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

In this paper, we investigate the non-confluence property of a class of stochastic differential equations with Markovian switching driven by fractional Brownian motion with Hurst parameter \(H\in (1/2,1)\). By using the generalized Itô formula and stopping time techniques, we obtain some sufficient conditions ensuring the non-confluence property for the considered equations. Additionally, we present

In this paper, we introduce the concept of Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic processes in the square mean sense, and establish some basic results on the function space of such processes like completeness, convolution and composition theorems. Under the situation that the functions forcing are Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic and

This article aims to investigate the semi-classical analog of the general Caputo-type diffusion equation with time-dependent diffusion coefficient associated with the discrete Schrödinger operator, \(\mathcal {H}_{\hbar ,V}:=-\hbar ^{-2}\mathcal {L}_{\hbar }+V\) on the lattice \(\hbar \mathbb {Z}^{n},\) where V is a positive multiplication operator and \(\mathcal {L}_{\hbar }\) is the discrete Laplacian

This paper study a class of time-space fractional reaction-diffusion equations with nonlocal initial conditions and construct an abstract theory in fractional power spaces to discuss the results related S-asymptotically \(\omega \)-periodic mild solutions. When the coefficients are sufficiently small, under the condition that the nonlinear term can grow any number of orders, we discuss the existence

In this paper, the averaging principle for stochastic Caputo fractional differential equations with the nonlinear terms satisfying the non-Lipschitz condition is considered. The work in the article is roughly divided into three parts. Firstly, we establish a generalized Gronwall inequality with singular integral kernel which is a key part in our analysis. Secondly, we discuss the existence and uniqueness

This paper has two primary objectives. The first one is to demonstrate that the solutions of master equation $$\begin{aligned} (\partial _t-\Delta )^s u(x,t) =f(u(x, t)), \,\,(x, t)\in B_1(0)\times \mathbb {R}, \end{aligned}$$ subject to the vanishing exterior condition, are radially symmetric and strictly decreasing with respect to the origin in \(B_1(0)\) for any \(t\in \mathbb {R}\). Another one

The paper investigates fractional differential equations involving the Liouville derivative. Solution to these equations under a boundary condition inside the time interval are derived in explicit form, their uniqueness is established using integral transforms technique.

Bearing in mind an increasing popularity of the fractional calculus the main aim of this paper is to derive several new representation formulae for the cumulative distribution function (cdf) of the McKay \(I_\nu \) Bessel distribution including the Grünwald-Letnikov fractional derivative; also, two connection formulae between cdf of the McKay \(I_\nu \) random variable and the so–called Neumann series

We investigate the fractional stochastic heat equation, driven by a random noise which admits a covariance measure structure with respect to the time variable and has a spatial covariance given by the Riesz kernel. This class of process includes White-colored noise, fractional colored noise and other related processes. We give a sufficient condition for the existence of the mild solution and we establish

The causal shift-invariant convolution is studied from the point of view of inversion. Abel’s algorithm, used in the tautochrone problem, is considered and Sonin’s existence condition is deduced. To generate pairs of functions verifying Sonin’s condition, the class of Mittag-Leffler type functions is used. In particular, functions that are impulse responses of ARMA(N,N) systems serve as a basis. The

The Volterra integral equation associated with autonomous Caputo fractional differential equation (FDE) of order \(\alpha \in (0,1)\) in \({\mathbb {R}}^d\) was shown by the authors [4] to generate a semi-group on the space \({\mathfrak {C}}\) of continuous functions \(f:{\mathbb {R}}^+\rightarrow {\mathbb {R}}^d\) with the topology uniform convergence on compact subsets. It serves as a semi-dynamical

The paper studies the fractional order version of the reinforced membrane problem introduced in [A. Henrot and H. Maillot, 2001]. Existence and uniqueness of the solutions of the corresponding non-local equations has been proven for the relaxed problem. In addition, for the radial symmetric case the existence of the optimal domain has been shown.

Let \(p,q\in [1,\infty )\), s be a nonnegative integer, \(\alpha \in \mathbb {R}\), and \(\mathcal {X}\) be \(\mathbb {R}^n\) or a cube \(Q_0\subsetneqq \mathbb {R}^n\). In this article, the authors introduce the localized special John–Nirenberg–Campanato spaces via congruent cubes, \(jn_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathcal {X})\), and show that, when \(p\in (1,\infty )\), the predual of \(jn_{(p

We study the fractional Fokker-Planck-Kolmogorov equation with the fractional index \(\alpha \in [1,2)\) and use a vector-valued Calderón-Zygmund theorem to obtain the existence and uniqueness of \(L^p([0,T];{{\mathcal {C}}}_b^{\alpha +\beta }({{\mathbb {R}}}^d))\cap W^{1,p}([0,T];{{\mathcal {C}}}_b^\beta ({{\mathbb {R}}}^d))\) solution under the assumptions that the drift coefficient and nonhomogeneous

A necessary and sufficient condition for the existence or nonexistence of global solutions to the following fractional reaction-diffusion equations $$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}=\Delta _{\alpha } u + \psi (t)f(u),\,\,&{} \text{ in } \mathbb {R}^{N}\times (0,\infty ),\\ u(\cdot ,0)=u_{0}\ge 0,\,\,&{} \text{ in } \mathbb {R}^{N}, \end{array}\right. } \end{aligned}$$ has not been

The maximal \(B_{p,q}^{s}\)-regularity properties of a fractional convolution elliptic equation is studied. Particularly, it is proven that the operator generated by this nonlocal elliptic equation is sectorial in \( B_{p,q}^{s}\) and also is a generator of an analytic semigroup. Moreover, well-posedeness of nonlocal fractional parabolic equation in Besov spaces is obtained. Then by using the \(B_{p

The theory of fractal surfaces, in its basic setting, asserts the existence of a bivariate continuous function defined on a triangular domain. The extant literature on the construction of fractal surfaces over triangular domains use certain assumptions for the construction and deal primarily with the interpolation aspects. Working in the framework of fractal surfaces over triangular domains, this note