutions on initial-boundary value conditions.

Survey Paper

March 9, 2016

Atypical Case of the Dielectric Relaxation Responses and its Fractional Kinetic Equation

Aleksander Stanislavsky, Karina Weron

Page range: 212-228

Abstract

We present a probabilistic model of the microscopic scenario of dielectric relaxation relating to the atypical case of two-power-law responses.The surveyed approach extends the cluster model concept used for the description of the typical, Havriliak-Negami (HN) law. Within the proposed framework, all empirical two-power-law relaxation patterns may be derived. Their relaxation functions are expressed in terms of the three-parameter Mittag-Leffler function, and the kinetic equation takes the pseudodifferential form generalizing the Riemann-Louiville fractional calculus. This provides a clue to explain the universality observed in relaxation phenomena.

Research Paper

March 9, 2016

Operator Method for Construction of Solutions of Linear Fractional Differential Equations with Constant Coefficients

Ravshan Ashurov, Alberto Cabada, Batirkhan Turmetov

Page range: 229-252

Abstract

One of the effective methods to find explicit solutions of differential equations is the method based on the operator representation of solutions. The essence of this method is to construct a series, whose members are the relevant iteration operators acting to some classes of sufficiently smooth functions. This method is widely used in the works of B. Bondarenko for construction of solutions of differential equations of integer order. In this paper, the operator method is applied to construct solutions of linear differential equations with constant coefficients and with Caputo fractional derivatives. Then the fundamental solutions are used to obtain the unique solution of the Cauchy problem, where the initial conditions are given in terms of the unknown function and its derivatives of integer order. Comparison is made with the use of Mikusinski operational calculus for solving similar problems.

Research Article

March 9, 2016

On the Existence and Multiplicity of Solutions for Dirichlet’s problem for Fractional Differential equations

Diego Averna, Stepan Tersian, Elisabetta Tornatore

Page range: 253-266

Abstract

In this paper, by using variational methods and critical point theorems, we prove the existence and multiplicity of solutions for boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. Our results extend the second order boundary value problem to the non integer case. Moreover, some conditions to determinate nonnegative solutions are presented and examples are given to illustrate our results.

Research Paper

March 9, 2016

Approximate controllability for semilinear composite fractional relaxation equations

Zhenbin Fan, Qixiang Dong, Gang Li

Page range: 267-284

Abstract

We consider a control system governed by a semilinear composite fractional relaxation equation in Hilbert space. We first prove that the system has a mild solution. Then, we investigate the approximate controllability of the relaxation equation under the assumption that the corresponding linear system is approximately controllable. An example is also given to illustrate our results.

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Please note that, from the 2022 volume onwards, Fractional Calculus and Applied Analysis is published by Springer Nature. For more information, please see the new journal homepage here. (please put under ‘here’ the link to Springer's website: https://www.springer.com/journal/13540.