This study aims to address the topic of finite-time synchronization within a specific subset of fractional-order Degn–Harrison reaction–diffusion systems. To achieve this goal, we begin with the introduction of a novel lemma specific for finite-time stability analysis. Diverging from existing criteria, this lemma represents a significant extension of prior findings, laying the groundwork for subsequent investigations. Building upon this foundation, we proceed to develop efficient dependent linear controllers designed to orchestrate finite-time synchronization. Leveraging the power of a Lyapunov function, we derive new, robust conditions that ensure the attainment of synchronization within a predefined time frame. This innovative approach not only enhances our understanding of finite-time synchronization, but also offers practical solutions for its realization in complex systems. To validate the efficacy and applicability of our proposed methodology, extensive numerical simulations are conducted. Through this comprehensive analysis, we aim to contribute valuable insights to the field of fractional-order reaction–diffusion systems while paving the way for practical implementations in real-world applications.

In the context of reaching the best way to control the movement of autonomous cars linearly and angularly, making them more stable and balanced on different roads and ensuring that they avoid road obstacles, this manuscript chiefly aims to reach the optimal approach for a fractional-order PID controller (or $P{I}^{\gamma }{D}^{\rho }$-controller) instead of the already classical one used to provide smooth automatic parking for electrical autonomous cars. The fractional-order $P{I}^{\gamma }{D}^{\rho }$-controller is based on the particle swarm optimization (PSO) algorithm for its design, with two different approximations: Oustaloup’s approximation and the continued fractional expansion (CFE) approximation. Our approaches to the fractional-order PID using the results of the PSO algorithm are compared with the classical PID that was designed using the results of the Cohen–Coon, Ziegler–Nichols and bacteria foraging algorithms. The scheme represented by the proposed $P{I}^{\gamma }{D}^{\rho }$-controller can provide the system of the autonomous vehicle with more stable results than that of the PID controller.

In this article, we introduce an analytical method for solving both non-separable linear and non-linear partial differential equations, for which separation of variables method does not work. This method is based on the theory of tensor product in Banach spaces coupled with some properties of atoms operators. We provide some illustrative examples. For more information see https://ejde.math.txstate.edu/Volumes/2024/28/abstr.html

This work purposes to establish a novel analytical method to obtain the atomic solutions of partial differential equations. This will be carried out by employing the tensor product theory in Banach spaces coupled with some properties of atoms operators. Some illustrative examples will be provided to validate our findings.

In this paper, we discuss the atomic solutions of the abstract Cauchy problem in two variables. The proposed abstract Cauchy problem is assumed to be non-homogeneous and in tensor product form. We utilize some results from the theory of tensor product of Banach spaces to obtain solutions in the atom operator form. The approach is applied for both ordinary and α-conformable fractional derivatives.

A general Ostrowski’s type inequality for double integrals is given. We utilize function whose partial derivative of order four exists and is bounded.