A $ \psi $-fractal-fractional Atangana–Baleanu model for a nonlinear memristive integrator circuit: analysis and numerical approximation

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.