Search Results (271)

This work investigates a sophisticated class of neutral fuzzy fractional functional differential equations (N3FDEs), where the fractional order $\alpha$ satisfies $0<\alpha \le 1$. We present a comprehensive analysis of the existence, uniqueness, and well-posedness of solutions under the generalized Hukuhara framework. First, we examine the existence and uniqueness of solutions under the generalized Hukuhara framework, providing an refined iterative formula for linear systems. We further verify the system’s well-posedness, proving that solutions remain stable and respond continuously to changes in initial data and parameters. Second, we introduce a novel spectral Vieta–Lucas projection method to approximate the solution. By leveraging the unique properties of Vieta–Lucas polynomials, we transform complex memory-dependent fuzzy equations into a streamlined algebraic system. Finally, numerical examples and error analysis show the method is accurate and efficient. Full article

This study presents results on the well-posedness, Ulam–Hyers stability, and $\mathrm{m}$th moment averaging principle for the Itô–Doob fractional stochastic system within the framework of $\eta$-Caputo fractional derivatives. We demonstrate well-posedness using the fixed-point approach. A generalized Grönwall inequality is employed to establish sufficient conditions for Ulam–Hyers stability. Furthermore, we establish the averaging principle that facilitates obtaining a simplified averaged system from the original complex, multiple time-scale system. Finally, numerical simulations using the Euler–Maruyama method are provided to support the theoretical findings. Full article

In this paper, we investigate the well-posedness and stability of a class of coupled systems of deformable fractional differential equations in Banach spaces. The deformable fractional derivative, which interpolates continuously between a function and its classical derivative through a single scalar parameter, provides a flexible and tractable framework for modeling complex dynamical phenomena with memory effects. By employing Perov’s fixed-point theorem under matrix contractive conditions, we establish the existence and uniqueness of solutions for the considered coupled system. The existence of at least one solution under broader growth conditions is then proved via the nonlinear alternative of Leray–Schauder type. Furthermore, the continuous dependence of solutions on initial data is rigorously established, confirming the well-posedness of the system. Hyers–Ulam stability and generalized Hyers–Ulam–Rassias stability results are also derived, providing quantitative estimates relevant to numerical approximation and applied analysis. Three illustrative examples are presented to demonstrate the applicability and effectiveness of the theoretical results. Full article

We propose a reformulated notion of Levitin–Polyak (abbreviated as LP) well posedness for fuzzy optimization problems formulated in the fuzzy order-preserving (FOP) setting, where minimizing sequences are governed by a total ordering defined on fuzzy intervals. Under this formulation, we present verifiable sufficient conditions that guarantee LP well-posed behavior. These conditions are derived using ranking mechanisms that maintain interval order relations and ensure solution comparability. One central contribution is an equivalence-based theoretical characterization of LP well posedness obtained through an examination of the topological properties of the approximate solution mapping, particularly its closed-graph structure and upper semicontinuity. In addition, convergence of approximating solution sequences is investigated under the upper Hausdorff metric, leading to stability results for the associated solution sets. The established criteria provide a comprehensive framework for analyzing the convergence performance of algorithms designed for fuzzy optimization environments. Full article

Eddy-current inspection of anisotropic composites, such as aeronautical CFRP, demands models that ensure mathematical rigor, tensorial consistency, and clear energetic interpretation. This work presents a novel unified variational framework with a multiplicative tensor perturbation for the time-harmonic eddy-current problem in anisotropic media with defective regions. The formulation is posed in the natural spaces $H(\mathrm{curl};\Omega )\times H^1\left({\Omega }_c\right)$, and the well-posedness is established via the Lax–Milgram theorem under physically consistent assumptions on permeability and conductivity. The sesquilinear form admits a Hermitian decomposition that separates dissipative and reactive contributions, revealing the energetic structure of the weak formulation. Defects are modeled through multiplicative modifications of the baseline anisotropic conductivity tensor. This congruence-based approach preserves symmetry and positive definiteness, ensuring non-negative Joule losses and structural stability, allowing a modular representation of subsurface delamination, fiber breakage, conductive inclusions, and distributed porosity within a single tensorial framework. A central result of the present formulation is the reconstruction of the complex power functional from the evaluation of the weak form at the solution, showing that the active dissipated power and the magnetic reactive power arise directly from the same integral terms. Through the complex Poynting theorem, the quadratic form is linked to the internal complex power, establishing a direct connection between the variational formulation and measurable quantities such as probe impedance variations. Simulations of realistic layered CFRP configurations, including single- and multi-defect scenarios, confirm that, compared with additive perturbations, the multiplicative model provides enhanced energetic contrast, particularly in strongly anisotropic and interacting defect conditions. Agreement with experimental measurements, supported by a quantitative comparison of dissipated power variations obtained from controlled EC experiments, corroborates the physical relevance and robustness of the proposed complex power functional. Full article

This paper presents a new perspective on the energy decay of a nonlocal truncated Timoshenko–Ehrenfest system. By using the non local elasticity theory, this model is a generalization of the standard truncated Timoshenko system. The well-posedness of the proposed model is established via the Faedo-Galerkin method. Energy stability and decay properties are then derived using suitable multiplier techniques. Finally, a numerical scheme is constructed, and the exponential decay of the discrete energy is given special attention. Numerical simulations are provided to illustrate and validate the theoretical results. Full article

This paper presents unified counterexamples for which standard elliptic regularity results break down for linear elliptic equations with highly singular coefficients in dimensions $d\ge 3$. First, we establish well-posedness for the case where the drift vector field has merely $L^2$-integrability but can be expressed as the gradient of a bounded potential function. Subsequently, we investigate the critical endpoint cases of known regularity results where coefficients or data satisfy borderline integrability conditions. By using a single, explicit function, $\rho \left(x\right)=ln(2+{\displaystyle \frac{1}{\parallel x\parallel }})$, we present counterexamples to the regularity of solutions for divergence form equations and stationary Fokker–Planck equations. Full article

We develop a local, patchwise geometric framework that embeds a broad class of potential Hamiltonian dynamical systems into a family of Riemannian Hamilton patches built over an underlying Gutzwiller manifold. We adopt a conformal (Jacobi) ansatz and a frame-adapted reconstruction procedure, through which we construct, on each patch, a pulled-back metric, along with a reduced (truncated) connection (not a metric-compatible connection) and a corresponding dynamical curvature tensor governing geodesic deviation in the Hamilton coordinates. Then, using the Poisson–Hodge reconstruction, we reconstruct coordinate potentials, enforcing harmonic obstructions, and along with exactness and Jacobian nondegeneracy conditions, we obtain explicit elliptic bounds that control the connection and curvature residuals. On the basis of this construction, we formalize the notion of a Hamilton manifold such that reparametrized geodesics approximate Newton trajectories with controlled acceleration and tolerances. As a generalized structural framework, to promote the local Jacobi reconstructions to a coherent dynamical evolution and provide a dynamical closure, we introduce a patchwise hyperbolic geometric flow for the pullback metric coupled to a kinetic (Vlasov) closure that controls reconstruction and curvature residuals. Under natural regularity, ellipticity, and overlap-tolerance assumptions, together with precise estimates that control the reconstruction and curvature errors, we establish short-time well-posedness of the coupled Vlasov–hyperbolic geometric flow that defines the patchwise Hamilton manifold. Motivated by this construction of the Hamilton manifold with atlas-dependent time, we propose convergence and stability conjectures for dissipative and conservative (non-dissipative) hyperbolic geometric flows. On a single patch, these conjectures characterize local orbital stability (in the sense of coercivity modulo symmetry) and identify local linear instability when unstable linear modes are present. On a finite atlas (the Hamilton manifold with atlas-dependent time), we state conjectures under which local stability propagates to global stability, provided that overlap residuals remain uniformly sufficiently small. The framework identifies the geometric origin of local instability diagnostics used in Hamiltonian mechanics and outlines a practical strategy for verifying stability or instability, numerically or analytically, on finite coverings of configuration space (the Hamilton manifold). Full article

Despite its importance in modeling subdiffusion in fractal and heterogeneous media, a rigorous and computational scheme for solving the fractional diffusion equation on generalized comb structures over unbounded domains has remained elusive, mainly due to the nonlocal memory effect and slow spatial decay of solutions. To the best of our knowledge, we address this long-standing gap by presenting a fully integrated framework that simultaneously resolves both challenges. We derive the governing equation from constitutive relations and establish exact absorbing boundary conditions (ABCs) for the multi-skeleton comb model, a result absent in prior work. A transparent Dirichlet-to-Neumann (DtN) map, constructed via Laplace analysis, rigorously handles skeletal Dirac delta singularities and eliminates spurious reflections without empirical parameters. Furthermore, we propose a novel structure-preserving finite difference scheme that applies the sum-of-exponentials (SOE) approximation not only to the interior Caputo derivative but also to the convolution kernels arising from the ABCs. This yields a dramatic reduction in computational complexity, from quadratic $O\left(N_t^2\right)$ to quasi-linear $O(N_{t}log{N}_t)$, while preserving the physics of anomalous transport. We prove the well-posedness, unconditional stability, and convergence of the method. Numerical results confirm theoretical error estimates and show excellent agreement between simulated particle distributions, mean square displacement profiles, and exact asymptotics, validating both accuracy and robustness. The speedup (CPU time ratio Direct/Fast) is about $1.00\times$–$1.23\times$ for $N_t=5000$ in our tests. Our approach sets a new benchmark for simulating anomalous dynamics in fractal-inspired media. Full article

In this work, we address the nonhomogeneous initial-boundary value problem for the coupled Hirota equation posed on the finite interval $[0,L]$. To investigate the well-posedness of this problem, we first adopt an appropriate transformation, namely the Laplace transform, which is tailored to the specific characteristics of the problem, and further obtain an explicit solution formula for the linear inhomogeneous coupled system. Subsequently, the local well-posedness of the original nonhomogeneous initial-boundary value problem in $X_{s,T}\times X_{s,T}\left(X_{s,T}=C(0,T;H^s(0,1))\cap L^2(0,T;H^{s+1}(0,1))\right)$ is rigorously established through the combination of this explicit formula, the contraction mapping principle and energy estimates. Full article

This article deals with the global existence and uniqueness of solutions to a fractional-order SIQR epidemic model, alongside its intricate chaotic and complex dynamics as functions of the fractional order. The well-posedness of the model solutions, including global existence, uniqueness, and positivity, is established by constructing appropriate Lyapunov functions. The local and global stability analyses are conducted for both the disease-free and endemic equilibria of the model. An asymptotic solution of the system in the form of series is derived by the Laplace–Adomian decomposition method (L–ADM), and its convergence is rigorously proved. Subsequently, numerical analysis determines and interprets the optimal truncation order of this asymptotic solution. Numerical simulations are performed based on the asymptotic solution, and the dynamics and chaos of the dynamic system with respect to the fractional order are analyzed and illustrated in terms of the maximum Lyapunov exponent and structural complexity. Finally, a local sensitivity analysis is conducted for each state variable with respect to the model parameters. Full article

This paper develops a functional operator-theoretic framework for nonlinear Erdelyi–Kober (EK) fractional dynamical systems formulated in Banach spaces endowed with the compact-open topology. Within this setting, sufficient conditions for existence, uniqueness, and Ulam–Hyers stability of solutions are established using the Banach and Schaefer fixed-point theorems. The continuity, boundedness, and Lipschitz properties of the associated nonlinear operators are analyzed to ensure well-posedness of the fractional system. As a constructive complement to the theoretical results, a power series iterative method (PSIM) is employed to obtain an explicit fractional series representation of the solution in the case $0<\alpha <1$. The applicability of the theoretical framework is illustrated through a nonlinear fractional dynamical Belousov–Zhabotinsky system (DBZS), where the assumptions of the main theorems are verified and the solution is constructed via the proposed series scheme. The results provide a coherent link between abstract fixed-point analysis and a constructive semi-analytical representation of solutions for EK fractional systems. Full article

Electrical impedance tomography (EIT) provides noninvasive, high-temporal-resolution imaging for medical and industrial applications. However, accurate image reconstruction remains challenging due to the severe ill-posedness and nonlinearity of the inverse problem, as well as the limited robustness of existing single-source learning-based methods in real measurement scenarios. To address these limitations, a data-constrained and physics-guided Multi-Source Conditional Diffusion Model (MS-CDM) is proposed for EIT image reconstruction. Unlike conventional conditional diffusion methods that rely on a single measurement or an image prior, MS-CDM utilizes boundary voltage measurements as data-driven constraints and incorporates coarse reconstructions as physics-guided structural priors. This multi-source conditioning strategy provides complementary guidance during the reverse diffusion process, enabling balanced recovery of fine boundary details and global topological consistency. To support this framework, a Hybrid Swin–Mamba Denoising U-Net is developed, combining hierarchical window-based self-attention for local spatial modeling with bidirectional state-space modeling for efficient global dependency capture. Extensive experiments on simulated datasets and three real EIT experimental platforms demonstrate that MS-CDM consistently outperforms state-of-the-art numerical, supervised, and diffusion-based methods in terms of reconstruction accuracy, structural consistency, and noise robustness. Moreover, the proposed model exhibits robust cross-system applicability without system-specific retraining under multi-protocol training, highlighting its practical applicability in diverse real-world EIT scenarios. Full article

In this study, we investigate the upper- and lower-bound approximations of numerical eigenvalues derived by weak Galerkin spectral element methods on arbitrary convex quadrilateral meshes for the Laplace eigenvalue problem. Firstly, the Piola transformation is employed to construct the approximation space for weak gradients on each convex quadrilateral element, while a one-to-one mapping is used to establish the approximation space for weak functions. Subsequently, based on the weak Galerkin spectral element approximation space defined on convex quadrilateral meshes, a Galerkin approximation scheme is formulated, and its well-posedness is then analyzed. Furthermore, numerical experiments are performed on arbitrary convex quadrilateral meshes of the square and L-shaped domains to explore the upper- and lower-bound approximations of numerical eigenvalues. Numerical findings indicate that the presented method not only obtains optimal orders of convergence with respect to both the mesh size and the polynomial degree, but also provides upper- and lower-bound approximations for the reference eigenvalues by proper choices of polynomial degrees in approximation spaces and parameters of the approximation scheme in both h-version and p-version weak Galerkin spectral element methods. This study offers new perspectives and methodologies for the high-precision numerical solution of eigenvalue problems in elliptic equations. Full article

This paper presents a novel mathematical framework for anthrax transmission by integrating Caputo fractional derivatives, distributed delays, and a Beddington–DeAngelis incidence function. The proposed model captures memory effects in disease progression, temporal heterogeneities in pathogen release, and saturation phenomena in host–pathogen interactions. We establish the well-posedness of the system and derive the basic reproduction number ${\mathcal{R}}_0$, which serves as a sharp threshold for disease dynamics: when ${\mathcal{R}}_0\le 1$, the disease-free equilibrium is globally asymptotically stable; when ${\mathcal{R}}_0>1$, a unique endemic equilibrium emerges and is globally stable. Theoretical analysis demonstrates that the fractional order modulates convergence rates through memory effects, while distributed delays influence oscillatory behaviors and time to equilibrium. Numerical simulations validate these findings and illustrate the impacts of key parameters on disease transmission. The results provide a scientific foundation for designing targeted public health interventions in anthrax control. Full article