We establish the existence and uniqueness of a renormalized solution to an evolution equation featuring the non-local fractional p(x, y)-Laplacian and nonnegative \(L^1\)-data. The definition of renormalized solutions is adapted to the non-local nature to bypass the use of chain rules which is unavailable. The fractional p(x, y)-Laplacian well encapsulates the fractional p-Laplacian with a constant

The current study discusses the approximate solutions for a class of fractional stochastic integro-differential equation with short memory driven by non-instantaneous impulses (NIIs) defined on a separable Hilbert space. The approximation to the nonlinear functions is obtained using orthogonal projection operator. The existence and convergence of the sequence of approximate solutions is proved using

In this paper, we consider the long time behavior of solutions of the three dimensional (3D) generalized Navier-Stokes equations with damping. This family of 3D generalized Navier-Stokes equations with damping can be viewed as an interpolation model between subcritical (if \(\alpha >\frac{5}{4}\)), critical (if \(\alpha =\frac{5}{4}\)), and supercritical dissipations (if \(\alpha <\frac{5}{4}\)) and

This study introduces a novel family of exponential sampling type neural network Kantorovich operators, leveraging Hadamard fractional integrals to significantly enhance function approximation capabilities. By incorporating a flexible parameter \(\alpha \), derived from fractional Hadamard integrals, and utilizing exponential sampling, introduced to tackle exponentially sampled data, our operators

This paper is devoted to exploring a new class of Hilfer fractional stochastic switched dynamical systems with the Rosenblatt process and abrupt changes, where the abrupt changes occur suddenly at specific points and extend over finite time intervals. Initially, we established solvability outcomes for the proposed switched dynamical systems by employing the fractional calculus, fixed point method,

In this paper, we study a multivariate version of the generalized counting process (GCP) and discuss its various time-changed variants. The time is changed using random processes such as the stable subordinator, inverse stable subordinator, and their composition, tempered stable subordinator, gamma subordinator etc. Several distributional properties that include the probability generating function

This paper aims to study the eigenvalue problems of a mixed local and nonlocal operator in the exterior of a nonempty, bounded, simply connected domain \(\varOmega \subset {\mathbb {R}}^N\) with Lipschitz boundary \(\partial \varOmega \ne \emptyset \). By employing the variational methods combined with the Ljusternik-Schnirelmann principle, we prove the existence of a non-decreasing sequence of eigenvalues

In this paper, we focus on a type of fractional nonlinear Schrödinger equation with odd periodic boundary conditions, where the nonlinearity periodically depending on spatial variable x. By an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional reversible systems with unbounded perturbation, we obtain that there exists a lot of smooth quasi-periodic solutions with small amplitude

We deal with the boundary value problem $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u(x)= \lambda f(u(x)), & x\in \Omega ,\\ u(x)=0, & x\in \mathbb {R}^N \setminus \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \) is an open and bounded subset of \(\mathbb {R}^N\) with smooth boundary, \((-\Delta )^s\), \(s\in (0,1)\) denotes the fractional Laplacian, \(\lambda \ge 0\) and

In the present paper, we study inverse problems related to determining the time-dependent coefficient and unknown source function of fractional heat equations. Our approach shows that having just one set of data at an observation point ensures the existence of a weak solution for the inverse problem. Furthermore, if there is an additional datum at the observation point, it leads to a specific formula

This paper aims to establish the existence of a weak solution for the following problem: $$\begin{aligned} (-\Delta )^{s}_{\mathcal {H}}u(x) +V(x)h(x,x,|u|)u(x)=\left( \int _{{\mathbb R}^{N}}\dfrac{K(y)F(u(y))}{|x-y|^\lambda }\,\textrm{d}y\right) K(x)f(u(x)), \end{aligned}$$ in \({\mathbb R}^{N}\) where \(N\ge 1\), \(s\in (0,1), \lambda \in (0,N), \mathcal {H}(x,y,t)=\int _{0}^{|t|} h(x,y,r)r\ dr,\)

In this paper, we are concerned with a two-term fractional differential equation with functional boundary conditions. We discuss the existence of two kinds of solutions with respect to this type of equation. In this sense, for a class of two-term problems with specific boundary conditions, we use Matlab software to calculate the eigenvalues of the boundary value problems with Riemann-Liouville fractional

In this paper, we investigate the existence of mild solutions of the nonlinear fractional Rayleigh-Stokes problem for a generalized second grade fluid on an infinite interval. We firstly show the boundedness and continuity of solution operator. And then, by using a generalized Arzelà-Ascoli theorem and some new techniques, we get the compactness on the infinite interval. Moreover, we prove the existence

This paper delves into the existence of three weak solutions for a fractional \((\varphi , \psi )\)-like system involving the fractional \(\varphi \) Laplacian and the fractional \(\psi \) Laplacian respectively within the fractional Orlicz-Sobolev space. The proof is achieved by the well-known Bonanno-Marano techniques.

In this work, the exact solutions of time fractional Fokker-Planck equation are investigated using the symmetry approach. Also, the convergence of the reported solutions is proved along with the graphical interpretation of the obtained solutions.

Being based on coupling by change of measure and an approximation technique, the Harnack inequalities for a class of stochastic functional differential equations driven by fractional Ornstein-Uhlenbeck process with Hurst parameter \(0

In this investigation, we conduct a rigorous analysis of a class of non-homogeneous generalized double phase problems, characterized by the inclusion of the fractional \(\phi _{x ,y}^i(\cdot )\)-Laplacian operator (where \(i=1,2\)) and a Choquard-logarithmic nonlinearity, along with a real parameter. Our methodology involves establishing a set of precise conditions related to the Choquard nonlinearities

We consider fractional partial differential equations posed on the full space \(\mathbb {R}^d\). Using the well-known Caffarelli-Silvestre extension to \(\mathbb {R}^d \times \mathbb {R}^+\) as equivalent definition, we derive existence and uniqueness of weak solutions. We show that solutions to a truncated extension problem on \(\mathbb {R}^d \times (0,\mathcal {Y})\) converge to the solution of the

We consider multi-term fractional differential equations with continuous variable coefficients and differential operators of Erdélyi–Kober type and multiple independent fractional orders. We solve such equations in a general framework, obtaining explicit solutions in the form of uniformly convergent series. By considering several particular cases, we verify the consistency of our results with others

This paper investigates the existence of infinitely many pairs of nontrivial solutions for a class of nonsmooth fractional Hamiltonian systems, where the energy functional associated with the system is not continuously differentiable and does not satisfy the Palais-Smale condition. By considering a potential function of the form \(V(t,x)=-K(t,x)+W(t,x)\), where K and W are continuously differentiable

In this paper, we deal with a time dependent inverse source problem for a nonlinear p-Laplacian pseudoparabolic equation containing a fractional derivative in time of order \(\alpha \in (0,1)\). Moreover, the equation is perturbed by a power-law damping (reaction) term, which, depending on whether its sign is positive or negative, may account for the presence of a source or an absorption within the

This paper systematically treats the asymptotic behavior of many (linear/nonlinear) classes of higher-order fractional differential equations with multiple terms. To do this, we utilize the characteristics of Caputo fractional differentiable functions, the comparison principle, counterfactual reasoning, and the spectral analysis method (concerning the integral presentations of basic solutions). Some

In this note, we consider a pseudo-differential operator \(T_a\) defined as $$\begin{aligned} T_a f(x)=\int _{\mathbb {R}^n}e^{2\pi ix\cdot \xi }a(x,\xi )\widehat{f}(\xi )d\xi . \end{aligned}$$ It is well-known that \(T_a\) is not bounded on \(L^2\) in general when a belongs to the forbidden Hörmander class \(S^{n(\rho -1)/2}_{\rho ,1},0\le \rho \le 1\). In this note, when \(s>0,0\le \rho \le 1,1\le

This paper investigates the Cauchy problem of time-space fractional Keller-Segel-Navier-Stokes system in \({\mathbb {R}}^d~(d\ge 2)\), which describes both memory effect and Lévy process of the system. The local and global existence of mild solutions are obtained by the \(L^p-L^q\) estimates of Mittag-Leffler operators combined with Banach fixed point theorem and Banach implicit function theorem, respectively

In this article, using that the fractional Laplacian can be factored into a product of the divergence operator, a Riesz potential operator and the gradient operator, we introduce an anomalous fractional diffusion operator, involving a matrix K(x), suitable when anomalous diffusion is being studied in a non homogeneous medium. For the case of K(x) a constant, symmetric positive definite matrix we show

There is a well-established theory that links semi-Markov chains having Mittag-Leffler waiting times to time-fractional equations. We here go beyond the semi-Markov setting, by defining some non-Markovian chains whose waiting times, although marginally Mittag-Leffler, are assumed to be stochastically dependent. This creates a long memory tail in the evolution, unlike what happens for semi-Markov processes

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