35 Results Found

In this paper, we introduce grand Herz–Morrey spaces with variable exponent and prove the boundedness of Riesz potential operators in these spaces.

In this paper, we study the two weight commutators theorem of Riesz potential on an arbitrary homogeneous group $\mathbb{H}$ of dimension N. Moreover, in accordance with the results in the Euclidean space, we acquire the quantitative weighted bound on homogeneo...

Symmetry properties of a nonlinear two-dimensional space-fractional diffusion equation with the Riesz potential of the order $\alpha \in (0,1)$ are studied. Lie point symmetry group classification of this equation is performed with resp...

In this paper, our first objective is to define the idea of grand variable Herz spaces. Then, our main goal is to prove boundedness results for operators, including the rough Riesz potential operator of variable order and the fractional Hardy operato...

In this paper, we give the sufficient conditions for the compactness of sets in generalized Morrey spaces $M_p^{w(·)}$. This result is an analogue of the well-known Fréchet–Kolmogorov theorem on the compactness of a set in Lebesgue spac...

In this paper, we establish sufficient conditions for the pre-compactness of sets in the global Morrey-type spaces $G{M}_{p\theta }^{w(·)}$. Our main result is the compactness of the commutators of the Riesz potential $\left[b,I_{\alpha }\right]$ in global Morrey-type sp...

This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces $L^{\mathtt{p}(·)}$. To the best of our knowledge, the boundedness of such...

Let $\mathcal{G}$ be a homogeneous Lie group of dimension $\varkappa$. In this article, we investigate the mapping properties of the Wolff-type potential ${\mathcal{W}}_{\vartheta ,2}$ on $\mathcal{G}$, associated with a homogeneous quasi-norm, and show that it coincides, up to an explicit dime...

An extension of the general fractional calculus (GFC) is proposed as a generalization of the Riesz fractional calculus, which was suggested by Marsel Riesz in 1949. The proposed Riesz form of GFC can be considered as an extension GFC from the positiv...

The bilateral tempered fractional derivatives are introduced generalising previous works on the one-sided tempered fractional derivatives and the two-sided fractional derivatives. An analysis of the tempered Riesz potential is done and shows that it...

The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian op...

A general approach to solving the Dirichlet problem, both for bounded $3D$ domains and for their unbounded complements, in terms of the fractional ($3D$) Poisson equation, is presented. Lauren Schwartz class solutions are sought for tempered distribution...

We consider the principle of least action in the context of fractional calculus. Namely, we derive the fractional Euler–Lagrange equation and the general equation of motion with the composition of the left and right fractional derivatives defin...

When $2-1/Q<p\le 2$, we establish the $C_{\mathrm{loc}}^{0,1}$ and $C_{\mathrm{loc}}^{1,\alpha }$-regularities of weak solutions to quasi-linear p-Laplacian type non-homogeneous equations in the Heisenberg group ${\mathbb{H}}^n$, where $Q=2n+2$ is the homogeneous dimension of ${\mathbb{H}}^n$.

In this article, we investigate the regularization and qualitative properties of parabolic Ginzburg–Landau equations in variable exponent Herz spaces. These spaces capture both local and global behavior, providing a natural framework for our an...

The method of deriving fractional-order differential equations using Riesz fractional-order calculus is a new breakthrough, and it is possible to use the inverse scattering transform (IST) to solve the analytical solutions of the derived equations. H...

The Wheeler–DeWitt equation for a flat and compact Friedmann–Lemaître–Robertson–Walker cosmology at the pre-inflation epoch is studied in the contexts of the standard and fractional quantum cosmology. Working within the...

In the present work, we explored the dynamics of single kinks, kink–anti-kink pairs and bound states in the prototypical fractional Klein–Gordon example of the sine-Gordon equation. In particular, we modified the order $\beta$ of the tempo...

In this study, the bending behavior of beams is investigated using the fractional Euler–Bernoulli beam model. This model is developed based on fractional calculus, particularly employing the Riesz–Caputo derivatives, and is capable of acc...

In this paper, we consider evolution equations in the abstract Hilbert space under the special conditions imposed on the operator at the right-hand side of the equation. We establish the method that allows us to formulate the existence and uniqueness...

In this paper, we consider the following Kirchhoff-type equation: ${\{}_{u\in H^1\left({\mathbb{R}}^N\right),}^{-(a+b{\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x)\Delta u+V\left(x\right)u=(I_{\alpha }\ast F\left(u\right))f\left(u\right)+\lambda g\left(u\right),\mathrm{in}{\mathbb{R}}^N,}$ where $a>0$, $b\ge 0$, $\lambda >0$, $\alpha \in (N-2,N)$, $N\ge 3$, $V:{\mathbb{R}}^N\to \mathbb{R}$ is a potential function and $I_{\alpha }$ is a Riesz potential of order $\alpha \in (N-2,N)$. U...

In this paper, the authors investigate a partial inverse spectral problem for Sturm–Liouville operators with frozen arguments on star-shaped graphs. The problem is to reconstruct the potential on one edge from the known potentials on the other...

In this article, we investigate the existence of a nontrivial solution for the nonlinear Choquard equation with upper critical exponent see Equation (6). The Riesz potential in this case has never been studied. We establish the existence of the groun...

Let be the high-order Schrödinger operator ${(-\Delta )}^2+V^2,$ where V is a non-negative potential satisfying the reverse Hölder inequality ($R{H}_q$), with $q>n/2$ and $n\ge 5$. In this paper, we prove that when $0<\alpha \le 2-n/q$, th...

This paper develops novel results in the harmonic analysis of Bessel operators, extending their theory to higher-dimensional and non-Euclidean spaces. We present a refined framework for Hardy spaces associated with Bessel operators, emphasizing atomi...

For the first time, a new dissipation-preserving scheme is proposed and analyzed to solve a Caputo–Riesz time-space-fractional multidimensional nonlinear wave equation with generalized potential. We consider initial conditions and impose homoge...

In this paper, we consider the time-dependent Schrödinger equation: $i\frac{\partial \psi (x,t)}{\partial t}=\frac{1}{2}{(-\mathrm{\Delta })}^{\frac{\alpha }{2}}\psi (x,t)+V\left(x\right)\psi (x,t),$ $x\in \mathbb{R},t>0$ with the Riesz space-fractional derivative of order...

The article produces a brief review of some recent results which predict stable propagation of solitons and solitary vortices in models based on the nonlinear Schrödinger equation (NLSE) including fractional one-dimensional or two-dimensional diffrac...

In this work, we apply fractional calculus to study quantum cosmology. Specifically, our Wheeler-DeWitt (WDW) equation includes a Friedman-Robertson-Walker (FRW) geometry, a radiation fluid, a positive cosmological constant ($\mathsf{\Lambda }$), and an ad-hoc...

This paper is devoted to the study of the following fractional Choquard equation with prescribed $L^2$ norm: $\left\{\begin{array}{c}{(-\mathrm{\Delta })}^{s}u+\mu u=\left(I_{\alpha }\ast F\left(u\right)\right)F^{\prime }\left(u\right)\mathrm{in}{\mathbb{R}}^N,\hfill \\ {\parallel u\parallel }_{L^2\left({\mathbb{R}}^N\right)}=a,\hfill \end{array}\right.$ where $N\ge 2$, $s\in (0,1)$, $I_{\alpha }$ is the Riesz potentia...

The present paper investigates the following inhomogeneous generalized Hartree equation $i\dot{u}+\Delta u=\pm {\left|u\right|}^{p-2}{\left|x\right|}^b(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b)u$, where the wave function is $u:=u(t,x):\mathbb{R}\times {\mathbb{R}}^N\to \mathbb{C}$, with $N\ge 2$. In addition, th...

As a model that possesses both the potentialities of Caputo time fractional diffusion equation (Caputo-TFDE) and symmetric two-sided space fractional diffusion equation (Riesz-SFDE), time-space fractional diffusion equation (TSFDE) is widely applied...

A conceptually consistent understanding is sought for the interactions sampled by the imaginary cubic oscillator with potential $V^{\left(ICO\right)}\left(x\right)=\mathrm{i}{x}^3$, which is by itself not acceptable as a meaningful quantum model due to a combination of its non-Hermiticity...

In this paper, we study a class of nonlocal Schrödinger–Poisson–Slater equations: $-\Delta u+u+\lambda \left(I_{\alpha }\ast {\left|u\right|}^q\right){\left|u\right|}^{q-2}u={\left|u\right|}^{p-2}u$, where $q,p>1$, $\lambda >0$, and $I_{\alpha }$ is the Riesz potential. We obtai...

Recent studies have emphasized the importance of the long-distance diffusion model in characterizing tracer transport occurring within both subsurface and surface environments, particularly in heterogeneous systems. Long-distance diffusion, often ref...