In this study, a new robust predictive control technique is investigated for uncertain fractional-order descriptor systems. Using the properties of fractional calculus and the construction of an appropriate Lyapunov function, the sufficient conditions to guarantee the existence of a robust predictive controller are given by minimizing the worst-case optimization problem. The new robust predictive controller

In this paper, we introduce a novel approach to the fractional Dunkl–Laplacian within a framework derived from specific reflection symmetries in Euclidean spaces. Our primary contributions include pointwise formulas, Bochner subordination, and addressing an extension problem for the fractional Dunkl–Laplacian.

In this paper, we consider the following Kirchhoff-Schrödinger-Poisson equation: $$\begin{aligned} \begin{aligned} \left\{ \begin{array}{lc} \left( a+b[u]_s^2\right) (-\varDelta )^s u+V_\lambda (x) u+\phi u=|u|^{p-2}u+|u|^{2_s^*-2} u &{}{} \text { in } {\mathbb {R}}^3, \\ (-\varDelta )^t \phi =u^2 &{}{} \text { in } {\mathbb {R}}^3, \end{array}\right. \end{aligned} \end{aligned}$$ where \(s \in \left(

This article investigates time optimal controls for the Cauchy problem of Hilfer fractional evolution equations. At first, by employing the fixed point technique and the operator semigroup theory, an existence theorem is obtained. Then the existence of time optimal control pair is studied by applying an approximate technique. An example is given as applications in the last section.

This paper is a continuation of the recent article “Multi-parametric Le Roy function”, Fract. Calc. Appl. Anal. 26(5), 54–69 (2023), https://doi.org/10.1007/s13540-022-00119-y, by S. Rogosin and M. Dubatovskaya. Here we present further analytic properties of the Le Roy function depending on several parameters. Using the Mellin-Barnes representations we determine relations of the multi-parametric Le

We consider a parametric elliptic problem governed by the spectral Neumann fractional Laplacian on a bounded domain of \(\mathbb {R}^N\), \(N\ge 2\), with a general nonlinearity. This problem is related to the existence of steady states for Keller-Segel systems in which the diffusion of the chemical is nonlocal. By variational arguments we prove the existence of a weak solution as a local minimum of

We establish some fixed-time decay estimates in Lebesgue spaces for the fractional heat propagator \(e^{-tH^{\beta }}\), \(t, \beta >0\), associated with the harmonic oscillator \(H=-\Delta + |x|^2\). We then prove some local and global wellposedness results for nonlinear fractional heat equations.

The present paper deals with the study of the approximation properties of the well-known sampling Kantorovich (SK) operators in “Sobolev-like settings”. More precisely, a convergence theorem in case of functions belonging to the usual Sobolev spaces for the SK operators has been established. In order to get such a result, suitable Strang-Fix type conditions have been required on the kernel functions

In this paper, the averaging principle for stochastic Caputo fractional differential equations (SCFDEs) 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

In this paper, we mainly make research on the approximation of continuous functions in the view of the fractal structure based on previous studies. Initially, fractal dimensions and the Hölder continuity of the linear combination of continuous functions have been explored and dense subsets of the space of continuous functions have also been studied. Then, it has been proved that the order of finding

We introduce a class of fractional Dirac type operators with time variable coefficients by means of a Witt basis, the Djrbashian–Caputo fractional derivative and the fractional Laplacian, both operators defined with respect to some given functions. Direct and inverse fractional Cauchy type problems are studied for the introduced operators. We give explicit solutions of the considered fractional Cauchy

In this paper, we study a nonlinear system involving a generalized tempered fractional p-Laplacian in \({\mathbb {R}}^{n}\): $$\begin{aligned} \left\{ \begin{array}{ll} (-\varDelta -\lambda _{f})_{p}^{s}u(x)+\omega u(x)=C_{n,t}(|x|^{2t-n}*u^{q})u^{q-1}, &{}x\in {\mathbb {R}}^{n},\\ u(x)>0,&{}x\in {\mathbb {R}}^{n}, \end{array} \right. \end{aligned}$$ where \(02,\ p-10\). By using the direct method

In the work presented in this paper, a study for a class of space-time fractional Langevin diffusion equations with two different time-fractional derivatives and a spatial fractional Laplacian was conducted. Firstly, we investigate the well-posedness and regularity of the mild solutions for a nonlocal problem by using several distinctly different approaches in the following three cases with the source

In this paper, we study the boundedness of p-adic Riesz potential on two-weighted Morrey–Herz spaces with both power weight and Muckenhoupt weight. Moreover, the boundedness for commutators of p-adic Riesz potential with symbols in Lipschitz spaces and weighted central BMO spaces on such spaces are also established.

This paper studies two efficient numerical methods for the generalized tempered integrodifferential equation with respect to another function. The proposed methods approximate the unknown solution through two phases. First, the backward Euler (BE) method and first-order interpolation quadrature rule are adopted to approximate the temporal derivative and generalized tempered integral term to construct

In this review paper we try to describe some recent results on modeling and analysis of economic risks by using the techniques of fractional calculus. The use of fractional order operators in the risk theory is due to the presence of long and short memory in most of economic models and their nonlocality over time. We emphasize on the use and interpretation of the Dzherbashian-Caputo fractional derivative

In this paper, we study the cyclic anti-periodic boundary value problem of a nonlinear tripled system of fractional Langevin equations. By means of Krasnoselskii fixed point theorem and Banach contraction mapping theorem, we obtain sufficient conditions for the existence and uniqueness of results. Moreover, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of the proposed problem. Finally

We consider the homogenization for time-fractional diffusion equations in a periodic structure. First, we derive the homogenized time-fractional diffusion equations. Next, we prove the stability in determining a constant diffusion coefficient by minimum data. Moreover, we investigate the inverse problems of estimating the homogenized diffusion coefficient by the data for non-homogenized structure.

In this paper, we study the boundedness for the fractional integral operator \(I_\alpha \) on generalized Morrey spaces associated with ball Banach function spaces. Moreover, the boundeness for the commutator of \(I_\alpha \) on these spaces is also established. Our main results extend various previous results on some concrete generalized Morrey spaces associated with ball Banach function spaces.

An elementary inequality is discussed for autonomous Caputo fractional differential equation (FDE) of order \(\alpha \in (0,1)\) in \(\mathbb {R}^d\) for which the vector field satisfies a dissipativity condition. This inequality is fundamental for investigating qualitative and dynamical properties of such equations. Here its use and effectiveness are illustrated to show the global existence and uniqueness

In the present paper, different uncertainty principles for hypercomplex functions associated with the fractional Fourier transform (FRFT) are investigated. First, the inverse formula and Plancherel theorem of the FRFT are obtained. Then, a new uncertainty principle for the FRFT of hypercomplex functions in the time and FRFT domains is explored. Moreover, an uncertainty principle in two FRFT domains

It is known that at least ten equivalent definitions of the fractional Laplacian exist in an unbounded domain. Here we derive a further equivalent definition that is based on the Mellin transform and it can be used when the fractional Laplacian is applied to radial functions. The main finding is tested in the case of the space-fractional diffusion equation. The one-dimensional case is also considered

By the Schauder fixed point theorem and generalized Ascoli-Arzela theorem, we present that the Riemann-Liouville fractional differential equation has at least one globally attractive solution and \(x(t)=x_{0}t^{\beta -1}+o(t^{\beta -\gamma _{1}})\) as \(t\rightarrow +\infty \), where \(\beta<\gamma _{1}<\gamma <1\). The novelty in this paper is that the global solution of Riemann-Liouville fractional

We consider a general class of linear integro-differential equations with Caputo fractional derivatives and weakly singular kernels. First, the underlying initial value problem is reformulated as an integral equation and the possible singular behavior of its exact solution is determined. After that, using a suitable smoothing transformation and spline collocation techniques, the numerical solution

This paper is concerned with the approximate controllability of a class of semi-linear neutral integro-differential equation with nonlocal condition. The basic tools of this study are the theory of resolvent operators of linear neutral integro-differential equation and fractional powers. The fundamental solution of linear neutral integro-differential equations is also constructed to deal with the non-uniform

In this contribution, a wave equation with a time-dependent variable-order fractional damping term and a nonlinear source is considered. Avoiding the circumstances of expressing the nonlinear variable-order fractional wave equations via closed-form expressions in terms of special functions, we investigate the existence and uniqueness of this problem with Rothe’s method. First, the weak formulation

In recent years, much attention has been paid to the study of forward and inverse problems for the Rayleigh-Stokes equation in connection with the importance of this equation for applications. This equation plays an important role, in particular, in the study of the behavior of certain non-Newtonian fluids. The equation includes a fractional derivative of order \(\alpha \), which is used to describe

The main objective of this paper is analysis of the initial-boundary value problems for the linear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type time-fractional derivative acting in the fractional Sobolev spaces. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. First we

This paper is concerned with the multiplicity of solutions for the fractional Kirchhoff type equations with a critical nonlocal term $$\begin{aligned} {\left\{ \begin{array}{ll}\displaystyle M\left( \Vert u\Vert ^2_Z\right) \mathscr {L}_K u=\lambda f(x,u)+ \left( \int _{\varOmega }\frac{|u(y)|^{2^*_{\alpha ,s}}}{|x-y|^{\alpha }}dy\right) |u|^{2^*_{\alpha ,s}-2}u,&{} x\in \varOmega , \\ u(x)=0, &{}x\in

In this paper, we study the initial-boundary value problem for a fractional pseudo-parabolic p-Laplacian type equation. First, by means of the imbedding theorems, the theory of potential wells and the Galerkin method, we prove the existence and uniqueness of global solutions with subcritical initial energy, critical initial energy and supercritical initial energy, respectively. Next, we establish the

In this article, we study the existence and uniqueness of a weak solution to the fractional single-phase lag heat equation. This model contains the terms \(\mathcal {D}_t^\alpha (\partial _{t}u)\) and \(\mathcal {D}_t^\alpha u \) (with \(\alpha \in (0,1)\)), where \(\mathcal {D}_t^\alpha \) denotes the Caputo fractional differential operator (in time) of order \(\alpha \). We consider homogeneous Dirichlet

In this article, we establish a new tempered sequence space related to tempered sequence \(\ell _{p}^{\alpha },\) \(p\ge 1\) and obtain the Hausdorff measure of noncompactness of this space. By using this measure of noncompactness with Darbo’s fixed point theorem, we investigate the existence results for an infinite system of Langevin fractional differential equations involving a generalized derivative

We consider linear time-varying control problems involving the Caputo fractional derivative. By introducing the controllability Gramian matrix, we prove that the invertibility of the Gramian is a necessary and sufficient condition for controllability. Furthermore, we prove equivalence between controllability and observability of the associated adjoint problem. Using methods from linear control theory

We establish a novel definition of the fractional gradient and divergence of order \(\alpha \in (0, 1)\) through the use of Riesz potential on homogeneous Carnot groups. We introduce and investigate the distributional fractional Sobolev space and the space of fractional BV functions in this context. Additionally, we provide a definition of fractional Caccioppoli sets on homogeneous Carnot groups and

We study the asymptotic behavior of solutions to various Dirichlet sublinear-type problems involving the fractional Laplacian when the fractional parameter s tends to zero. Depending on the type on nonlinearity, positive solutions may converge to a characteristic function or to a positive solution of a limit nonlinear problem in terms of the logarithmic Laplacian, that is, the pseudodifferential operator

In this study, we investigate nonlocal problems for a class of the Sobolev type fractional delay impulsive integrodifferential inclusions of order \({r} \in (1,2)\) via sectorial operator in Hilbert space. The results are obtained by engaging of mixed Volterra-Fredholm integrodifferential systems, sectorial operator of type \((P,\kappa ,{r},\gamma )\) and Martelli’s fixed point theorem under the assumption

Nonstationary fractional-order systems represent a new class of dynamic systems characterized by time-varying parameters as well as memory effect and hereditary properties. Differential game described by system of linear nonstationary differential equations of fractional order is treated in the paper. The game involves two players, one of which tries to bring the system’s trajectory to a terminal set