Peijun Zhang

This paper develops a unified qualitative framework for four classes of Liouville–Caputo fractional difference equations arising from different combinations of fractional sums and integer-order differences. Based on the equivalence between initial/final value problems and Volterra-type summation equations, sufficient conditions for the existence and uniqueness of solutions are established by applying the Banach contraction mapping principle together with refined combinatorial estimates. Furthermore, the continuous dependence of solutions on prescribed initial or final data is investigated. By deriving explicit error estimates through a discrete fractional Gronwall-type inequality, we prove that Lipschitz solutions depend continuously on perturbations of boundary data. Numerical experiments for a representative case are presented to verify the theoretical results, including the influence of the fractional order and the sensitivity with respect to boundary data, while additional examples illustrate the applicability of the framework. The obtained results extend the unified discrete fractional calculus framework by providing a rigorous well-posedness analysis and offering a theoretical foundation for further applications of discrete fractional models with memory effects and diverse boundary conditions.

This study investigates the primary resonance response of a composite micropipe reinforced with graphene nanoplatelets conveying fluid, with applications in remote self-powered micro-sensors, precise drug delivery, the safety of chemical reactions, ecological assessments, and the transport of cells. A sequential pseudo-nonlinear normal mode (NNM) scheme, combined with the modified Rauscher technique and harmonic balance method (HBM), is employed to derive the frequency response. The gyroscopically coupled equations are systematically reduced, at each stage, to a single degree-of-freedom equation through NNM framework, treating the system as a two-dimensional invariant manifold. The HBM generates nonlinear algebraic equations, which are solved using an arclength continuation method with an adaptive arc length. Critical to this analysis, Hill’s method is implemented to determine the stability of periodic solutions along the frequency response branches, enabling classification of solution branches as stable or unstable. This stability classification reveals that the bandwidth of multi-valued response coincides with the region of unstable periodic solutions, with stability transitions occurring at the saddle-node bifurcation points (turning points) of the frequency response curves. Parametric studies demonstrate that an increase in the slenderness ratio, radius ratio, flow velocity, and flow mass density and considering laminar flow instead of turbulent flow widen the multi-valued response bandwidth, while micro-scale contribution narrows it. A comprehensive sensitivity analysis reveals that the multi-valued response bandwidth and maximum frequency response amplitude are least sensitive to nanocomposite weight fraction, flow mass density, and micro-scale parameter, while exhibiting the highest sensitivity to slenderness ratio and radius ratio. The primary resonance characteristics show moderate sensitivity to the flow speed. These findings can be implemented for the design of remote self-powered micro-sensors, system identification, and structural health monitoring applications.

This study introduces a refined multi-objective optimization framework using the Non-dominated Sorting Genetic Algorithm II (NSGA-II). It strategically positions solid stiffening rings in spherical sandwich shells to enhance a defined set of natural frequencies. The NSGA-II employs fast non-dominated sorting, crowding distance assignment, and elitist selection to maintain solution diversity and convergence. This method efficiently explores the design space, balancing conflicting objectives to produce a well-distributed Pareto front. Optimized configurations offer potential for aerospace applications, including lightweight satellite structures and high-performance pressure vessels that demand superior modal characteristics. The sandwich shells consist of Graphene Platelet-Reinforced Nanocomposite (GPLRC) face sheets and a porous core. The model is based on First-Order Shear Deformation Theory (FSDT) and Sanders’ strain relations for deep shells, with linear elasticity for each layer. Stiffening rings are modeled as artificial circumferential springs. The 2D Ritz method with a combination of Chebyshev and trigonometric polynomials minimizes the system’s Lagrangian to derive the governing equations. Solving the eigenvalue problem provides the natural frequencies. This NSGA-II-based approach delivers a robust Pareto front of optimal ring placements, advancing frequency-based optimization in spherical sandwich shell structures. Results demonstrate that the optimized configurations achieve significant frequency enhancement, confirming the effectiveness of the proposed approach in structural vibration control.

Fractional calculus has emerged as a powerful tool for characterizing non-classical dynamic phenomena, yet its discretization remains fragmented, with existing studies primarily focusing on single combinations of variable bounds and difference directions. To address this gap, this paper proposes a unified theoretical framework for discrete fractional calculus by systematically introducing four novel Caputo fractional difference definitions, which integrate variable upper/lower-limit sums with forward/backward difference operations. First, we rigorously derive the fundamental properties of these four definitions, including the commutativity of fractional sums and differences, and their consistency with integer-order difference operations. Second, we construct fractional difference equations for each definition, establish their equivalence to Volterra sum equations, and provide explicit solutions and strict proofs for their corresponding initial and final value problems. To validate the theoretical results, we design four targeted computational cases and numerical simulations, confirm the consistency between theoretical solutions and numerical results, and intuitively demonstrate the long-memory effect of fractional-order discrete systems. Furthermore, we present a concise comparison of the four definitions, clarifying their suitability for discrete systems with distinct boundary conditions and dynamic characteristics. This work not only completes the theoretical system of discrete fractional calculus with variable bounds but also provides standardized and targeted mathematical tools for modeling complex discrete dynamic processes, laying a solid foundation for the practical application of discrete fractional calculus in fields such as engineering control, infectious disease modeling, and economic dynamics.

1. In recent years, substantial research efforts have been dedicated to study dynamic behavior of trapezoidal plates. Among these, Rahmani et al. (2024) studied free vibration behavior of cantileve...