Fractional Space Research Articles

Concentration-compactness type principle for systems with critical terms in [formula omitted

In this paper, we obtain some important variants of the Lions and Chabrowski Concentration-compactness principle, for nonlinear systems in the context of fractional Sobolev spaces with variable exponents. As an application of the results, we show the existence and asymptotic behaviour of nontrivial solutions for elliptic systems involving a new class of general nonlocal integrodifferential operators with variable exponents and critical growth conditions in RN.

Estimates for negative eigenvalues of Schrödinger operators on unbounded fractal spaces

We study an asymptotic formula for the number of negative eigenvalues of Schrödinger operators on unbounded fractal spaces, which admit a cellular decomposition. We first give some sufficient conditions for Weyl-type asymptotic formula to hold. Second, we verify these conditions for the infinite blowup of Sierpiński gasket and unbounded generalized Sierpiński carpets. Finally, we demonstrate how the result can be applied to the infinite blowup of certain fractals associated with iterated function systems with overlaps, including those defining the classical infinite Bernoulli convolution with golden ratio.

A New Perspective on the Stochastic Fractional Order Materialized by the Exact Solutions of Allen-Cahn Equation

Stochastic fractional differential equations are among the most significant and recent equations in physical mathematics. Consequently, several scholars have recently been interested in these equations to develop analytical approximations. In this study, we highlight the stochastic fractional space Allen-Cahn equation (SFACE) as a major application of this class. In addition, we utilize the simplest equation method (SEM) with a dual sense of Brownian motion to convert the presented equation into an ordinary differential equation (ODE) and apply an effective computational technique to obtain exact solutions. By carefully comparing the derived solutions with solutions from other articles, we prove the distinction of these solutions for their diversity and the discovery of new solutions for SFACE that appear in many scientific fields, such as mathematical biology, quantum mechanics, and plasma physics. The results introduced in this article were obtained by plotting several graphs and examining how noise affects exact solutions using Mathematica and MATLAB software packages.

Fractional Sobolev spaces with power weights

We investigate the form of the closure of the smooth, compactly supported functions $C_{c}^{\infty}(\Omega)$ in the weighted fractional Sobolev space $W^{s,p;\,w,v}(\Omega)$ for bounded $\Omega$. We focus on the weights $w,\,v$ being powers of the distance to the boundary of the domain. Our results depend on the lower and upper Assouad codimension of the boundary of $\Omega$. For such weights we also prove the comparability between the full weighted fractional Gagliardo seminorm and the truncated one.

Emergence of fractal cosmic space from fractional quantum gravity

Emergence of fractal cosmic space from fractional quantum gravity

Advancing Fractional Riesz Derivatives through Dunkl Operators

The aim of this work is to introduce a novel concept, Riesz–Dunkl fractional derivatives, within the context of Dunkl-type operators. A particularly noteworthy revelation is that when a specific parameter κ equals zero, the Riesz–Dunkl fractional derivative smoothly reduces to both the well-known Riesz fractional derivative and the fractional second-order derivative. Furthermore, we introduce a new concept: the fractional Sobolev space. This space is defined and characterized using the versatile framework of the Dunkl transform.

Correlation between Convolution Kernel Function and Error Function of Bone Fractal Operators

This article studies the convolutional kernel function of fractal operators in bone fibers. On the basis of the micro-nano composite structure of compact bone, we abstracted the physical fractal space of bone fibers and derived the fractal operators. The article aims to construct the convolutional analytical expression of bone fractal operators and proves that the error function is the core component of the convolution kernel function in the fractal operators. In other words, bone mechanics is the fractional mechanics controlled by error function.

Mathematical Analysis on Spherical Shell of Permeable Material in NID Space

In this paper, we have studied the magnetic shielding effect of a spherical shell analytically in fractional dimensional space (FDS). The Laplacian equation in fractional space predicts the complex phenomena of physics. This is a boundary value problem that has been solved by the separation variable method mathematically by taking low frequency w = 0. Electric potential is obtained in fractional dimensional space for the three regions, namely outside the spherical shell, between the shell and hollow sphere and inside the sphere. Also, the induced dipole moment has been derived. We obtain a general solution that reduces to the classical results by setting fractional parameter α = 3 which takes its value (2 < α ≤ 3).

Advances in Ostrowski-Mercer Like Inequalities within Fractal Space

The main idea of the current investigation is to explore some new aspects of Ostrowski’s type integral inequalities implementing the generalized Jensen–Mercer inequality established for generalized s-convexity in fractal space. To proceed further with this task, we construct a new generalized integral equality for first-order local differentiable functions, which will serve as an auxiliary result to restore some new bounds for Ostrowski inequality. We establish our desired results by employing the equality, some renowned generalized integral inequalities like Hölder’s, power mean, Yang-Hölder’s, bounded characteristics of the functions and considering generalized s-convexity characteristics of functions. Also, in support of our main findings, we deliver specific applications to means, and numerical integration and graphical visualization are also presented here.

Fractal calculus approach to diffusion on fractal combs

In this paper, we present a generalization of diffusion on fractal combs using fractal calculus. We introduce the concept of a fractal comb and its associated staircase function. To handle functions supported on these combs, we define derivatives and integrals using the staircase function. We then derive the Fokker–Planck equation for a fractal comb with dimension α, incorporating fractal time, and provide its solution. Additionally, we explore α-dimensional and (2α)-dimensional Brownian motion on fractal combs with drift and fractal time. We calculate the corresponding fractal mean square displacement for these processes. Furthermore, we propose and solve the heat equation on an α-dimensional fractal comb space. To illustrate our findings, we include graphs that showcase the specific details and outcomes of our results.

Local well‐posedness of the higher‐order nonlinear Schrödinger equation on the half‐line: Single‐boundary condition case

AbstractWe establish local well‐posedness in the sense of Hadamard for a certain third‐order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher‐order nonlinear Schrödinger equation, formulated on the half‐line . We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at . Our functional framework centers around fractional Sobolev spaces with respect to the spatial variable. We treat both high regularity () and low regularity () solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial‐boundary value problems, as it involves proving boundary‐type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher‐order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; and (iii) complicated oscillatory kernels in the weak solution formula for the linear initial‐boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial‐boundary value problem for a partial differential equation associated with a multiterm linear differential operator.

A compact embedding result and its applications to a nonlocal Schrödinger equation

AbstractCompactness is a fundamental issue in nonlinear analysis. Assume and with , we consider a subspace of , which is the space of invariant functions corresponding to a subgroup G of , where is a kind of function spaces including fractional Sobolev spaces . We show that the embeddings are compact for provided Ω is compatible with G, where is the fractional Sobolev critical exponent. Moreover, the existence and multiplicity of radial and nonradial solutions were obtained of the following nonlocal Schrödinger equation: where is an integro‐differential operator has order 2s and can be given by and the kernel K satisfies the following properties: there is and such that for any ; , where .

Role of dynamic contrast-enhanced MRI in predicting severe acute radiation-induced rectal injury in patients with rectal cancer.

To explore the potential of dynamic contrast-enhanced MRI (DCE-MRI) quantitative parameters in predicting severe acute radiation-induced rectal injury (RRI) in rectal cancer. This retrospective study enrolled 49 patients with rectal cancer who underwent neoadjuvant chemoradiotherapy and rectal MRI including a DCE-MRI sequence from November 2014 to March 2021. Two radiologists independently measured DCE-MRI quantitative parameters, including the forward volume transfer constant (Ktrans), rate constant (kep), fractional extravascular extracellular space volume (ve), and the thickness of the rectal wall farthest away from the tumor. These parameters were compared between mild and severe acute RRI groups based on histopathological assessment. Receiver operating characteristic curve analysis was performed to analyze statistically significant parameters. Forty-nine patients (mean age, 54years ± 12 [standard deviation]; 37 men) were enrolled, including 25 patients with severe acute RRI. Ktrans was lower in severe acute RRI group than mild acute RRI group (0.032min-1 vs 0.054min-1; p = 0.008), but difference of other parameters (kep, ve and rectal wall thickness) was not significant between these two groups (all p > 0.05). The area under the receiver operating characteristic curve of Ktrans was 0.72 (95% confidence interval: 0.57, 0.84). With a Ktrans cutoff value of 0.047min-1, the sensitivity and specificity for severe acute RRI prediction were 80% and 54%, respectively. Ktrans demonstrated moderate diagnostic performance in predicting severe acute RRI. Dynamic contrast-enhanced MRI can provide non-invasive and objective evidence for perioperative management and treatment strategies in rectal cancer patients with acute radiation-induced rectal injury. • To our knowledge, this study is the first to evaluate the predictive value of contrast-enhanced MRI (DCE-MRI) quantitative parameters for severe acute radiation-induced rectal injury (RRI) in patients with rectal cancer. • Forward volume transfer constant (Ktrans), derived from DCE-MRI, exhibited moderate diagnostic performance (AUC = 0.72) in predicting severe acute RRI of rectal cancer, with a sensitivity of 80% and specificity of 54%. • DCE-MRI is a promising imaging marker for distinguishing the severity of acute RRI in patients with rectal cancer.

Stability of Hardy Littlewood Sobolev inequality under bubbling

In this note we will generalize the results deduced in Figalli and Glaudo (Arch Ration Mech Anal 237(1):201–258, 2020) and Deng et al. (Sharp quantitative estimates of Struwe’s Decomposition. Preprint http://arxiv.org/abs/2103.15360, 2021) to fractional Sobolev spaces. In particular we will show that for sin (0,1), n>2s and nu in mathbb {N} there exists constants delta = delta (n,s,nu )>0 and C=C(n,s,nu )>0 such that for any function uin dot{H}^s(mathbb {R}^n) satisfying,u-∑i=1νU~iH˙s≤δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\| u-\\sum _{i=1}^{\ u } \ ilde{U}_{i}\\right\\| _{\\dot{H}^s} \\le \\delta \\end{aligned}$$\\end{document}where tilde{U}_{1}, tilde{U}_{2},ldots tilde{U}_{nu } is a delta -interacting family of Talenti bubbles, there exists a family of Talenti bubbles U_{1}, U_{2},ldots U_{nu } such that u-∑i=1νUiH˙s≤CΓif2s<n<6s,Γ|logΓ|12ifn=6s,Γp2ifn>6s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\| u-\\sum _{i=1}^{\ u } U_{i}\\right\\| _{\\dot{H}^s} \\le C\\left\\{ \\begin{array}{ll} \\Gamma &{} \ ext{ if } 2s< n < 6s,\\\\ \\Gamma |\\log \\Gamma |^{\\frac{1}{2}} &{} \ ext{ if } n=6s, \\\\ \\Gamma ^{\\frac{p}{2}} &{} \ ext{ if } n > 6s \\end{array}\\right. \\end{aligned}$$\\end{document}for Gamma =left| Delta u+u|u|^{p-1}right| _{H^{-s}} and p=2^*-1=frac{n+2s}{n-2s}.

Continuity of attractors for singularly perturbed semilinear problems with nonlinear boundary conditions and large diffusion

We exhibit singularly perturbed parabolic problems with large diffusion and nonhomogeneous boundary conditions for which the asymptotic behavior can be described by a one-dimensional ordinary differential equation. We estimate the continuity of attractors in Hausdorff’s metric by the rate of convergence of resolvent operators. Moreover, we will show explicitly how this estimate of continuity varies exponentially with the fractional power spaces Xα for α in an appropriate interval.

A general fractal structure approach to the study of the thermodynamic properties of liquids and its applications

The calculation of liquid heat capacity in its general form is one of the most challenging subjects in condensed matter physics due to the dynamical disorder of liquids, in contrast to the solid phase, and the strong interactions involved, unlike the gas phase. Despite these difficulties, a phonon theory of liquids has been proposed, although the discussion has been limited to a Debye-type density of states. In the present paper, we adopt a new approach: rather than focusing on the Debye model, we extend the Debye-type density of states to fractal spaces and obtain an expression for the liquid heat capacity based on a fractal model. We develop a new method for calculating the heat capacity of liquids using the fractal concept of substances. To verify the proposed approach, formulas are derived for the temperature dependence of the heat capacity of liquid Hg for various fractal dimensions. Our approach can be applied to both the quantum and classical regimes and can be reduced to the phonon theory of liquids and solids in the limiting cases.

Global existence and convergence results for a class of nonlinear time fractional diffusion equation

This paper investigates Cauchy problems of nonlinear parabolic equation with a Caputo fractional derivative. When the initial datum is sufficiently small in some appropriate spaces, we demonstrate the existence in global time and uniqueness of a mild solution in fractional Sobolev spaces using some novel techniques. Under some suitable assumptions on the initial datum, we show that the mild solution of the time fractional parabolic equation converges to the mild solution of the classical problem when . Under some appropriate assumptions on the initial datum, we show that the mild solution of the time fractional diffusion equation converges to the mild solution of the classical problem when . Our theoretical results can be widely applied to many different equations such as the Hamilton–Jacobi equation, the Navier–Stokes equation in two cases: the fractional derivative and the classical derivative. Our paper also provides a completely new answer to the related open problem of convergence of solutions to fractional diffusion equations as the order of fractional derivative approaches 1−.

Fractal Operators and Fractional-Order Mechanics of Bone

In recent years, the concept of physical fractal space has been abstracted from muscle/ligament fibers, nerve fibers and blood flows. In the physical fractal space, bio-fractal mechanics may be set up. Surprisingly, the concepts and methods developed in the above bio-fractal mechanics are of universalities, i.e., the physical fractal spaces, fractal operators and fractional-order mechanics universally exist in various biological systems such as bones. This paper will focus on the bones in which the physical fractal space is abstracted, the fractal operators are derived and the fractional-order mechanics is established.

EM wave propagation within plasma-filled rectangular waveguide using fractional space and LFD

This paper aims to investigate the propagation of the electromagnetic (EM) within the rectangular waveguide that is filled with plasma. A rapid Cherenkov free electron laser ( C-FEL) beam was injected into the plasma to excite its natural oscillations and, therefore, an EM wave was generated. We focused on TM-mode propagation through this waveguide. Exact solutions of the EM wave equation have been found using both the Laplacian operator in the fractional D-dimensional space and the local fractional derivative (LFD). The fractional solutions have been converted into classical results to simulate the usual behavior of the waves. It has been found that the well-known Bessel, Neumann, and Mittage–Leffler functions are observed and their propagation is directly proportional to fractional parameters.

Weighted fractional Sobolev spaces as interpolation spaces in bounded domains

AbstractWe characterize the real interpolation space between a weighted space and a weighted Sobolev space in arbitrary bounded domains in , with weights that are positive powers of the distance to the boundary.