In this paper, which can be considered as an extension of our previous publication (Liu and Yu in Fract Calc Appl Anal 25:2040-2061, 2022) in same journal, we analyze the stability and synchronization for the discrete fractional-order neural network systems with mixed time delays. By new techniques, we give the proof of the discrete fractional-order Halanay inequality with mixed time delays, which

We consider a p-fractional Choquard-type equation $$\begin{aligned} (-\varDelta )_p^s u+a|u|^{p-2}u=b(K*F(u))F'(u)+\varepsilon _g |u|^{p_g-2}u \quad \text {in } \mathbb {R}^N, \end{aligned}$$ where \(0

We are concerned with a space-time fractional parabolic initial-boundary value problem of Sturm-Liouville type in a general star graph with mixed Dirichlet and Neumann boundary controls. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. Using the notion of no-regret control introduced by Lions, we prove the existence, uniqueness, and characterize the

We introduce a novel class of stochastic partial differential equations (SPDEs) driven by space-only fractional Lévy noise. In contrast to the prevalent focus on space-time noise in the existing literature, our work explores the unique challenges and opportunities presented by purely spatial perturbations. We establish the existence and uniqueness of the solution to the stochastic heat equation by

We introduce an efficient discretisation of a novel fractional-order adaptive exponential (FrAdEx) integrate-and-fire model, which is used to study the fractional-order dynamics of neuronal activities. The discretisation is based on an extension of L1-type methods that can accurately handle exponential growth and the spiking mechanism of the model. This new method is implicit and uses adaptive time

In this paper, we investigate various classes of the abstract multi-term fractional difference equations and the abstract higher-order difference equations with integer order derivatives. The abstract difference equations under our consideration can be unsolvable with respect to the highest derivative. We use the Riemann-Liouville and Caputo fractional derivatives, provide some new applications of

In this article, we consider a multi-term \(\varphi \)-Caputo fractional dynamical system with non-instantaneous impulses. Firstly, we derive the solution for the linear \(\varphi \)-Caputo fractional differential equation by using the generalized Laplace transform. Then, some necessary and sufficient conditions have been examined for the controllability of the linear multi-term \(\varphi \)-Caputo

The fractional Laplacian and fractional gradient are operators which play fundamental role in modeling of anomalous diffusion in d-dimensional space with the fractional exponent \(\alpha \in (1,2)\). The principal-value integrals are split into singular and regular parts where we avoid using any weight function for the approximation in the singularity neighborhood. The resulting approximation coefficients

Let \(\{e^{-t{\mathcal {L}}^{\alpha }}\}_{t>0}\) be the heat semigroup related to the fractional Schrödinger operator \(\mathcal {L}^{\alpha }:=(-\varDelta +V)^{\alpha }\) with \(\alpha \in (0,1)\), where V is a non-negative potential belonging to the reverse Hölder class. In this paper, we analyze the convergence of the following type of the series $$\begin{aligned} T_{N,t}^{\alpha ,\beta }(f)=\sum

In this paper, we introduce a concept of the nth-level general fractional derivatives that combine the Djrbashian–Nersessian fractional derivatives and the general fractional derivatives with the Sonin kernels in one definition. Then some basic properties of these fractional derivatives including the fundamental theorems of fractional calculus and a formula for their Laplace transform are presented

This study examines the controllability criteria for linear and semilinear fractional dynamical systems with delays in both state and control variables in the framework of the Caputo fractional derivative. To establish the controllability criteria for linear fractional dynamical systems, the study derives necessary and sufficient conditions by employing the positive definiteness of the Grammian matrix

This study focuses on the existence, uniqueness, and quenching behavior of solution to the time-space fractional Kawarada problem, where the time derivative is the Caputo-Hadamard derivative and the spatial derivative is the fractional Laplacian. The mild solution represented by Fox H-function, based on the fundamental solution, is considered in space \(C\left( [a, T], L^r(\mathbb {R}^d)\right) \)

In this paper, we are concerned with 2D non-autonomous Reissner-Mindlin-Timoshenko plate systems with Laplacian damping terms and nonlinear sources terms. The global well-posedness is proved by using the theory of maximal monotone operators. And then we get the Lipschtiz stability of the solution. By establishing the existence of pullback absorbing sets and pullback asymptotic compactness of the process

This article studies topological properties of the solution set for a class of Caputo fractional delayed evolution inclusions. Firstly, in the scenario when the cosine family is noncompact, the compactness and \(R_{\delta }\)-property are obtained for the mild solution set. Then, as an application of the above obtained results, the approximative controllability is demonstrated. Finally, an example

In this paper, we provide new multiplicity results for a class of impulsive fractional Schrödinger-Kirchhoff-type equations involving p-Laplacian and Riemann-Liouville derivatives. By using the variational method and critical point theory, we obtain that the impulsive fractional problem has infinitely many solutions under appropriate hypotheses when the parameter \(\lambda \) lies in different intervals

We study the existence of nonnegative solutions to the following nonlocal elliptic problem involving singularity $$\begin{aligned} \mathfrak {M}\left( \int _{Q}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right) (-\Delta )_{p}^{s} u&=\frac{\lambda }{u^{\gamma }}+u^{p_s^*-1}~\text {in}~\Omega ,\\ u&>0~\text {in}~\Omega ,\\ u&=0~\text {in}~\mathbb {R}^N\setminus \Omega , \end{aligned}$$ where \(\mathfrak {M}\)

The paper addresses structural properties of distributed order controllers. A Distributed Order PID (DOPID) controller is a control structure in which a continuum of “differintegral” actions of orders between -1 and 1 are integrated together, and where relative contributions of different orders is determined by a weighting function. This stands in sharp contrast to conventional proportional-integral-derivative

This paper investigates the existence and multiplicity of positive solutions to the following semilinear problem: where \(f\in C([0,\infty ),{\mathbb {R}})\) represents an oscillating nonlinearity that satisfies a type of area condition. Our main analytical tools include variational methods and the sub-supersolution method.

In this study, we establish an explicit representation of solutions to \(\psi \)-Hilfer type linear fractional differential equations with variable coefficients in weighted spaces. Furthermore, we prove the existence and uniqueness of solutions for these equations. As a special case, we derive corresponding results for \(\psi \)-fractional differential equations with variable coefficients. To demonstrate

In this paper, we study the well-posedness and discretization of the space-time fractional parabolic equations (STFPEs) of the Sturm-Liouville type on a metric star graph. The considered problem involves the fractional time derivative in the Caputo sense, and the spatial fractional derivative is of the Sturm-Liouville type consisting of the composition of the right-sided Caputo derivative and left-sided

In this paper we study an evolution problem (FDIVHVI) which constitutes of the nonlinear fractional differential inclusion with damping driven by a variational-hemivariational inequality (VHVI) in Banach spaces. More precisely, first, it is shown that the solution set for VHVI is nonempty, bounded, convex and closed under the surjectivity theorem and the Minty formula. Then, we introduce an associated

In this note, we present a new uniqueness criterion for nonlinear fractional differential equations, which can be seen as an improvement of the result given by Ferreira [Bull. London Math. Soc. 45, 930–934 (2013)].

In this paper, Cauchy problem for incompressible Navier-Stokes equations with time fractional differential operator and fractional Laplacian in \(\mathbb {R}^n\) (\(n\ge 2\)) is investigated. The global and local existence and uniqueness of mild solutions are obtained with the help of Banach fixed point theorem when the initial data belongs to \(L^{p_{c}}(\mathbb {R}^n)\) \((p_c=\frac{n}{\alpha -1})\)

The solution and source term of nonlinear fractional differential equations (NFDEs) with initial values generally have the initial singularity. As is known that numerical methods for NFDEs usually occur the phenomenon of order reduction due to the existence of initial singularity. In this paper, an improved fractional predictor-corrector (PC) method is developed for NFDEs based on the technique of

This paper investigates stochastic averaging principle for a class of mixed slow-fast stochastic differential equations driven simultaneously by a multidimensional standard Brownian motion and a multidimensional fractional Brownian motion with Hurst parameter \(1/2

In this paper, a quasi-reversibility method is used to solve an inverse spatial source problem of multi-term time-space fractional parabolic equation by observation at the terminal measurement data. We are mainly concerned with the case where the time source can be changed sign, which is practically important but has not been well explored in literature. Under certain conditions on the time source

Our main concern is the existence of a new \(PC_{2-v}\)-mild solution for Hilfer fractional impulsive evolution equations of order \(\alpha \in (1,2)\) and \(\beta \in [0,1]\) as well as the approximate controllability problem in Banach spaces. Firstly, under the condition that the operator A is the infinitesimal generator of a cosine family, we give a new representation of \(PC_{2-v}\)-mild solution

In this paper, we establish sufficient conditions in order to guarantee the existence and uniqueness of discrete weighted pseudo S-asymptotically \(\omega \)-periodic solution to the semilinear fractional difference equation $$\begin{aligned} {\left\{ \begin{array}{ll} _C\nabla ^{\alpha } u^n=Au^n+g^n(u^n), \quad n\ge 2,\\ u^0=x_0 \in X, \quad u^1=x_1\in X, \\ \end{array}\right. } \end{aligned}$$ where

In this work, we consider an inverse problem of determining the coefficient at the lower term of a fractional diffusion equation. The direct problem is the initial-boundary problem for this equation with non-local initial and homogeneous Dirichlet conditions. To determine the unknown coefficient, an overdetermination condition of the integral form is specified with respect to the solution of the direct

This paper is devoted to the continuity of the weak solution for tempered fractional general diffusion equations driven by tempered fractional Brownian motion (TFBM). Based on the Feynman-Kac formula (1.2), by using the Itô isometry for the stochastic integral with respect to TFBM, Parseval’s identity and some ingenious calculations, we establish the continuities of the solution with respect to Hurst

In regular dynamics, discrete maps are model presentations of discrete dynamical systems, and they may approximate continuous dynamical systems. Maps are used to investigate general properties of dynamical systems and to model various natural and socioeconomic systems. They are also used in engineering. Many natural and almost all socioeconomic systems possess memory which, in many cases, is power-law-like

In this work, we employ a biorthogonal approach to construct the infinite-dimensional Fractional Pascal measure \(\mu ^{(\alpha )}_{_{\sigma }}, 0 < \alpha \le 1\), defined on the tempered distributions space \(\mathcal {E}'\) over \(\mathbb {R} \times \mathbb {R}^{*}_{+}\). The Hilbert space \(L^{2}(\mu ^{(\alpha )}_{_{\sigma }})\) is characterized using a set of generalized Appell polynomials \(\mathbb

This paper is to obtain sufficient conditions for the uniqueness and existence of solutions to a new nonlinear fractional partial integro-differential equation with boundary conditions. Our analysis relies on an equivalent implicit integral equation in series obtained from an inverse operator, the multivariate Mittag-Leffler function, Leray-Schauder’s fixed point theorem as well as Banach’s contractive

We use a generalized Riemann-Liouville type derivative of an abstract function of one variable and existence of a weak solution to an abstract fractional parabolic problem on [0, T] containing Riemann-Liouville derivative of a function of one variable and spectral fractional powers of a weak Dirichlet-Laplace operator to study existence of a strong solution to this problem. Our goal in this regard

We propose an alternative definition of a mixed partial derivative in the Caputo sense for functions of two variables defined on the rectangle \(P=[0,a]\times [0,b]\) (\(a>0, b>0\)). We give an integral representation of functions possessing such a derivative. Moreover, we study the existence and uniqueness of a solution, as well as the Ulam–Hyers type stability of a fractional counterpart of a nonlinear

In this paper, our goal is to study the following class of Hardy–Hénon type problems $$\begin{aligned} \left\{ \begin{array}{rclcl}\displaystyle (-\Delta )^{1/2} u& =& \lambda |x|^{\mu } u+|x|^{\alpha }f(u)& \text{ in }& (-1,1),\\ u& =& 0& \text{ on }& \mathbb {R}\setminus (-1,1), \end{array}\right. \end{aligned}$$ when \(\mu \ge \alpha {>-1}\), and the nonlinearity f has exponential critical growth

In this paper, we investigate the optimal control of a semi-linear fractional PDEs involving the spectral diffusion operator, or the realization of the integral fractional Laplace operator with the zero Dirichlet exterior condition, both of order s with \(s\in (0,1)\). The state equation contains a non-smooth nonlinearity, and the objective functional is convex in the control variable but contains

This paper addresses the questions of well-posedness to fractional m-Laplacian reaction diffusion equation with singular potential term and logarithmic nonlinearity: $$\begin{aligned} \left| x\right| ^{-2s}\partial _t u+(-\varDelta )_{m}^{s} u+ (-\varDelta )^{s} \partial _t u\!=\!u|u|^{-2} R(u), \end{aligned}$$ where \(R(u)=\left| u\right| ^{r}\ln (|u|)\). Guided by the made assumptions, we arrive

Here we introduce a notion of fractional k-dimensional measure, \(0\le k

In this paper, an important property of fractional order operators involving discontinuous functions is discussed, First, a pioneering work of impulsive fractional differential equations is recalled to illuminate the incorrectness of notation \({^C_{t_k}D}^{q}_t\). Second, a class of piecewise-defined equations with Caputo fractional derivative is contrastively investigated, and it is revealed that

We investigate a semilinear problem for a fractional diffusion equation with variable order Caputo fractional derivative \(\left( \partial _t^{\beta (t)} u\right) (t)\) subject to homogeneous Dirichlet boundary conditions. The right-hand side of the governing PDE is nonlinear (Lipschitz continuous) and it contains a weakly singular Volterra operator. The whole process takes place in a bounded Lipschitz

This research aims to investigate the stabilization problem of the Riemann-Liouville spatial \(\beta \)-fractional output with order \(\beta \in (0,\ 1)\) for a class of bilinear dynamical systems with a time Caputo \(\alpha \)-fractional derivative. Initially, we provide definitions and establish the well-posedness of the problem addressed. Furthermore, we introduce a feedback control strategy that

This paper studies the properties of weak solutions to a class of space-time fractional parabolic-elliptic Keller-Segel equations with logistic source terms in \({\mathbb {R}}^{n}\), \(n\ge 2\). The global existence and \(L^{\infty }\)-bound of weak solutions are established. We mainly divide the damping coefficient into two cases: (i) \(b>1-\frac{\alpha }{n}\), for any initial value and birth rate;

In the present paper, to fill the gap of the effect of singularity arising from multiple fractional derivatives on numerical analysis, the regularity and high order difference scheme for time fractional telegraph equations are taken into consideration. Firstly, the analytic solution is obtained by employing Laplace transform, and its regularity is then deduced. Secondly, by the technic of decomposition

In general fractional calculus (GFC), the counterpart of the fractional time derivative is a differential-convolution operator whose integral kernel satisfies some additional conditions, under which the Cauchy problem for the corresponding time-fractional equation is not only well-posed, but has properties similar to those of classical evolution equations of mathematical physics. In this work, we develop

In this paper, we consider the initial value problems for multi-term time-fractional wave equations in the framework of Besov spaces, which can be described the Couette flow of viscoelastic fluid. Considering the initial data in Besov spaces, we obtain some results about the local well-posedness and the blow-up of mild solutions for the proposed problem. Further, we extend these results to Besov–Morrey

We introduce and study the spaces of fractionally absolutely continuous functions of two variables of any order and the fractional Sobolev type spaces of functions of two variables. Our approach is based on the Riemann-Liouville fractional integrals and derivatives. We investigate relations between these spaces as well as between the Riemann-Liouville and weak derivatives.

In this paper, we study a class of non-autonomous lower critical fractional Choquard equation with a pure-power nonlinear perturbation. Under some reasonable assumptions on the potential function h, we prove the existence and discuss asymptotic behavior of ground state solutions for our problem. Meanwhile, we also prove that the number of normalized solutions is at least the number of global maximum

In this paper, we consider the following Schrödinger system involving the tempered fractional p-Laplacian $$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} & (-\varDelta -\lambda )^s_p u(x)+au^{p-1}(x)=f(u(x),v(x)),\\ & (-\varDelta -\lambda )^t_p v(x)+bv^{p-1}(x)=g(u(x),v(x)), \end{aligned} \end{array}\right. } \end{aligned}$$ where \(n \ge 2\), \(a, b>0\), \(2

We study a nonlinear, nonlocal Dirichlet problem driven by the fractional p-Laplacian, involving a \((p-1)\)-sublinear reaction. By means of a weak comparison principle we prove uniqueness of the solution. Also, comparing the problem to ’asymptotic’ weighted eigenvalue problems for the same operator, we prove a necessary and sufficient condition for the existence of a solution. Our work extends classical

This survey aims to present various approaches to non-integer integrals and derivatives and their practical implementation within Wolfram Mathematica. It begins by short discussion of historical moments and applications related to fractional calculus. Different methods for handling non-integer powers of differentiation operators are presented, along with generalizations of fractional integrals and