Search Results (179)

In this paper, based on the Schauder fixed point theorem, the (generalized) Darbo fixed point theorem, and the (generalized) Banach contraction mapping principle, we study the mixed-type fractional impulse evolution equation with non-local and delay terms, and obtain the existence and uniqueness theorems under whether the operator is compact or not. The order of the derivative in this paper is $0<\alpha <1$, this fractional order introduces a series of problems concerning compactness, continuity, and convergence. We overcome these problems using methods such as H$\ddot{o}$lder inequality and Minkowski inequality. Moreover, under the condition of the non-compact measure, the non-negative constant is extended to an unbounded Lebesgue-integrable function. In addition, when obtaining the uniqueness of the solution through the (generalized) Banach contraction mapping principle, the non-negative constant L in the Lipschitz condition is extended to an unbounded Lebesgue integrable function. Finally, a case study is conducted to demonstrate the validity of the theoretical results. Full article

In this paper, we introduce a new class of contractions, called dual connected-image contractions, for a pair of self-mappings on a metric space endowed with a directed graph. This concept extends the notion of connected-image contractions by incorporating the interaction between two mappings through the graph structure. By employing a class of auxiliary functions, we establish existence and uniqueness results for common fixed points under this generalized contractive framework. The proposed approach not only unifies and extends several known results in the literature but also provides greater flexibility in handling nonlinear problems. As an application, the theoretical results are applied to a class of nonlinear fractional differential equations with nonlocal integral boundary conditions, where the existence of solutions is established by reformulating the problems as equivalent integral equations and applying the developed common fixed point framework. Full article

Fractional differential operators provide an effective approach for modeling complex technological processes, particularly physical phenomena in continuum mechanics characterized by memory and non-local effects. Different types of fractional derivatives require different numerical approximation schemes; in this study, the Caputo and Grünwald–Letnikov derivatives are considered. The aim of this work was to develop and validate a fractional differential model of longitudinal oscillations in a suspended monorail system that accounts for nonlinear and memory-dependent effects. In contrast to classical integer-order approaches, the proposed framework incorporates multiscale surface irregularity effects, including rail roughness, friction, and other disturbances influencing system dynamics, through a fractional-order formulation. A fractional differential mathematical model describing the motion of longitudinal oscillations of a large-sized cargo transported along a suspended monorail is proposed. A numerical algorithm based on finite-difference approximation of fractional operators was developed for its implementation. The scientific contribution lies in integrating multiscale surface irregularity effects into a fractional-order modeling framework to improve the accuracy of dynamic response prediction. Numerical experiments demonstrated the effectiveness of the approach, and the results were validated through comparison with existing models of monorail dynamics. Additionally, statistical validation based on correlation analysis confirmed good agreement with the experimental data. The proposed model can be applied to the design and optimization of suspended transport systems, improving vibration control, reliability, and operational safety under real dynamic loading conditions. Full article

This paper investigates the optimal harvesting of a nonlinear, size-structured population governed by a first-order transport equation with nonlocal environmental crowding feedback and exogenous inflow. First, we establish finite-horizon well-posedness for the controlled state system in an $L^1$ framework, proving the existence, uniqueness, positivity, and continuous dependence of weak solutions. Second, we show that the infinite-dimensional stationary problem reduces exactly to a scalar nonlinear closure equation, yielding existence and conditional uniqueness results for stationary states. Within this equilibrium framework, we distinguish the persistence of the forced system from intrinsic demographic self-replacement and introduce size-continuous per-recruit and spawning-potential diagnostics. Finally, we formulate a partial differential equation (PDE)-constrained optimal harvesting problem. Under a compactness assumption on the control-to-state map, we establish the existence of optimal controls. We then formally derive a Pontryagin-type first-order optimality system for the harvesting problem. The variation of the nonlocal environmental feedback produces a coupled integral source term in the adjoint equation. The associated pointwise maximization condition yields a bang–bang harvesting structure, while a monotone size-threshold policy is shown to require an additional single-crossing assumption on the switching function. These hypotheses are illustrated using a fisheries model with density-dependent von Bertalanffy growth. Full article

The inverse problem under consideration concerns a one-dimensional parabolic equation whose thermal diffusivity takes the quadratic-in-space form $a_s\left(\tau \right){\kappa }^2+b_s\left(\tau \right)\kappa +c_s\left(\tau \right)$. The unknowns are three time-dependent coefficients $a_s\left(\tau \right),b_s\left(\tau \right),c_s\left(\tau \right)$ together with the temperature field $T(\kappa ,\tau )$. The direct problem supplies initial data, Neumann boundary conditions, and three over-determination conditions: two boundary temperatures and the spatial integral of T. We prove two theorems. The first theorem establishes the local-in-time existence of a solution under explicit regularity and sign conditions on the given data $\xi ,{\nu }_k,{\delta }_{\ell },\theta$ and compatibility at $\tau =0$. The second theorem guarantees the uniqueness of this solution. Despite uniqueness, the inverse reconstruction remains ill-posed: small perturbations in the over-specified data can cause large deviations in the recovered coefficients. For the forward model, we implement two numerical schemes: (i) a Crank–Nicolson finite difference methodology (CN-FDM) on a uniform grid and (ii) a semi-discretized Crank–Nicolson approach combined with Bell spectral collocation in space (CN–Bell). The inverse step minimizes a Tikhonov-regularized least-squares functional using MATLAB’s (R2026a) lsqnonlin. Two numerical examples (smooth and non-smooth), tested with both exact synthetic data and artificially added noise, demonstrate stable and accurate coefficient reconstructions. The framework applies directly to heat conduction and porous media flow where diffusivity varies quadratically in space. Full article

Fractional-order dynamic systems, due to their long memory and nonlocality, have significant advantages in describing the dynamic behavior of complex engineering systems. However, existing identification methods often struggle to balance modeling accuracy and model structural complexity under conditions of strong nonlinearity, strong coupling, and multiple operating conditions. To address the challenge of modeling complex fractional-order systems with strong nonlinearity, strong coupling, and multiple operating conditions, this paper proposes a fuzzy multi-model modeling and identification method based on the decomposition-synthesis approach. First, a fractional-order fuzzy multi-model structure is constructed to characterize the dynamic characteristics of such complex systems. Second, an improved SKFCM hybrid clustering algorithm is proposed, combining K-means clustering and satisfactory fuzzy C-means clustering. This optimizes the cluster center selection strategy and overcomes the shortcomings of traditional satisfactory FCM algorithms, such as random initial membership and unreasonable cluster center selection, thus achieving a reasonable determination of the number of local models. Finally, the least-squares and Levenberg–Marquardt algorithms are integrated to interactively identify local model parameters, system fractional-order, and fuzzy scheduling function parameters, solving key difficulties such as unknown fractional-order s in antecedent variables, numerous parameter couplings, and difficulty in determining the antecedent space. Through academic examples and simulations of a robotic arm system, the proposed method effectively achieves high-precision modeling of complex fractional-order systems, demonstrating strong feasibility and superiority. Full article

This paper advances a comprehensive controllability framework for Prabhakar fractional differential systems ($\mathcal{PFDS}$s) of order $\eta \in (1,2)$ with nonlocal initial conditions, where the second-order setting requires the joint specification of both an initial state and an initial velocity. Explicit solution representations for four structurally distinct classes of second-order Prabhakar systems are derived via the Laplace transform method and Neumann series expansions, revealing that the placement of the forcing term directly in the system or under the Prabhakar fractional integral operator produces fundamentally different convolution kernels. For linear integro-differential systems, necessary and sufficient controllability conditions are established through a Gramian rank criterion with an explicit norm-bounded control law, while for nonlinear systems, sufficient conditions are obtained via the Schauder fixed-point theorem under an asymptotic growth condition. Three numerical examples validate the theory: a three-dimensional linear system and a two-dimensional nonlinear integro-differential system achieve terminal errors of order ${10}^{-12}$ and ${10}^{-7}$, respectively, and a Prabhakar fractional mass–spring–damper system with viscoelastic hereditary damping demonstrates direct physical relevance, with all theoretical conditions verified and a terminal error of $7.42\times {10}^{-5}$ confirming precise rest-position steering by the Gramian-based control law. Full article

In this paper, we develop a Legendre spectral operational matrix method for the numerical solution of two-dimensional Volterra–Fredholm integro-differential equations subject to mixed boundary conditions. The proposed approach transforms the physical domain onto a reference square and approximates the unknown solution using a tensor-product Legendre polynomial expansion. Exact operational matrices for differentiation and lower-limit integration are constructed, allowing the original integro-differential problem to be reduced systematically to a finite-dimensional algebraic system for the spectral coefficients. The formulation provides a unified treatment of differential, Volterra, and Fredholm operators within a single spectral framework and avoids complicated discretizations of multidimensional integral terms. For a specialized linear form of the problem, rigorous convergence estimates are established in both $L^2$ and $L^{\infty }$ norms under suitable regularity assumptions on the coefficients and kernels. The analysis shows that the dominant convergence behavior is governed by the differential operator, while the integral terms contribute only higher-order consistency effects. Several benchmark examples involving both linear and nonlinear two-dimensional integro-differential equations are presented to demonstrate the performance of the proposed method. Numerical results exhibit rapid spectral-type error decay as the polynomial degree increases, with the numerical errors approaching machine precision for moderate truncation orders. These results confirm the accuracy, efficiency, and reliability of the proposed Legendre spectral operational matrix framework for solving a broad class of multidimensional integro-differential equations with nonlocal operators. Full article

This work develops an analytical framework for nonlinear fractional partial differential equations that combine Kirchhoff-type terms, double-phase operators, and $\psi$-Hilfer fractional derivatives. This paper investigates two classes of problems involving variable-exponent growth conditions. The first problem analyzes general nonlinear sources and formulates the solution as a fixed point of a nonlinear operator. Precisely, by proving that the functional energy is coercive, hemicontinuous, and strictly monotone, we establish the existence and the uniqueness of weak solutions via monotone operator theory. The second problem incorporates a convection-type nonlinearity, which breaks variational structure and requires the more robust theory of pseudomonotone operators. Under suitable growth and mixed-order assumptions on the nonlinearity, we prove the existence of at least one weak solution. The main tools are grounded in variable-exponent Lebesgue and Musielak–Orlicz–Sobolev spaces, with compact embeddings, modular estimates, and fractional integral identities playing a key role in the proofs. We note that the results contribute to the mathematical modeling of phenomena involving nonlocal elasticity, viscoelastic materials, phase-transition media, and fractional dynamical systems where the stiffness of the medium depends on the total deformation (Kirchhoff effect) and the energy density alternates between distinct growth regimes (double-phase). The $\psi$-Hilfer derivative enhances the scope by enabling models with tunable memory and hereditary effects. Full article

This paper investigates a new class of mixed impulsive fractional boundary value problems (BVPs) in which the mixing occurs both in the governing fractional differential equations—through the combined presence of $\psi$-Caputo and quantum (q-difference) fractional derivatives—and in the boundary conditions formulated via fractional integral constraints. By incorporating two distinct operators within the same dynamical framework, the proposed model is capable of capturing both memory effects and discrete-scale behaviors inherent in complex hybrid systems. Using the Banach contraction mapping principle and the Leray–Schauder nonlinear alternative, sufficient conditions ensuring the existence and uniqueness of solutions are established. The theoretical results unify and extend several known fractional models. Owing to its flexible structure, the proposed framework may serve as a useful mathematical tool for modeling impulsive phenomena in systems where non-local memory and scale-transition mechanisms coexist, such as in engineering, physics, and applied sciences. Finally, numerical examples are provided to illustrate the applicability and qualitative behavior of the solutions. Full article

Thisarticle develops and analyzes a fractional-order Leslie–Gower prey–predator–parasite system incorporating two discrete delays and nonlocal spatial diffusion. The model’s central novelty lies in the simultaneous integration of three biologically realistic features that have not previously been combined: (i) fractional-order memory effects via a Caputo derivative of order $\alpha \in (0,1]$, (ii) two distinct biological delays—an infection transmission delay ${\tau }_1$ and a predator handling delay ${\tau }_2$—and (iii) nonlocal spatial dispersal modeled through fractional Laplacian operators ${(-\Delta )}^{\gamma /2}$. This triple integration enables the model to capture long-range temporal memory, delayed biological responses, and nonlocal spatial interactions simultaneously, offering insights into dynamics that are challenging to capture with classical integer-order or single-delay formulations. The fractional Laplacian generalizes classical diffusion by allowing long-range dispersal events (Lévy flights), where individuals can occasionally move over large distances with heavy-tailed step-size distributions—a phenomenon observed in many animal movement patterns but absent from standard diffusion models. We provide rigorous proofs of solution existence, uniqueness, non-negativity, and boundedness in both temporal and spatiotemporal settings. Local asymptotic stability conditions are derived for all feasible equilibrium states via characteristic equation analysis. The coexistence equilibrium undergoes a Hopf bifurcation when either delay crosses a critical threshold, with fractional order $\alpha$ modulating the bifurcation point and post-bifurcation oscillation frequency. A Lyapunov functional demonstrates global asymptotic stability of the infection-free equilibrium under biologically interpretable conditions. Turing instability analysis reveals conditions for spontaneous pattern formation, with the fractional exponent $\gamma$ controlling pattern wavelength and correlation length. Numerical simulations validate theoretical predictions, including spatial patterns, traveling waves, and chaos. To bridge theory with potential applications, we outline a statistical framework for parameter estimation and uncertainty quantification, suggesting that $\beta$, $\alpha$, and ${\tau }_1$ may be priority targets for parameter estimation. Full article

This study examines the bending of cross-ply laminated composite nanoplates coupled to a piezoelectric fiber-reinforced composite layer via the nonlocal strain gradient theory. The aim is to accurately capture size-dependent impacts and electromechanical interaction in nanoscale composite structures. The mechanical response is modeled utilizing a refined four-variable shear deformation theory, with the governing equilibrium equations developed using the virtual work assumption. The nanoplate is examined under simply supported boundary conditions exposed to both mechanical loading and applied electric voltage. A detailed parametric investigation is done to assess the contribution of non-local and strain gradient factors, imposed voltage, and geometric ratios on the bending behavior. The results show that the nonlocal parameter generates a softening result, increasing deflection, whereas the strain gradient parameter raises stiffness and minimizes deformation. Moreover, the applied voltage successfully controls the bending response by electromechanical actuation, underlining the potential of PFRC-integrated nanoplates in smart nanoscale systems. Full article

This article investigates the numerical solution of the two-dimensional Poisson equation defined over a rectangular domain subject to a double integral nonlocal boundary condition. We propose a finite difference scheme by discretizing the integral term using the two-dimensional trapezoidal rule. The main difficulty of this problem is that, in the non-classical case, we cannot use the method of separation of variables and decompose the problem into one-dimensional problems. Our approach involves reducing the integral boundary condition from the complete domain to the interior points and strategically partitioning the computational domain into the boundary and interior points. We propose a method that allows us to find a solution by solving the Poisson equation with classical boundary conditions, and using the solutions found to construct a solution to a problem with a nonlocal integral condition. This method requires solving a linear system whose dimension is much smaller than the original. Under certain conditions on the kernel, the proposed method is correct. Full article

This paper develops an operator-reduction framework for a class of coupled implicit Caputo-Hadamard fractional differential systems subject to nonlocal Hadamard integral constraints. The system, involving fractional derivatives in both state and auxiliary variables, is resolved through a pointwise contraction argument that eliminates the auxiliary components and reduces the problem to a two-dimensional fixed-point operator acting on a Banach space of continuous functions. This reduction overcomes the compactness obstruction that arises in direct multi-component formulations. Under explicit growth and smallness conditions, the existence of at least one solution is established via Mönch’s fixed-point theorem. By imposing strengthened Lipschitz hypotheses, the reduced operator becomes a strict contraction on an invariant ball, yielding uniqueness and Ulam-Hyers stability with explicit constant $C_{UH}=1/(1-\Lambda )$. A fully computed example demonstrates the verifiability of the theoretical assumptions and illustrates how the smallness condition $\Lambda <1$ governs both existence and stability. The results establish a systematic operator-based approach for implicit Caputo-Hadamard systems with nonlocal integral constraints. Full article

Tidal-scale wave modulation is a critical yet complex process in macro-tidal estuaries. This study investigates semidiurnal wave modulations in the Changjiang River Estuary (CRE) using unique, long-term in situ observations and high-resolution ADCIRC–SWAN coupled simulations. Pronounced semidiurnal signals are identified in significant wave height (H_s), mean wave period, and wave direction. Observational results demonstrate that the modulation intensity is highest in Hangzhou Bay and the CRE mouth, decreasing gradually offshore. A key finding is that semidiurnal H_s maxima systematically coincide with peak flood currents and precede high water by approximately three hours. Long-term records confirm that this modulation persists year-round and intensifies during energetic events such as typhoons. The expression of the tidal signal depends on wave composition: wind-sea-dominated conditions exhibit stronger period modulation, whereas swell-dominated conditions favor coherent H_s modulation as kinematic tidal effects remain more apparent in the absence of strong local wind forcing. Numerical sensitivity experiments demonstrate that tidal currents are the primary driver of the observed wave modulation, while water-level effects are largely confined to shallow shoals. The results highlight that accurately reproducing the observed frequency–directional structure requires the inclusion of current-induced Doppler shifts and refraction. Beyond the classical following-current effects, the analysis suggests that the spatial deceleration of currents along the wave path acts as a kinematic trap that focuses wave action and sustains H_s intensification. This mechanism provides a physically plausible explanation for the observed phase relationship and points to the non-local nature of estuarine wave dynamics, where the wave state appears as an integrated response to cumulative current gradients along the propagation path. These findings emphasize the necessity of incorporating wave–current coupling in future coastal modeling and hazard forecasting. Full article