Fractional Calculus and Applied Analysis

Published by De Gruyter

Volume 21 Issue 2

Issue of Fractional Calculus and Applied Analysis

Contents
Journal Overview

Publicly Available June 9, 2018

Frontmatter

Page range: i-v

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Editorial

Publicly Available June 9, 2018

FCAA related news, events and books (FCAA–volume 21–2–2018)

Virginia Kiryakova

Page range: 267-275

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Research Paper

June 9, 2018

Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients

Adam Kubica, Masahiro Yamamoto

Page range: 276-311

Abstract

We discuss an initial-boundary value problem for a fractional diffusion equation with Caputo time-fractional derivative where the coefficients are dependent on spatial and time variables and the zero Dirichlet boundary condition is attached. We prove the unique existence of weak and regular solutions.

Research Paper

June 9, 2018

Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms

Xiao-Li Ding, Juan J. Nieto

Page range: 312-335

Abstract

In this paper, we consider the analytical solutions of multi-term time-space fractional partial differential equations with nonlocal damping terms for general mixed Robin boundary conditions on a finite domain. Firstly, method of reduction to integral equations is used to obtain the analytical solutions of multi-term time fractional differential equations with integral terms. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the multi-term time-space fractional partial differential equations with nonlocal damping terms to the multi-term time fractional differential equations with integral terms. By applying the obtained analytical solutions to the resulting multi-term time fractional differential equations with integral terms, the desired analytical solutions of the multi-term time-space fractional partial differential equations with nonlocal damping terms are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.

Research Paper

June 9, 2018

Fractional generalizations of Zakai equation and some solution methods

Sabir Umarov, Fred Daum, Kenric Nelson

Page range: 336-353

Abstract

The paper discusses fractional generalizations of Zakai equations arising in filtering problems. The derivation of the fractional Zakai equation, existence and uniqueness of its solution, as well as some methods of solution to the fractional filtering problem, including fractional version of the particle flow method, are presented.

Research Paper

June 9, 2018

Stability analysis of impulsive fractional difference equations

Guo–Cheng Wu, Dumitru Baleanu

Page range: 354-375

Abstract

We revisit motivation of the fractional difference equations and some recent applications to image encryption. Then stability of impulsive fractional difference equations is investigated in this paper. The fractional sum equation is considered and impulsive effects are introduced into discrete fractional calculus. A class of impulsive fractional difference equations are proposed. A discrete comparison principle is given and asymptotic stability of nonlinear fractional difference equation are discussed. Finally, an impulsive Mittag–Leffler stability is defined. The numerical result is provided to support the analysis.

Research Paper

June 9, 2018

Mellin convolutions, statistical distributions and fractional calculus

A. M. Mathai

Page range: 376-398

Abstract

This paper shows that meaningful interpretations for Mellin convolutions of products and ratios involving two, three or more functions, can be given through statistical distribution theory of products and ratios involving two, three or more real scalar random variables or general multivariate situations. This paper shows that the approach through statistical distributions can also establish connection to fractional integrals, reaction-rate probability integrals in nuclear reaction-rate theory, Krätzel integrals and Krätzel transform in applied analysis, continuous mixtures, Bayesian analysis etc. This paper shows that the theory of Mellin convolutions, currently available for two functions, can be extended to many functions through statistical distributions. As illustrative examples, products and ratios of generalized gamma variables, which lead to Krätzel integrals, reaction-rate probability integrals, inverse Gaussian density etc, and type-1 beta variables, which lead to various types of fractional integrals and fractional calculus in general, are considered.

Research Paper

June 9, 2018

Fractional wavelet frames in L²(ℝ)

Firdous A. Shah, Lokenath Debnath

Page range: 399-422

Abstract

The objective of this paper is to construct fractional wavelet frames in L 2 (ℝ). A necessary condition and four sufficient conditions for fractional wavelet frames are given by virtue of fractional Fourier transform. The proposed inequalities generalize all the classical wavelet inequalities when θ = π /2. An example is presented at the end.

Research Paper

June 9, 2018

Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions

Bashir Ahmad, Rodica Luca

Page range: 423-441

Abstract

We study the existence of solutions for a system of nonlinear Caputo fractional differential equations with coupled boundary conditions involving Riemann-Liouville fractional integrals, by using the Schauder fixed point theorem and the nonlinear alternative of Leray-Schauder type. Two examples are given to support our main results.

Research Paper

June 9, 2018

Two point fractional boundary value problems with a fractional boundary condition

Jeffrey W. Lyons, Jeffrey T. Neugebauer

Page range: 442-461

Abstract

In this paper, we employ Krasnoseľskii’s fixed point theorem to show the existence of positive solutions to three different two point fractional boundary value problems with fractional boundary conditions. Also, nonexistence results are given.

Research Paper

June 9, 2018

Large deviation principle for a space-time fractional stochastic heat equation with fractional noise

Litan Yan, Xiuwei Yin

Page range: 462-485

Abstract

In this paper, we consider the large deviation principle for a class of space-time fractional stochastic heat equation ∂tβuε(t,x)=−ν(−Δ)α2uε(t,x)+It1−βf(uε(t,x))+εIt1−β[W˙H(t,x)],$$\begin{array}{} \displaystyle \partial^\beta_tu^\varepsilon(t,x)=-\nu(-\Delta)^{\frac\alpha 2}u^\varepsilon(t,x)+I_t^{1-\beta}f(u^\varepsilon(t,x))+ \sqrt{\varepsilon}I^{1-\beta}_t[\dot{W}^H(t,x)], \end{array}$$ where Ẇ H is a fractional white noise, ν > 0, β ∈ (0, 1), α ∈ (0, 2]. The operator ∂tβ$\begin{array}{} \displaystyle \partial^\beta_t \end{array}$ is the Caputo fractional integration operator, and −(−Δ)α2$\begin{array}{} \displaystyle -(-\Delta)^{\frac\alpha 2} \end{array}$ is the fractional power of Laplacian. Our proof is based on the weak convergence approach.

Research Paper

June 9, 2018

Extension of Mikhlin multiplier theorem to fractional derivatives and stable processes

Deniz Karlı

Page range: 486-508

Abstract

In this paper, we prove a new generalized Mikhlin multiplier theorem whose conditions are given with respect to fractional derivatives in integral forms with two different integration intervals. We also discuss the connection between fractional derivatives and stable processes and prove a version of Mikhlin theorem under a condition given in terms of the infinitesimal generator of symmetric stable process. The classical Mikhlin theorem is shown to be a corollary of this new generalized version in this paper.

Research Paper

June 9, 2018

Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications

Chuan-Jing Song, Yi Zhang

Page range: 509-526

Abstract

Noether theorem is an important aspect to study in dynamical systems. Noether symmetry and conserved quantity for the fractional Birkhoffian system are investigated. Firstly, fractional Pfaff actions and fractional Birkhoff equations in terms of combined Riemann-Liouville derivative, Riesz-Riemann-Liouville derivative, combined Caputo derivative and Riesz-Caputo derivative are reviewed. Secondly, the criteria of Noether symmetry within combined Riemann-Liouville derivative, Riesz-Riemann-Liouville derivative, combined Caputo derivative and Riesz-Caputo derivative are presented for the fractional Birkhoffian system, respectively. Thirdly, four corresponding conserved quantities are obtained. The classical Noether identity and conserved quantity are special cases of this paper. Finally, four fractional models, such as the fractional Whittaker model, the fractional Lotka biochemical oscillator model, the fractional Hénon-Heiles model and the fractional Hojman-Urrutia model are discussed as examples to illustrate the results.

Research Paper

June 9, 2018

Asymptotic behavior of mild solutions for nonlinear fractional difference equations

Zhinan Xia, Dingjiang Wang

Page range: 527-551

Abstract

In this paper, we establish some sufficient criteria for the existence, uniqueness of discrete weighted pseudo asymptotically periodic mild solutions and asymptotic behavior for nonlinear fractional difference equations in Banach space, where the nonlinear perturbation is Lipschitz type, or non-Lipschitz type. The results are a consequence of application of different fixed point theorems, namely, the Banach contraction mapping principle, Leray-Schauder alternative theorem and Matkowski’s fixed point technique.

Research Paper

June 9, 2018

Positive solutions to nonlinear systems involving fully nonlinear fractional operators

Pengcheng Niu, Leyun Wu, Xiaoxue Ji

Page range: 552-574

Abstract

In this paper we consider the following fractional system F(x,u(x),v(x),Fα(u(x)))=0,G(x,v(x),u(x),Gβ(v(x)))=0,$$\begin{array}{} \displaystyle \left\{ \begin{gathered} F(x,u(x),v(x),{\mathcal{F}_\alpha }(u(x))) = 0,\\ G(x,v(x),u(x),{\mathcal{G}_\beta }(v(x))) = 0, \\ \end{gathered} \right. \end{array}$$ where 0 < α , β < 2, 𝓕 α and 𝓖 β are the fully nonlinear fractional operators: Fα(u(x))=Cn,αPV∫Rnf(u(x)−u(y))x−yn+αdy,Gβ(v(x))=Cn,βPV∫Rng(v(x)−v(y))x−yn+βdy.$$\begin{array}{} \displaystyle {\mathcal{F}_\alpha }(u(x)) = {C_{n,\alpha }}PV\int_{{\mathbb{R}^n}} {\frac{{f(u(x) - u(y))}} {{{{\left| {x - y} \right|}^{n + \alpha }}}}dy} ,\\ \displaystyle{\mathcal{G}_\beta }(v(x)) = {C_{n,\beta }}PV\int_{{\mathbb{R}^n}} {\frac{{g(v(x) - v(y))}} {{{{\left| {x - y} \right|}^{n + \beta }}}}dy} . \end{array}$$ A decay at infinity principle and a narrow region principle for solutions to the system are established. Based on these principles, we prove the radial symmetry and monotonicity of positive solutions to the system in the whole space and a unit ball respectively, and the nonexistence in a half space by generalizing the direct method of moving planes to the nonlinear system.

About this journal

Objective
Fractional Calculus and Applied Analysis (FCAA, abbreviated in the World databases as Fract. Calc. Appl. Anal. or FRACT CALC APPL ANAL) is a specialized international journal for theory and applications of an important branch of Mathematical Analysis (Calculus) where differentiations and integrations can be of arbitrary non-integer order. The high standards of its contents are guaranteed by the prominent members of Editorial Board and the expertise of invited external reviewers, and proven by the recently achieved high values of impact factor (JIF) and impact rang (SJR), launching the journal to top places of the ranking lists of Thomson Reuters and Scopus.

Topics

Fractional Calculus (FC)
Special Functions and Integral Transforms, related to FC
Fractional Order Differential and Integral Equations and Systems
Mathematical Models of Phenomena, described by the above topics

Algebraic Analysis, Operational and Convolutional Calculi
Generalized Functions, Harmonic Analysis
Numerical and Approximation Methods, Computational Procedures and Algorithms, related to the Primary FCAA topics
Fractional Stochastic Processes
Fractal and Integral Geometry

Differential and Integral Equations, Problems of Mathematical Physics
Control Theory, Mechanics, Probability and Statistics, Finances, Engineering, etc.

If revealing connections between Fractional Calculus and the above-mentioned topics to model problems of the real physical and social world

Article formats
Research Papers, Survey Papers, Discussion Surveys and Notes, Short Notes, Open Problems, Archives (renewal publication of old and hardly accessible or in other languages articles of significant interest for the journal’s audience), Reviews on Books, Proceedings, etc., Editorial Notes with announcements for New Books, Recent and Forthcoming Meetings, Anniversaries and other events related to FCAA

> Information on submission process

Volume 21 Issue 2

Issue of Fractional Calculus and Applied Analysis

Frontmatter

FCAA related news, events and books (FCAA–volume 21–2–2018)

Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients

Abstract

Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms

Abstract

Fractional generalizations of Zakai equation and some solution methods

Abstract

Stability analysis of impulsive fractional difference equations

Abstract

Mellin convolutions, statistical distributions and fractional calculus

Abstract

Fractional wavelet frames in L2(ℝ)

Abstract

Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions

Abstract

Two point fractional boundary value problems with a fractional boundary condition

Abstract

Large deviation principle for a space-time fractional stochastic heat equation with fractional noise

Abstract

Extension of Mikhlin multiplier theorem to fractional derivatives and stable processes

Abstract

Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications

Abstract

Asymptotic behavior of mild solutions for nonlinear fractional difference equations

Abstract

Positive solutions to nonlinear systems involving fully nonlinear fractional operators

Abstract

Fractional wavelet frames in L²(ℝ)