
显示样式: 排序: IF: - GO 导出
-
Fractional derivatives and cauchy problem for differential equations of fractional order Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31 M.M. Dzherbashian; A.B. Nersesian
Editorial Note: This is a paper by M.M. Djrbashian and A.B. Nersesian of 1968, that was published in Russian.
-
Mkhitar Djrbashian and his contribution to Fractional Calculus Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31 Sergei Rogosin; Maryna Dubatovskaya
This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the
-
Implementation of fractional optimal control problems in real-world applications Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31 Neelam Singha
In this article, we aim to analyze a mathematical model of tumor growth as a problem of fractional optimal control. The considered fractional-order model describes the interaction of effector-immune cells and tumor cells, including combined chemo-immunotherapy. We deduce the necessary optimality conditions together with implementing the Adomian decomposition method on the suggested fractional-order
-
Fractional nonlinear stochastic heat equation with variable thermal conductivity Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31 Miloš Japundžić; Danijela Rajter-Ćirić
We consider a nonlinear stochastic heat equation with Riesz space-fractional derivative and variable thermal conductivity, on infinite domain. First we approximate the original problem by regularizing the Riesz space-fractional derivative. Then we prove that the approximate problem has almost surely a unique solution within a Colombeau generalized stochastic process space. In our solving procedure
-
Bounded solutions of second order of accuracy difference schemes for semilinear fractional schrödinger equations Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31 Allaberen Ashyralyev; Betul Hicdurmaz
The present paper deals with initial value problem (IVP) for semilinear fractional Schrödinger integro-differential equation
-
Nakhushev extremum principle for a class of integro-differential operators Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31 Arsen Pskhu
We investigate extreme properties of a class of integro-differential operators. We prove an assertion that extends the Nakhushev extremum principle, known for fractional Riemann-Liouville derivatives, to integro-differential operators with kernels of a general form. We establish the weighted extremum principle for convolution operators and the Riemann-Liouville fractional derivative. In addition, as
-
Uniqueness for an inverse source problem of determining a space-dependent source in a non-autonomous time-fractional diffusion equation Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31 Marian Slodička
We study uniqueness of a solution for an inverse source problem arising in linear time-fractional diffusion equations with time-dependent coefficients. We consider source term in a separated form h(t)f (x). The unknown source f (x) is recovered from the final time measurement u (x, T). A new uniqueness result is formulated in under the assumption that h ∈ C ([0, T]) and 0 ≢ h ≥ 0. No monotonicity in
-
Determination of time-dependent sources and parameters of nonlocal diffusion and wave equations from final data Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31 Jaan Janno
Two inverse problems with final overdetermination for diffusion and wave equations containing the Caputo fractional time derivative and a fractional Laplacian of distributed order are considered. They are: 1) the problem to reconstruct a time-dependent source term; 2) the problem to recover simultaneously the source term, the order of the time derivative and the fractional Laplacian. Uniqueness of
-
Multidimensional van der Corput-Type estimates involving Mittag-Leffler functions Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31 Michael Ruzhansky; Berikbol T. Torebek
The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with
-
Determination of the order of fractional derivative for subdiffusion equations Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31 Ravshan Ashurov; Sabir Umarov
The identification of the right order of the equation in applied fractional modeling plays an important role. In this paper we consider an inverse problem for determining the order of time fractional derivative in a subdiffusion equation with an arbitrary second order elliptic differential operator. We prove that the additional information about the solution at a fixed time instant at a monitoring
-
A survey on numerical methods for spectral Space-Fractional diffusion problems Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31 Stanislav Harizanov; Raytcho Lazarov; Svetozar Margenov
The survey is devoted to numerical solution of the equation Aαu=f, 0 < α<1, where A is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain Ω in ℝd. The fractional power Aα is a non-local operator and is defined though the spectrum of A. Due to growing interest and demand in applications of sub-diffusion models to physics and engineering
-
Wave propagation dynamics in a fractional Zener model with stochastic excitation Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31 Teodor Atanacković; Stevan Pilipović; Dora Seleši
Equations of motion for a Zener model describing a viscoelastic rod are investigated and conditions ensuring the existence, uniqueness and regularity properties of solutions are obtained. Restrictions on the coefficients in the constitutive equation are determined by a weak form of the dissipation inequality. Various stochastic processes related to the Karhunen-Loéve expansion theorem are presented
-
FCAA special 2020 conferences' issue (FCAA–Volume 23–6–2020) Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31 Virginia Kiryakova
Journal Name: Fractional Calculus and Applied Analysis Volume: 23 Issue: 6 Pages: 1561-1569
-
Frontmatter Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-12-31
Journal Name: Fractional Calculus and Applied Analysis Volume: 23 Issue: 6 Pages: i-v
-
Cauchy problem for general time fractional diffusion equation Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Chung-Sik Sin
In the present work, we consider the Cauchy problem for the time fractional diffusion equation involving the general Caputo-type differential operator proposed by Kochubei []. First, the existence, the positivity and the long time behavior of solutions of the diffusion equation without source term are established by using the Fourier analysis technique. Then, based on the representation of the solution
-
Experimental investigation of fractional order behavior in an oscillating disk Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Richard Mark French; Rajarshi Choudhuri; Jose Garcia-Bravo; Jordan Petty
An experimental investigation is presented of an oscillating disk submerged in water. The system is adapted from one modeled by the Bagley-Torvik equation, being modified to more closely approximate idealizations in the derivation. The modified system eliminates alignment problems that would violate the assumption of fluid forces being due only to shear in the fluid. An implicit finite difference model
-
Approximate controllability for stochastic fractional hemivariational inequalities of degenerate type Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Yatian Pei; Yong-Kui Chang
This paper is mainly concerned with stochastic fractional hemivariational inequalities of degenerate (or Sobolev) type in Caputo and Riemann-Liouville derivatives with order (1, 2), respectively. Based upon some properties of fractional resolvent family and generalized directional derivative of a locally Lipschitz function, some sufficient conditions are established for the existence and approximate
-
State dependent versions of the space-time fractional poisson process Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Kuldeep Kumar Kataria; Palaniappan Vellaisamy
In this paper, we introduce and study two counting processes by considering state dependency on the order of fractional derivative as well as on the exponent of backward shift operator involved in the governing difference-differential equations of the state probabilities of space-time fractional Poisson process. The Adomian decomposition method is employed to obtain their state probabilities and then
-
The asymptotic behavior of solutions of discrete nonlinear fractional equations Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Mustafa Bayram; Aydin Secer; Hakan Adiguzel
In this study, we consider a class of nonlinear fractional difference equations following form:
-
Trace inequalities for fractional integrals in mixed norm grand lebesgue spaces Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Vakhtang Kokilashvili; Alexander Meskhi
D. Adams type trace inequalities for multiple fractional integral operators in grand Lebesgue spaces with mixed norms are established. Operators under consideration contain multiple fractional integrals defined on the product of quasi-metric measure spaces, and one-sided multiple potentials. In the case when we deal with operators defined on bounded sets, the established conditions are simultaneously
-
Multivariate fractional phase–type distributions Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Hansjörg Albrecher; Martin Bladt; Mogens Bladt
We extend the Kulkarni class of multivariate phase–type distributions in a natural time–fractional way to construct a new class of multivariate distributions with heavy-tailed Mittag-Leffler(ML)-distributed marginals. The approach relies on assigning rewards to a non–Markovian jump process with ML sojourn times. This new class complements an earlier multivariate ML construction [] and in contrast to
-
Regularity results for nonlocal evolution Venttsel’ problems Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Simone Creo; Maria Rosaria Lancia; Alexander I. Nazarov
We consider parabolic nonlocal Venttsel’ problems in polygonal and piecewise smooth two-dimensional domains and study existence, uniqueness and regularity in (anisotropic) weighted Sobolev spaces of the solution. The nonlocal term can be regarded as a regional fractional Laplacian on the boundary. The regularity results deeply rely on a priori estimates, obtained via the so-called Munchhausen trick
-
Analysis of fractional integro-differential equations with nonlocal Erdélyi-Kober type integral boundary conditions Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Palanisamy Duraisamy; Thangaraj Nandha Gopal; Muthaiah Subramanian
In this article, we study the existence and uniqueness of solutions for nonlinear fractional integro-differential equations with nonlocal Erdélyi-Kober type integral boundary conditions. The existence results are based on Krasnoselskii’s and Schaefer’s fixed point theorems, whereas the uniqueness result is based on the contraction mapping principle. Examples illustrating the applicability of our main
-
Applications of Erdélyi-Kober fractional integral for solving time-fractional Tricomi-Keldysh type equation Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Kangqun Zhang
In this paper we consider Cauchy problem of time-fractional Tricomi-Keldysh type equation. Based on the theory of a Erdélyi-Kober fractional integral operator, the formal solution of the inhomogeneous differential equation involving hyper-Bessel operator is presented with Mittag-Leffler function, then nonlinear equations are considered by applying Gronwall-type inequalities. At last, we establish the
-
Two high-order time discretization schemes for subdiffusion problems with nonsmooth data Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Yanyong Wang; Yubin Yan; Yan Yang
Two new high-order time discretization schemes for solving subdiffusion problems with nonsmooth data are developed based on the corrections of the existing time discretization schemes in literature. Without the corrections, the schemes have only a first order of accuracy for both smooth and nonsmooth data. After correcting some starting steps and some weights of the schemes, the optimal convergence
-
Fractional fractals Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 J.A. Tenreiro Machado; Daniel Cao Labora
This paper introduces the notion of “fractional fractals”. The main idea is to establish a connection between the classical iterated function system and the first order truncation of the Gründwald-Letnikov fractional derivative. This allows us to consider higher order truncations, and also to study the limit sets for these higher order systems. We prove several results involving the existence and dimension
-
Fractional integro-differential equations in Wiener spaces Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Vu Kim Tuan
In this paper we study the global solvability of several ordinary and partial fractional integro-differential equations in the Wiener space of functions with bounded square averages.
-
Integrability properties of integral transforms via morrey spaces Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Natasha Samko
We show that integrability properties of integral transforms with kernel depending on the product of arguments (which include in particular, popular Laplace, Hankel, Mittag-Leffler transforms and various others) are better described in terms of Morrey spaces than in terms of Lebesgue spaces. Mapping properties of integral transforms of such a type in Lebesgue spaces, including weight setting, are known
-
Tempered relaxation equation and related generalized stable processes Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Luisa Beghin; Janusz Gajda
Fractional relaxation equations, as well as relaxation functions time-changed by independent stochastic processes have been widely studied (see, for example, [], [] and []). We start here by proving that the upper-incomplete Gamma function satisfies the tempered-relaxation equation (of index ρ ∈ (0, 1)); thanks to this explicit form of the solution, we can then derive its spectral distribution, which
-
FCAA related news, events and books (FCAA–Volume 23–5–2020) Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13 Virginia Kiryakova
Journal Name: Fractional Calculus and Applied Analysis Volume: 23 Issue: 5 Pages: 1241-1247
-
Frontmatter Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-11-13
Journal Name: Fractional Calculus and Applied Analysis Volume: 23 Issue: 5 Pages: i-v
-
Semilinear fractional elliptic problems with mixed Dirichlet-Neumann boundary conditions Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 José Carmona; Eduardo Colorado; Tommaso Leonori; Alejandro Ortega
We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator and a concave-convex term, together with mixed Dirichlet-Neumann boundary conditions.
-
Attractivity for differential equations of fractional order and ψ-Hilfer type Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 J. Vanterler da C. Sousa; Mouffak Benchohra; Gaston M. N’Guérékata
This paper investigates the overall solution attractivity of the fractional differential equation involving the ψ-Hilfer fractional derivative and using the Krasnoselskii’s fixed point theorem. We highlight some particular cases of the results presented here, especially involving the Riemann-Liouville, thus illustrating the broad class of fractional derivatives to which these results can be applied
-
Asymptotics of fundamental solutions for time fractional equations with convolution kernels Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 Yuri Kondratiev; Andrey Piatnitski; Elena Zhizhina
The paper deals with the large time asymptotic of the fundamental solution for a time fractional evolution equation with a convolution type operator. In this equation we use a Caputo time derivative of order α ∈ (0, 1), and assume that the convolution kernel of the spatial operator is symmetric, integrable and shows a super-exponential decay at infinity. Under these assumptions we describe the point-wise
-
On representation formulas for solutions of linear differential equations with Caputo fractional derivatives Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 Mikhail I. Gomoyunov
In the paper, a linear differential equation with variable coefficients and a Caputo fractional derivative is considered. For this equation, a Cauchy problem is studied, when an initial condition is given at an intermediate point that does not necessarily coincide with the initial point of the fractional differential operator. A detailed analysis of basic properties of the fundamental solution matrix
-
Fractional abstract Cauchy problem on complex interpolation scales Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 Andriy Lopushansky; Oleh Lopushansky; Anna Szpila
An fractional abstract Cauchy problem generated by a sectorial operator is investigated. An inequality of coercivity type for its solution with respect to a complex interpolation scale generated by a sectorial operator with the same parameters is established. An application to differential parabolic initial-boundary value problems in bounded domains with a fractional time derivative is shown.
-
Initial-value / Nonlocal Cauchy problems for fractional differential equations involving ψ-Hilfer multivariable operators Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 Jin Liang; Yunyi Mu; Ti-Jun Xiao
In this paper, we investigate two types of problems (the initial-value problem and nonlocal Cauchy problem) for fractional differential equations involving ψ-Hilfer derivative in multivariable case (ψ-m-Hilfer derivative). First we propose and discuss ψ-fractional integral, ψ-fractional derivative and ψ-Hilfer type fractional derivative of a multivariable function f : ℝm → ℝ (m is a positive integer)
-
On the harmonic extension approach to fractional powers in Banach spaces Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 Jan Meichsner; Christian Seifert
We show that fractional powers of general sectorial operators on Banach spaces can be obtained by the harmonic extension approach. Moreover, for the corresponding second order ordinary differential equation with incomplete data describing the harmonic extension we prove existence and uniqueness of a bounded solution (i.e., of the harmonic extension).
-
Global solutions and blowing-up solutions for a nonautonomous and nonlocal in space reaction-diffusion system with Dirichlet boundary conditions Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 Marcos J. Ceballos-Lira; Aroldo Pérez
We give sufficient conditions for global existence and finite time blow up of positive solutions for a nonautonomous weakly coupled system with distinct fractional diffusions and Dirichlet boundary conditions. Our approach is based on the intrinsic ultracontractivity property of the semigroups associated to distinct fractional diffusions and the study of blow up of a particular system of nonautonomus
-
On a quantitative theory of limits: Estimating the speed of convergence Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 Renato Spigler
The classical “ε-δ” definition of limits is of little use to quantitative purposes, as is needed, for instance, for computational and applied mathematics. Things change whenever a realistic and computable estimate of the function δ(ε) is available. This may be the case for Lipschitz continuous and Hölder continuous functions, or more generally for functions admitting of a modulus of continuity. This
-
Applications of Hilfer-Prabhakar operator to option pricing financial model Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 Živorad Tomovski; Johan L. A. Dubbeldam; Jan Korbel
In this paper, we focus on option pricing models based on time-fractional diffusion with generalized Hilfer-Prabhakar derivative. It is demonstrated how the option is priced for fractional cases of European vanilla option pricing models. Series representations of the pricing formulas and the risk-neutral parameter under the time-fractional diffusion are also derived.
-
Nontrivial solutions of non-autonomous dirichlet fractional discrete problems Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 Alberto Cabada; Nikolay Dimitrov
In this paper, we introduce a two-point boundary value problem for a finite fractional difference equation with a perturbation term. By applying spectral theory, an associated Green’s function is constructed as a series of functions and some of its properties are obtained. Under suitable conditions on the nonlinear part of the equation, some existence and uniqueness results are deduced.
-
Erdélyi–Kober fractional integrals and radon transforms for mutually orthogonal affine planes Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 Boris Rubin; Yingzhan Wang
We apply Erdélyi–Kober fractional integrals to the study of Radon type transforms that take functions on the Grassmannian of j-dimensional affine planes in ℝn to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. We obtain explicit inversion formulas for these transforms in the class of radial functions under
-
Fractional derivatives and the fundamental theorem of fractional calculus Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 Yuri Luchko
In this paper, we address the one-parameter families of the fractional integrals and derivatives defined on a finite interval. First we remind the reader of the known fact that under some reasonable conditions, there exists precisely one unique family of the fractional integrals, namely, the well-known Riemann-Liouville fractional integrals. As to the fractional derivatives, their natural definition
-
FCAA related news, events and books (FCAA–VOLUME 23–4–2020) Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11 Virginia Kiryakova
Journal Name: Fractional Calculus and Applied Analysis Volume: 23 Issue: 4 Pages: 935-938
-
Frontmatter Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-09-11
Journal Name: Fractional Calculus and Applied Analysis Volume: 23 Issue: 4 Pages: i-v
-
Weak solvability of the variable-order subdiffusion equation Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Andrii Hulianytskyi
In this work, we study a new type of linear partial differential equations – the variable-order subdiffusion equation. Here the Laplace operator in space acts on the Riemann-Liouville time derivative of space-dependent order. We construct a variable-order Sobolev and prove the weak solvability of the initial-boundary value problem for this equation, which confirms the well-posedness of the problem
-
An averaging principle for stochastic differential equations of fractional order 0 < α < 1 Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Wenjing Xu; Wei Xu; Kai Lu
This paper presents an averaging principle for fractional stochastic differential equations in ℝn with fractional order 0 < α < 1. We obtain a time-averaged equation under suitable conditions, such that the solutions to original fractional equation can be approximated by solutions to simpler averaged equation. By mathematical manipulations, we show that the mild solution of two equations before and
-
Construction of fixed point operators for nonlinear difference equations of non integer order with impulses Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Syed Sabyel Haider; Mujeeb Ur Rehman
In this article, we establish a technique for transforming arbitrary real order delta difference equations with impulses to corresponding summation equations. The technique is applied to non-integer order delta difference equation with some boundary conditions. Furthermore, the summation formulation for impulsive fractional difference equation is utilized to construct fixed point operator which in
-
The 2-adic derivatives and fractal dimension of Takagi-like function on 2-series field Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Bo Wu
In this paper, we consider a Takagi-like function on 2-series field and give its 2-adic derivatives by applying Vladimirov operator. The 2-adic derivatives of Takagi-like function with order 0 < α < 1 exist and show some fractal feature. Furthermore, both box dimension and Hausdorff dimension of the graph of its derivatives are obtained and equal to 1 + α.
-
Variational approximation for fractional Sturm–Liouville problem Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Prashant K. Pandey; Rajesh K. Pandey; Om P. Agrawal
In this paper, we consider a regular Fractional Sturm–Liouville Problem (FSLP) of order μ (0 < μ < 1). We approximate the eigenvalues and eigenfunctions of the problem using a fractional variational approach. Recently, Klimek et al. [] presented the variational approach for FSLPs defined in terms of Caputo derivatives and obtained eigenvalues, eigenfunctions for a special range of fractional order
-
Multiple positive solutions for a nonlocal PDE with critical Sobolev-Hardy and singular nonlinearities via perturbation method Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Adel Daoues; Amani Hammami; Kamel Saoudi
In this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent
-
Maximum principles for a class of generalized time-fractional diffusion equations Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Shengda Zeng; Stanisław Migórski; Van Thien Nguyen; Yunru Bai
Two significant inequalities for generalized time fractional derivatives at extreme points are obtained. Then, we apply the inequalities to establish the maximum principles for multi-term time-space fractional variable-order operators. Finally, we employ the principles to investigate two kinds of diffusion equations involving generalized time-fractional Caputo derivatives and space-fractional Riesz-Caputo
-
Inverse problem for a multi-term fractional differential equation Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Muhammad Ali; Sara Aziz; Salman A. Malik
Inverse problem for a family of multi-term time fractional differential equation with non-local boundary conditions is studied. The spectral operator of the considered problem is non-self-adjoint and a bi-orthogonal set of functions is used to construct the solution. The operational calculus approach has been used to obtain the solution of the multi-term time fractional differential equations. Integral
-
The green function for a class of Caputo fractional differential equations with a convection term Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Zhanbing Bai; Sujing Sun; Zengji Du; YangQuan Chen
A class of two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order α ∈ (1, 2] with a convection term is considered. Its boundary conditions are of Robin type including the Dirichlet boundary conditions as a special case. An explicit formula for the associated Green function is obtained in terms of two-parameter Mittag-Leffler functions. This work improves
-
“Fuzzy” calculus: The link between quantum mechanics and discrete fractional operators Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Raoul R. Nigmatullin; Paolo Lino; Guido Maione
In this paper, based on the “fuzzy” calculus covering the continuous range of operations between two couples of arithmetic operations (+, –) and (×, :), a new form of the fractional integral is proposed occupying an intermediate position between the integral and derivative of the first order. This new form of the fractional integral satisfies the C1 criterion according to the Ross classification. The
-
Solution of linear fractional order systems with variable coefficients Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Ivan Matychyn; Viktoriia Onyshchenko
The paper deals with the initial value problem for linear systems of FDEs with variable coefficients involving Riemann–Liouville derivatives. The technique of the generalized Peano–Baker series is used to obtain the state-transition matrix. Explicit solutions are derived both in the homogeneous and inhomogeneous case. The theoretical results are supported by an example.
-
Degenerate Kirchhoff (p, q)–Fractional systems with critical nonlinearities Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Alessio Fiscella; Patrizia Pucci
This paper deals with the existence of nontrivial solutions for critical possibly degenerate Kirchhoff fractional (p, q) systems. For clarity, the results are first presented in the scalar case, and then extended into the vectorial framework. The main features and novelty of the paper are the (p, q) growth of the fractional operator, the double lack of compactness as well as the fact that the systems
-
Determination of the fractional order in semilinear subdiffusion equations Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Mykola Krasnoschok; Sergei Pereverzyev; Sergii V. Siryk; Nataliya Vasylyeva
We analyze the inverse boundary value-problem to determine the fractional order ν of nonautonomous semilinear subdiffusion equations with memory terms from observations of their solutions during small time. We obtain an explicit formula reconstructing the order. Based on the Tikhonov regularization scheme and the quasi-optimality criterion, we construct the computational algorithm to find the order
-
Generalized fractional Poisson process and related stochastic dynamics Fract. Calc. Appl. Anal. (IF 3.17) Pub Date : 2020-07-11 Thomas M. Michelitsch; Alejandro P. Riascos
We survey the ‘generalized fractional Poisson process’ (GFPP). The GFPP is a renewal process generalizing Laskin’s fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains