This study develops a generalized analytical framework for Space-Time Fractional Shallow Water Equations (ST-FSWE) to characterize the combined influence of memory effects and non-local dynamics in open-channel hydrodynamics. Formulated in the Caputo sense, the coupled nonlinear system yields a convergent recursive algebraic scheme via the Dual-Fractional Differential Transform Method (DF-DTM). The proposed framework is examined using three benchmark configurations: the Fractional Parabolic Bowl, the Fractional Sine Wave, and the Fractional Gaussian Hill. For the Parabolic Bowl, the model recovers the classical integer-order behavior in the limiting case α = β = 1. Across the three benchmark problems, the numerical investigations demonstrate that the temporal fractional order governs the memory-dependent evolution of the analytical flow solutions, whereas the spatial fractional order modifies their non-local spatial behavior. Furthermore, the Gaussian Hill benchmark reveals that the finite-order analytical approximation exhibits truncation-order dependence during the later stages of the evolution owing to the cumulative contribution of the retained higher-order correction terms. Overall, the proposed formulation provides a mathematically consistent analytical framework for investigating memory-dependent and non-local shallow water flow behavior while preserving consistency with the classical shallow water equations in the integer-order limit.
This paper addresses the fixed initial offset problem of iterative learning control (ILC) for phase-shift full-bridge series-resonant inverters (PSFB-SRIs). Considering the specific characteristics of PSFB-SRIs, a novel PD-type learning algorithm with forgetting factor is proposed. It is shown that the algorithm guarantees the output tracking errors are bounded in the presence of state disturbances, and asymptotic convergence to zero in their absence. Furthermore, an initial rectifying mechanism is introduced to eliminate the effect of fixed initial offsets. When the learning algorithm with the initial rectifying term is applied, the output tracking errors can tend to zero within a pre-specified interval under disturbance-free conditions. In the end, a simulation example is given to demonstrate the effectiveness of the proposed method.
Dendrimers and their derivatives, renowned for their unique molecular shapes and distinctive physicochemical properties, have been a focal point in interdisciplinary research for their applications. The Sombor index of a graph G, denoted as SO(G), is calculated by summing the square roots of d 2 (u) + d 2 (v) over the edges uv of the graph G, where d(u) is the degree of vertex u. This paper presents a novel approach to computing the Sombor index (SO) for a specific class of dendrimers by introducing a recursive formula for chain graphs. Our study not only resolves the Sombor index for chain graphs but also provides a foundational methodology that can be applied to other dendrimer types and chains of fullerenes. This work contributes to the understanding of topological indices in molecular modeling by offering an efficient recursive formula and demonstrating its broader applicability. The derived formula enhances the computational approach for degree-based indices and provides new insights into the structural properties of dendrimers and fullerene chains.
We examine the effects of viscoelasticity and shear-dependent viscosity in fluid flow over an exponentially stretching surface. The induced magnetic field is included due to high magnetic Reynolds numbers, which lead to significant interactions between the fluid velocity and the magnetic field. Activation energy and variable thermal conductivity are incorporated to assess their impact on heat transfer and fluid motion. Additionally, thermal slip and concentration effects are considered to provide a more comprehensive analysis of the boundary layer behavior. We develop the governing differential equations by applying the boundary layer approximation, reduce the system to coupled nonlinear partial differential equations. The resulting mathematical model is rendered dimensionless using suitable transformations. A numerical approach utilizing the bvp4c solver technique is carried out to obtain numerical computations of the resulting coupled ordinary differential equations. The effects of the relevant physical parameters are presented in both tabular and graphical formats. The Maxwell fluid factor increases the fluid elasticity and resistance to flow changes, resulting in lower fluid velocity within the boundary layer. As the Sutterby parameter increases, the effective viscosity becomes more prominent at lower shear rates, especially near the stretching surface. This leads to greater resistance to fluid movement and hinders the momentum transport from the surface into the fluid. Consequently, the fluid velocity decreases throughout the boundary layer. The insights obtained from present analysis are useful for thermal management systems and energy efficiency applications.
High-precision trajectory tracking for robotic manipulators is critical in complex industrial applications but remains challenging due to compound disturbances like nonlinear friction and model uncertainties. This paper proposes a novel composite control strategy that deeply integrates a nonlinear disturbance observer with a backstepping control framework, creating a unified solution with stability guarantees. Unlike conventional methods that separate disturbance rejection from stability control, our approach actively embeds real-time disturbance estimates into the backstepping controller’s design via dynamic feedforward compensation, enabling integrated observation and compensation. The main contributions are threefold: (1) establishing a high-fidelity dynamic model using Denavit-Hartenberg parameters and cross-product operators; (2) designing a composite controller that synergizes disturbance estimation with stability control; and (3) rigorously proving global asymptotic stability via Lyapunov theory while providing explicit parameter selection guidelines. Comprehensive numerical simulations demonstrate superior trajectory tracking accuracy and robustness compared to conventional methods, with faster error convergence and significantly reduced control chattering. This work provides both a theoretically rigorous control scheme for disturbed robotic systems and an extensible design framework for broader nonlinear control applications.
