Although wavelet neural networks (WNNs) combine the expressive capability of neural models with multiscale localization, there are currently few theoretical guarantees for their training. We investigate the weight decay (${\mathrm{L}}_2$ regularization) optimization dynamics of gradient descent (GD) for WNNs. Using explicit rates controlled by the spectrum of the regularized Gram matrix, we first demonstrate global linear convergence to the unique ridge solution for the feature regime when wavelet atoms are fixed and only the linear head is trained. Second, for fully trainable WNNs, we demonstrate linear rates in regions satisfying a Polyak–Łojasiewicz (PL) inequality and establish convergence of GD to stationary locations under standard smoothness and boundedness of wavelet parameters; weight decay enlarges these regions by suppressing flat directions. Third, we characterize the implicit bias in the over-parameterized neural tangent kernel (NTK) regime: GD converges to the minimum reproducing kernel Hilbert space (RKHS) norm interpolant associated with the WNN kernel with ${\mathrm{L}}_2$. In addition to an assessment process on synthetic regression, denoising, and ablations across λ and stepsize, we supplement the theory with useful recommendations on initialization, stepsize schedules, and regularization scales. Together, our findings give a principled prescription for dependable training that has broad applicability to signal processing applications and shed light on when and why ${\mathrm{L}}_2$-regularized GD is stable and quick for WNNs.

In this paper, we establish a new hybrid class of special polynomials, the Gould–Hopper–Fubini-based Appell polynomials. Using the monomiality principle, we derive their generating function and explore related properties and identities. We also investigate symmetry identities and obtain a determinant representation for these polynomials. Finally, we present and discuss results for several special cases of this family and support the derived results with computational studies and visual representations.

This paper presents a Galerkin finite element method based on cubic $\beta$-spline basis functions for the numerical solution of multi-term time-fractional differential equations. The spatial discretization is constructed using a parametric family of cubic $\beta$-splines, which provide enhanced flexibility and smoothness in the approximation space. For the temporal discretization, the Caputo fractional derivatives are approximated using the $L1$ scheme combined with a $\theta$-method to improve stability and numerical accuracy for multi-term fractional models. The resulting fully discrete formulation is solved efficiently and the source terms are evaluated using Gauss–Legendre quadrature to obtain accurate load vector approximations. The convergence behavior of the proposed scheme is investigated through mesh refinement analysis and numerical accuracy is assessed using standard error norms. Several benchmark test problems are considered to validate the proposed method. The numerical results demonstrate that the cubic $\beta$-spline Galerkin approach provides accurate and stable approximations, with error norms decreasing consistently as the mesh is refined. Comparisons with available exact solutions further confirm the efficiency and reliability of the proposed numerical scheme for solving complex multi-term time-fractional differential equations.

This work develops an Atangana–Baleanu (AB)–$ \psi $-fractal-fractional (ABFF) model for a nonlinear memristive integrator circuit using an outer-derivative construction. The Atangana–Baleanu derivative in Caputo (ABC) sense and the Atangana–Baleanu derivative in Riemann (ABR) sense are distinguished explicitly throughout. The raw outer derivative is a $ \psi $-weighted ABR operator; therefore, a classical initial-value problem is posed for the regularized state $ v-v(0) $, which is exactly equivalent to a weighted ABC equation. The resulting Volterra formulation contains the operational-clock density $ \omega_{\psi, \theta}(t) = \theta[\psi(t)-\psi(0)]^{\theta-1}\psi'(t) $ in both the instantaneous and hereditary terms. We establish explicit growth and local Lipschitz estimates, a Krasnosel'skii existence result, a continuation criterion, and finite-time uniqueness and continuous-dependence estimates based on a fractional Gronwall argument. Importantly, the revised uniqueness and Ulam–Hyers estimates require contraction only of the instantaneous AB term and no longer impose a smallness condition that grows like $ T^{\gamma} $. Generalized Ulam–Hyers–Rassias stability is reformulated in a weighted norm. A two-step product-integration method is derived, and a full first-order convergence theorem is proved under a bounded endpoint density. An exact scalar Volterra benchmark confirms the predicted order. For the oscillatory circuit, terminal values and the uninformative initial-state maximum norm are replaced by post-transient amplitudes and periods. A necessary initial-compatibility condition at $ t = 0 $ is identified explicitly. All reported fractional circuit simulations use endpoint-compatible bounded clocks with $ \omega_{\psi, \theta}(0)F(U_0) = 0 $; in particular, the base clock $ \psi_b(t) = t^2/(1+t) $ is used for the fractional-order and fractal-exponent studies. This removes an otherwise hidden incompatibility between the nonsingular ABC kernel and the prescribed non-equilibrium initial state.

This paper addresses a class of multi-point initial value problems with impulses and proportional delays. The framework is based on the Atangana–Baleanu–Caputo (ABC) fractional derivative, which allows the model to incorporate hereditary memory effects absent in standard integer-order systems. Using suitable fixed-point arguments, conditions ensuring the existence and uniqueness of solutions are derived. The reliability and robustness of the obtained solutions is analyzed through the Hyers–Ulam (H-U) method and generalized H-U stability. To demonstrate the theoretical findings, a general numerical example model and a fractional-order housefly population model are considered that incorporate impulsive effects and delay terms reflecting real ecological feedback. Numerical simulations illustrate how variations in the fractional order and impulse intensity influence the dynamic behavior of the adult population. The results reveal that impulsive interventions can effectively regulate population oscillations, while the fractional order governs the rate of decay and long-term stability of the system.

This work develops a fuzzy piecewise fractional derivative (FPFD) model for cancer-immune-angiogenesis dynamics under uncertainty. Five fuzzy state variables track tumor cells, immune effectors, vessel density, oxygen, and drug concentration. We employ fuzzy triangular numbers with $\alpha$-cut interval arithmetic using constrained fuzzy arithmetic model parametric uncertainty, with numerical values. Oxygen-dependent carrying capacity follows a Hill-type function; hypoxia-induced angiogenesis follows a decreasing Michaelis–Menten function. The model transitions at $t_1=50$ days from memoryless fuzzy classical derivative to fuzzy ABC fractional derivative of order $\psi$. The transition time $t_1=50$ days is biologically justified based on experimental observations of the angiogenic switch in solid tumors, which typically occurs within 4–8 weeks post-inoculation. Positivity, boundedness, Lipschitz continuity, existence, and uniqueness of fuzzy solutions are proved via Banach fixed-point theorem in a weighted norm. A basic reproduction number interval ${\mathcal{R}}_0=[{\underline{\mathcal{R}}}_0,{\overline{\mathcal{R}}}_0]$ is derived; local and global stability conditions are established for disease-free and endemic equilibria using fuzzy differential inclusions. Global sensitivity analysis using latin hypercube sampling with $N=500$ samples explores the range of possible outcomes across the fuzzy parameter support. In the numerical implementation, we use a fourth-order fuzzy Runge–Kutta method (Phase I), and a fractional Adams–Bashforth–Moulton predictor-corrector method (Phase II), ensuring preservation of fuzzy number characteristics.