Fractal and Fractional

In this work, we propose a new physics-informed neural network framework based on the method of separation of variables (SVPINN) to solve the distributed-order time-fractional advection–diffusion equation. We develop a new method for calculating the distributed-order derivative, which enables the fractional integral to be modeled by a network and directly solved by combining automatic differentiation technology. In this way, the approximation of the distributed-order derivative is integrated into the parameter training system of the network, and the data-driven adaptive learning mechanism is used to replace the numerical discretization scheme. In the SVPINN framework, we decompose the kernel function of the Caputo integral into three independent functions using the method of separation of variables, and apply a neural network as a surrogate model for the modified integral and the function related to the time variable. The new physical constraint generated by the modified integral serves as an extra supervised learning task for the network. We systematically evaluated the feasibility of the SVPINN on several numerical experiments and demonstrated its performance.

The family of stable distributions and, in particular, the $\alpha$-stable distribution increases its applicability in engineering sciences. Examination of industrial data shows that originally assumed Gaussian properties are not so often observed. Research shows that stable functions can cover much wider spectrum of cases. However, the estimations of $\alpha$-stable distribution factors may pose some limitations. One of the control engineering aspects, i.e., the assessment of controller performance, may be successfully addressed by L-moments and L-moment ratio diagrams (LMRD). Simultaneously, LMRDs are often used as a method for distribution, fitting with the method of moments (MOM). Unfortunately, the moments do not exist for $\alpha$-stable distribution. This research shows that, with the use of a Monte-Carlo analysis, this limitation may be overcome, and an efficient method to estimate statistical factors of the $\alpha$-stable distribution is proposed.

Fractional differential equations offer a natural framework for describing systems in which present states are influenced by the past. This work presents a nonlinear Caputo-type fractional differential equation (FDE) with a nonlocal initial condition and attempts to describe a model of memory-dependent behavioral adaptation. The proposed framework uses a fractional-order derivative to discuss the long-term memory effects. The existence and uniqueness of solutions are demonstrated by Banach’s and Krasnoselskii’s fixed-point theorems. Stability is analyzed through Ulam–Hyers and Ulam–Hyers–Rassias benchmarks, supported by sensitivity results on the kernel structure and fractional order. The model is further employed for behavioral despair and learned helplessness, capturing the role of delayed stimulus feedback in shaping cognitive adaptation. Numerical simulations based on the convolution-based fractional linear multistep (FVI–CQ) and Adams–Bashforth–Moulton (ABM) schemes confirm convergence and accuracy. The proposed setup provides a compact computational and mathematical paradigm for analyzing systems characterized by nonlocal feedback and persistent memory.