Search Results (816)

This study investigates a modified five-dimensional chaotic system by incorporating structural term adjustments and Caputo fractional-order differential operators. The modified system exhibits significantly enriched dynamic behaviors, including offset boosting, phase trajectory rotation, phase trajectory reversal, and contraction phenomena. Additionally, the system exhibits bidirectional transitions—conservative-to-dissipative transitions governed by initial conditions and dissipative-to-conservative transitions controlled by fractional order variations—along with a unique chaotic-to-quasiperiodic transition observed exclusively at low fractional orders. To validate the system’s physical realizability, a signal processing platform based on Digital Signal Processing (DSP) is implemented. Experimental measurements closely align with numerical simulations, confirming the system’s feasibility for practical applications. Full article

The primary objective of this work is to develop a comprehensive theory of weighted fractional Sobolev spaces within the framework of timescales. To this end, we first introduce a novel class of weighted fractional operators and rigorously define associated weighted integrable spaces on timescales, generalising classical notions to this non-uniform temporal domain. Building upon these foundations, we systematically investigate the fundamental functional-analytic properties of the resulting Sobolev spaces. Specifically, we establish their completeness under appropriate norms, prove reflexivity under appropriate duality pairings, and demonstrate separability under mild conditions on the weight functions. As a pivotal application of our theoretical framework, we derive two robust existence theorems for solutions to the proposed model. These results not only extend classical partial differential equation theory to timescales but also provide a versatile tool for analysing dynamic systems with heterogeneous temporal domains. Full article

It is a well-known fact that the Dzhrbashyan–Nersesyan fractional derivative includes as particular cases the fractional derivatives of Riemann–Liouville, Gerasimov–Caputo, and Hilfer. The notion of resolving a family of operators for a linear equation with the Dzhrbashyan–Nersesyan fractional derivative is introduced here. Hille–Yosida-type theorem on necessary and sufficient conditions of the existence of a strongly continuous resolving family of operators is proved using Phillips-type approximations. The conditions concern the location of the resolvent set and estimates for the resolvent of a linear closed operator A at the unknown function in the equation. The existence of a resolving family means the existence of a solution for the equation under consideration. For such equation with an operator A satisfying Hille–Yosida-type conditions the uniqueness of a solution is shown also. The obtained results are illustrated by an example for an equation of the considered form in a Banach space of sequences. It is shown that such a problem in a space of sequences is equivalent to some initial boundary value problems for partial differential equations. Thus, this paper obtains key results that make it possible to determine the properties of the initial value problem involving the Dzhrbashyan–Nersesyan derivative by examining the properties of the operator in the equation; the results prove the existence and uniqueness of the solution and the correctness of the problem. Full article

In this paper, we first introduce a parametric identity for generalized differentiable functions using a generalized fractal–fractional integral operators. Based on this identity, we establish several variants of parameterized inequalities for functions whose local fractional derivatives in absolute value satisfy generalized convexity conditions. Furthermore, we demonstrate that our main results reduce to well-known Ostrowski- and Simpson-type inequalities by selecting suitable parameters. These inequalities contribute to finding tight bounds for various integrals over fractal spaces. By comparing the classical Hölder and Power mean inequalities with their new generalized versions, we show that the improved forms yield sharper and more refined upper bounds. In particular, we illustrate that the generalizations of Hölder and Power mean inequalities provide better results when applied to fractal integrals, with their tighter bounds supported by graphical representations. Finally, a series of applications are discussed, including generalized special means, generalized probability density functions and generalized quadrature formulas, which highlight the practical significance of the proposed results in fractal analysis. Full article

This study introduces a paradigm shift in permanent magnet synchronous motor (PMSM) control through the development of hybrid dual fractional-order sliding mode control (HDFOSMC) architecture integrated with evolutionary parameter learning (EPL). Conventional PMSM control frameworks face critical limitations in ultra-precision applications due to their inability to reconcile dynamic agility with steady-state precision under time-varying parameters and compound disturbances. The proposed HDFOSMC framework addresses these challenges via two synergistic innovations: (1) a dual fractional-order sliding manifold that fuses the rapid transient response of non-integer-order differentiation with the small steady-state error capability of dual-integral compensation, and (2) an EPL mechanism enabling real-time adaptation to thermal drift, load mutations, and unmodeled nonlinearities. Validation can be obtained through the comparison of the results on PMSM testbenches, which demonstrate superior performance over traditional fractional-order sliding mode control (FOSMC). By integrating fractional-order theory, sliding mode control theory, and parameter self-tuning theory, this study proposes a novel control framework for PMSM. The developed system achieves high-precision performance under extreme operational uncertainties through this innovative theoretical synthesis and comparative results. Full article

We develop novel extensions in the theory of weighted Lorentz spaces. In particular, we generalize classical results by introducing variable-exponent Lorentz spaces, establish sharp constants and quantitative bounds for maximal operators, and extend the framework to encompass fractional maximal operators. Moreover, we analyze endpoint cases through the study of oscillation operators and reveal new connections with weighted Hardy spaces. These results provide a unifying approach that not only refines existing inequalities but also opens new avenues in harmonic analysis and partial differential equations. Full article

Background/Objectives: The aim of this study was to determine whether extracellular volume (ECV) fraction as determined by contrast-enhanced computed tomography (CECT) can help distinguish between autoimmune pancreatitis (AIP) and pancreatic ductal adenocarcinoma (PDAC). Methods: Participants comprised 101 patients, including 20 diagnosed with AIP (AIP group), 42 with histologically confirmed PDAC (PDAC group), and 39 without pancreatic disease (healthy group). Contrast enhancement (CE) was calculated as CT attenuation in Hounsfield units [HU] on equilibrium-phase CECT–CT attenuation on pre-contrast CT. The ECV fraction was calculated by measuring the region of interest within the pancreatic region and aorta on pre-contrast and equilibrium-phase CECT. CT measurements were compared among groups. CE and ECV fractions were also compared for diffuse (n = 12) and focal or segmental types (n = 8). Focal- or segmental-type AIP was defined as the involvement of one or two pancreas segments. Diagnostic efficacy was evaluated through receiver operating characteristic (ROC) analyses. Results: CE and ECV fractions differed significantly between the groups (p < 0.001 each). CE was significantly higher in the AIP group (56.8 ± 7.9 HU) than in the PDAC group (42.3 ± 17.0 HU, p < 0.001) or healthy group (32.2 ± 6.1 HU, p < 0.001). ECV fraction was significantly higher in the AIP group (47.2 ± 7.3%) than in the PDAC group (31.7 ± 12.0%, p < 0.001) or healthy group (27.5 ± 5.4%, p < 0.001). In the AIP group, no significant differences in CE (56.7 ± 8.2 HU vs. 56.9 ± 8.1 HU; p = 1.000) or ECV fraction (48.0 ± 5.6% vs. 46.6 ± 8.4%; p = 0.970) were seen between diffuse and focal or segmental types. Areas under the ROC curve for differentiating AIP from PDAC were 0.78 for CE and 0.86 for ECV fraction, showing no significant difference (p = 0.083). Conclusions: ECV fraction might be clinically useful in differentiating AIP from PDAC. Full article

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 operators has not been addressed in any existing functional setting. We establish their boundedness under appropriate log-Hölder continuity and growth conditions on the exponent function $\mathtt{p}(·)$. To highlight the novelty and practical relevance of the proposed operator, we conduct a comparative analysis demonstrating its effectiveness in addressing convergence, regularity, and stability of solutions to partial differential equations. We also provide non-trivial examples that illustrate not only these properties but also show that, under this operator, a broader class of functions becomes locally integrable. The exponential decay factor notably broadens the domain of boundedness compared to classical Riesz and Bessel–Riesz potentials, making the operator more versatile and robust. Additionally, we generalize earlier results on Sobolev-type inequalities previously studied in constant exponent spaces by extending them to the variable exponent setting through our fractional operator, which reduces to the classical Riesz potential when the decay parameter $\lambda =0$. Applications to elliptic PDEs are provided to illustrate the functional impact of our results. Furthermore, we develop several new structural properties tailored to variable exponent frameworks, reinforcing the strength and applicability of the proposed theory. Full article

This study presents an image inpainting model based on an energy functional that incorporates the $L^p$ norm of the fractional Laplacian operator as a regularization term and the $H^{-1}$ norm as a fidelity term. Using the properties of the fractional Laplacian operator, the $L^p$ norm is employed with an adjustable parameter p to enhance the operator’s ability to restore fine details in various types of images. The replacement of the conventional $L^2$ norm with the $H^{-1}$ norm enables better preservation of global structures in denoising and restoration tasks. This paper introduces a diffusion partial differential equation by adding an intermediate term and provides a theoretical proof of the existence and uniqueness of its solution in Sobolev spaces. Furthermore, it demonstrates that the solution converges to the minimizer of the energy functional as time approaches infinity. Numerical experiments that compare the proposed method with traditional and deep learning models validate its effectiveness in image inpainting tasks. Full article

Most solutions of fractional differential equations (FDEs) that model real-world phenomena in various fields of science, industry, and engineering are complex and cannot be solved analytically. This paper mainly aims to present some useful results for studying the qualitative properties of solutions of FDEs involving the new generalized Hattaf mixed (GHM) fractional derivative, which encompasses many types of fractional operators with both singular and non-singular kernels. In addition, this study also aims to unify and generalize existing results under a broader operator. Furthermore, the obtained results are applied to some linear systems arising from medicine. Full article

Integer-order models cannot characterize the dynamic behavior of the flexible two-link manipulator (FTLM) system accurately due to its viscoelastic characteristics and flexible oscillation. Hence, this paper proposes a fractional-order modeling method and identification algorithm for the FTLM system. Firstly, we exploit the memory and history-dependent properties of fractional calculus to describe the flexible link’s viscoelastic potential energy and viscous friction. Secondly, we establish a fractional-order differential equation for the flexible link based on the fractional-order Euler–Lagrange equation to characterize the flexible oscillation process accurately. Accordingly, we derive the fractional-order model of the FTLM system by analyzing the motor–link coupling as well as the symmetry of the system structure. Additionally, a system identification algorithm based on the multi-innovation integration operational matrix (MIOM) is proposed. The multi-innovation technique is combined with the least-squares algorithm to solve the operational matrix and achieve accurate system identification. Finally, experiments based on actual data are conducted to verify the effectiveness of the proposed modeling method and identification algorithm. The results show that the MIOM algorithm can improve system identification accuracy and that the fractional-order model can describe the dynamic behavior of the FTLM system more accurately than the integer-order model. Full article

This paper presents a novel generalization of the mth-order Laguerre and Laguerre-based Appell polynomials and examines their fundamental properties. By establishing quasi-monomiality, we derive key results, including recurrence relations, multiplicative and derivative operators, and the associated differential equation. Additionally, both series and determinant representations are provided for this new class of polynomials. Within this framework, several subpolynomial families are introduced and analyzed including the generalized mth-order Laguerre–Hermite Appell polynomials. Furthermore, the generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials are defined using fractional operators and we investigate their structural characteristics. New families are also constructed, such as the mth-order Laguerre–Gould–Hopper–based Bernoulli, Laguerre–Gould–Hopper–based Euler, and Laguerre–Gould–Hopper–based Genocchi polynomials, exploring their operational and algebraic properties. The results contribute to the broader theory of special functions and have potential applications in mathematical physics and the theory of differential equations. Full article

This article concerns a novel coupled implicit differential system under $\phi$–Riemann–Liouville ($\mathbb{RL}$) fractional derivatives with $\mathfrak{p}$-Laplacian operator and multi-point strip boundary conditions on unbounded domains. An applicable Banach space is introduced to define solutions on unbounded domains $[c,\infty )$. The explicit iterative solution’s existence and uniqueness ($\mathbb{E}a\mathbb{U}$) are established by employing the Banach fixed point strategy. The different types of Ulam–Hyers–Rassias ($\mathbb{UHR}$) stabilities are investigated. Ultimately, we provide a numerical application of a coupled $\phi$-$\mathbb{RL}$ fractional turbulent flow model to illustrate and test the effectiveness of our outcomes. Full article

This paper examines a family of operators that combine the features of fractional-order and classical operators. Our goal is to obtain results on their invertibility in function spaces, based on their inherent improving properties. The class of proportional operators we study is extensive and includes both fractional-order and classical operators. This leads to interesting function spaces in which we obtain the right- and left-handed properties of invertibility. Thus, we extend and unify results concerning fractional-order and proportional operators. To confirm the relevance of our results, we have supplemented the paper with a series of results on the equivalence of differential and integral forms for various problems, including terminal value problems. Full article

Grey models have attracted considerable attention as a time series forecasting tool in recent years. Nevertheless, the linear characteristics of the differential equations on which traditional grey models rely frequently result in inadequate predictive accuracy and applicability when addressing intricate nonlinear systems. This study introduces a conformable fractional order unbiased kernel-regularized nonhomogeneous grey model (CFUKRNGM) based on statistical learning theory to address these limitations. The proposed model initially uses a conformable fractional-order accumulation operator to derive distribution information from historical data. A novel regularization problem is then formulated, thereby eliminating the bias term from the kernel-regularized nonhomogeneous grey model (KRNGM). The parameter estimation of the CFUKRNGM model requires solving a linear equation with a lower order than the KRNGM model, and is automatically calibrated through the Bayesian optimization algorithm. Experimental results show that the CFUKRNGM model achieves superior prediction accuracy and greater generalization performance compared to both the KRNGM and traditional grey models. Full article