Search Results (111)

This work introduces a new category of proportional fractional integro-differential equations (PFIDEs) governed by functional boundary conditions. We derive verifiable sufficient criteria that guarantee the Ulam–Hyers Stability, existence and uniqueness of solutions to this problem. Our analytical approach leverages Babenko’s method to construct an inverse operator, which allows us to reformulate the differential problem into an equivalent integral equation. The analysis is then conducted using key mathematical tools, including contraction mapping principle of Banach, the Leray–Schauder alternative, and properties of multivariate Mittag–Leffler functions. The Ulam–Hyers Stability is rigorously examined to assess the system’s resilience to small perturbations. The applicability and effectiveness of the established theoretical results are demonstrated through two illustrative examples. This research provides a unified and adaptable framework that advances the analysis of complex fractional-order dynamical systems subject to nonlocal constraints. Full article

In this paper, we investigate a new class of nonlinear fractional boundary value problems (BVPs) involving $(k,\psi )$-Caputo fractional derivative operators subject to multipoint closed boundary conditions. Such a formulation of boundary data generalizes classical closure constraints in terms of nonlocal dependence of the unknown function at several interior points, giving rise to a flexible mechanism for describing physical and engineering phenomena governed by nonlocal and memory effects. The proposed problem is first transformed into an equivalent fixed-point formulation, enabling the application of standard analytical tools. Results concerning the existence and uniqueness of solutions to the problem are obtained through the application of fixed-point principles, specifically those of Banach, Krasnosel’skiĭ, and the Leray–Schauder nonlinear alternative. The obtained results extend and generalize several known findings. Illustrative examples are presented to demonstrate the applicability of the theoretical findings. Moreover, the introduction incorporates a succinct review of boundary value problems associated with fractional differential equations and inclusions subject to closed boundary conditions. Full article

This paper aims to explore the numerical solution of non-linear fractional-order Burgers’-Huxley equation based on Caputo’s formulation of fractional derivatives. The equation serves as a versatile tool for analyzing a wide range of physical, biological, and engineering systems, facilitating valuable insights into nonlinear dynamic phenomena. The fractional operator provides a comprehensive mathematical framework that effectively captures the non-locality, hereditary characteristics, and memory effects of various complex systems. The approximation of temporal differential operator is carried out through finite difference based L1 scheme, while spatial discretization is performed using modified cubic B-spline basis functions. The stability as well as convergence analysis of the approach are also presented. Additionally, some numerical test experiments are conducted to evaluate the computational efficiency of a modified fourth-order cubic B-spline (M43BS) approach. Finally, the results presented in the form of tables and graphs highlight the applicability and robustness of M43BS technique in solving fractional-order differential equations. The proposed methodology is preferred for its flexible nature, high accuracy, ease of implementation and the fact that it does not require unnecessary integration of weight functions, unlike other numerical methods such as Galerkin and spectral methods. Full article

This paper investigates the uniform continuity and strong continuity of the semigroups of the fractional integral operators of power functions. Using the Krasnoselskii’s fixed-point theorem, we have studied the non-local problem related to fractional differential equations involving power functions with multi-point integral boundary conditions and obtain the existence of the solution. Full article

A nonlocal transport–reaction system is proposed to model the coupled dynamics of stem and differentiated cell populations, structured by a continuous damage variable. The framework incorporates bidirectional transitions via differentiation and dedifferentiation, with nonlocal birth operators encoding damage redistribution upon division and Hill-type feedback regulation dependent on total populations. Global well-posedness of solutions in $C([0,\infty );L^1([0,\infty )\times L^1\left([0,\infty )\right))$ is established by combining the contraction mapping principle for local existence with a priori $L^1$ bounds for global existence, ensuring uniqueness and nonnegativity. Integration yields balance laws for total populations, reducing to a finite-dimensional autonomous ordinary differential equation (ODE) system under constant death rates. Linearization reveals a bifurcation threshold separating extinction, homeostasis, and unbounded growth. Under compensatory feedback, Dulac’s criterion precludes periodic orbits, and the Poincaré–Bendixson theorem confines bounded trajectories to equilibria or heteroclinics. Uniqueness implies global asymptotic stability. A scaling invariance for steady states under uniform feedback rescaling is identified. The analysis extends structured population theory to feedback-regulated compartments with nonlocal operators and reversible dedifferentiation, providing explicit stability criteria and linking an infinite-dimensional structured model to tractable low-dimensional reductions. Full article

This paper aims to discuss the positive mild solutions for nonlocal fractional variable exponent differential equations with concave and convex coefficients. Based on a specifically defined order cone, even under the influence of the $p(t)$-Laplacian operator and the fractional integral operator, we avoid making many assumptions on the nonlocal coefficient $\mathcal{A}$ and just require that $\mathcal{A}>0$ on a set of positive measures. Utilizing the fixed-point index theory on cones, some new results on the existence and multiplicity of positive mild solutions were obtained, which extend and enrich some previous research findings. Finally, numerical examples are used to verify the feasibility of our main results. Full article

We introduce the generalized fractional-order Dirac delta distribution ${\delta }_{\mathrm{GFODDF}}$, defined by applying the generalized fractional derivative (GFD) operator to the Heaviside function. This construction extends the classical Dirac delta to non-integer orders, allowing modeling of systems with memory and non-local effects. We establish fundamental properties—including shifting, scaling, evenness, derivative, and convolution—within a rigorous distributional framework and present explicit proofs. Applications are demonstrated by solving linear fractional differential equations and by modeling drug release with fractional kinetics, where the new delta captures impulse responses with long-term memory. Numerical illustrations confirm that ${\delta }_{\mathrm{GFODDF}}$ reduces to the classical delta when $\eta =1$, while providing additional flexibility for $0<\eta <1$. These results show that ${\delta }_{\mathrm{GFODDF}}$ is a powerful tool for fractional-order analysis in mathematics, physics, and biomedical engineering. Full article

This study presents an innovative numerical method for solving linear fractional differential equations (LFDEs) using modified Bernstein polynomial bases. The proposed approach effectively addresses the challenges posed by the nonlocal nature of fractional derivatives, providing a robust framework for handling multiple initial and boundary value constraints. By integrating the LFDEs and approximating the solutions with modified fractional-order Bernstein polynomials, we derive operational matrices to solve the resulting system numerically. The method’s accuracy is validated through several examples, showing excellent agreement between numerical and exact solutions. Comparative analysis with existing data further confirms the reliability of the approach, with absolute errors ranging from 10⁻¹⁸ to 10⁻⁴. The results highlight the method’s efficiency and versatility in modeling complex systems governed by fractional dynamics. This work offers a computationally efficient and accurate tool for fractional calculus applications in science and engineering, helping to bridge existing gaps in numerical techniques. Full article

This paper undertakes a rigorous analytical exposition of the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions, incorporating the Atangana–Baleanu fractional derivative in the Caputo configuration under the influence of Wiener process and Poissonian discontinuities. The system’s analytical landscape is further enriched by the incorporation of Clarke sub-differentials, facilitating the treatment of nonsmooth, nonconvex, and multivalued dynamics. The inherent complexity arising from the confluence of fractional memory, stochastic perturbations, and impulsive phenomena necessitates the deployment of a sophisticated apparatus from variational analysis, measurable selection theory, and multivalued fixed point frameworks within infinite-dimensional Banach spaces. This study delineates rigorous sufficient conditions, ensuring controllability under such hybrid influences, thereby generalizing classical paradigms to encompass nonlocal and discontinuous dynamical regimes. A precisely articulated exemplar is included to validate the theoretical constructs and demonstrate the operational efficacy of the proposed analytical methodology. Full article

As a critical technology of autonomous mining trucks, object detection directly determines system safety and operational reliability. However, autonomous mining trucks often work in dusty open-pit environments, in which dusty interference significantly degrades the accuracy of object detection. To overcome the problem mentioned above, a multi-branch feature interaction and location detection network (MBFILNet) is proposed in this study, consisting of multi-branch feature interaction with differential operation (MBFI-DO) and depthwise separable convolution-enhanced non-local attention (DSC-NLA). On one hand, MBFI-DO not only strengthens the extraction of channel-wise semantic features but also improves the representation of salient features of images with dusty interference. On the other hand, DSC-NLA is used to capture long-range spatial dependencies to focus on target-object structural information. Furthermore, a custom dataset called Dusty Open-pit Mining (DOM) is constructed, which is augmented using a cycle-consistent generative adversarial network (CycleGAN). Finally, a large number of experiments based on DOM are conducted to evaluate the performance of MBFILNet in dusty open-pit environments. The results show that MBFILNet achieves a mean Average Precision (mAP) of 72.0% based on the DOM dataset, representing a 1.3% increase compared to the Featenhancer model. Moreover, in comparison with YOLOv8, there is an astounding 2% increase in the mAP based on MBFILNet, demonstrating detection accuracy in dusty open-pit environments can be effectively improved with the method proposed in this paper. Full article

Exploring the dynamics of nonlinear nanofluidic flow-induced vibrations, this work focuses on single-walled branched carbon nanotubes (SWCNTs) operating in a thermal–magnetic environment. Carbon nanotubes (CNTs), renowned for their exceptional strength, conductivity, and flexibility, are modeled using Euler–Bernoulli beam theory alongside Eringen’s nonlocal elasticity to capture nanoscale effects for varying downstream angles. The intricate interactions between nanofluids and SWCNTs are analyzed using the Differential Transform Method (DTM) and validated through ANSYS simulations, where modal analysis reveals the vibrational characteristics of various geometries. To enhance predictive accuracy and system stability, machine learning algorithms, including XGBoost, CATBoost, Random Forest, and Artificial Neural Networks, are employed, offering a robust comparison for optimizing vibrational and thermo-magnetic performance. Key parameters such as nanotube geometry, magnetic flux density, and fluid flow dynamics are identified as critical to minimizing vibrational noise and improving structural stability. These insights advance applications in energy harvesting, biomedical devices like artificial muscles and nanosensors, and nanoscale fluid control systems. Overall, the study demonstrates the significant advantages of integrating machine learning with physics-based simulations for next-generation nanotechnology solutions. Full article

The Monge–Ampère operator, as a nonlinear operator embedded in parabolic differential equations, complicates the demonstration of maximal regularity for these equations. This research uses the Riesz fractional derivative to connect the Monge–Ampère operator with the fractional Laplacian operator. It is then possible to seek the maximal regularity of the parabolic Monge–Ampère equations by following an approach similar to that used for finding the maximal regularity of the parabolic fractional Laplacian operator. The maximal regularity of nonlocal parabolic Monge–Ampère equations guarantees the existence of solutions in the whole space. Based on these conditions, a modified sliding method, an enhancement of the moving planes method, is employed to establish the monotonicity property of the solutions for the nonlocal parabolic Monge–Ampère equations in the whole space. Full article

This paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional derivative operators, and $(k_2,{\psi }_2)$- Riemann–Liouville fractional integral operators. The problem considered in the present study is of a more general nature as the $(k_1,{\psi }_1)$-Hilfer fractional derivative operator specializes to several other fractional derivative operators by fixing the values of the function ${\psi }_1$ and the parameter $\beta$. Also the $(k_2,{\psi }_2)$-Riemann–Liouville fractional integral operator appearing in the multistrip boundary conditions is a generalized form of the ${\psi }_2$-Riemann–Liouville, $k_2$-Riemann–Liouville, and the usual Riemann–Liouville fractional integral operators (see the details in the paragraph after the formulation of the problem. Our study includes both convex and non-convex valued maps. In the upper semicontinuous case, we prove four existence results with the aid of the Leray–Schauder nonlinear alternative for multivalued maps, Mertelli’s fixed-point theorem, the nonlinear alternative for contractive maps, and Krasnoselskii’s multivalued fixed-point theorem when the multivalued map is convex-valued and $L^1$-Carathéodory. The lower semicontinuous case is discussed by making use of the nonlinear alternative of the Leray–Schauder type for single-valued maps together with Bressan and Colombo’s selection theorem for lower semicontinuous maps with decomposable values. Our final result for the Lipschitz case relies on the Covitz–Nadler fixed-point theorem for contractive multivalued maps. Examples are offered for illustrating the results presented in this study. Full article

This work examines a singular elliptic problem with a fractional and a non-local integrodifferential operator. The question of whether solutions exist is transformed into the existence of critical points of the associated functional energy, to be more specific. The existence of a critical point is then demonstrated by combining the variational method with some monotonicity arguments. After this, due to the singular non-linearity, we manually demonstrate that this critical point is a weak solution for such a problem. Full article

Water pollution is a significant threat for human health, particularly in developed countries. This study advances the mathematical understanding of WP transmission dynamics by developing a fractional–fractal derivative framework with non-singular kernels and the Mittage–Leffler function, which successfully preserves the non-local behavior of pollutants. The fractional–fractal derivatives in sense of the Atangana–Baleanu–Caputo formulation inherently captures the non-local and memory-dependent behavior of pollutant diffusion, addressing limitations of classical differential operators. A novel parameter, $\gamma$, is introduced to represent the recovery rate of water systems through treatment processes, explicitly modeling the bridge between natural purification mechanisms and engineered remediation efforts. Furthermore, this study establishes stability analysis, and the existence and uniqueness of the solution are established through fixed-point theory to ensure the mathematical stability of the system. Moreover, a numerical scheme based on the Newton polynomial is formulated, by obtaining significant simulations of pollution dynamics under various conditions. Graphical results show the effect of important parameters on pollutant evolution, providing useful information about the behavior of the system. Full article