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.

This paper proposes a novel numerical approach for handling fractional boundary value problems. Such an approach is established on the basis of two numerical formulas; the fractional central formula for approximating the Caputo differentiator of order $\alpha$ and the fractional central formula for approximating the Caputo differentiator of order $2\alpha$, where $0<\alpha \le 1$. The first formula is recalled here, whereas the second one is derived based on the generalized Taylor theorem. The stability of the proposed approach is investigated in view of some formulated results. In addition, several numerical examples are included to illustrate the efficiency and applicability of our approach.

A quantum mechanical system that mimics the behavior of a classical harmonic oscillator in the quantum domain is called a simple harmonic quantum oscillator. The time-independent Schrödinger equation describes the quantum harmonic oscillator, and its eigenstates are quantized energy values that correspond to various energy levels. In this work, we first fractionalize the time-independent Schrödinger equation, and then we solve the generated problem with the use of the Adomian decomposition approach. It has been shown that fractional quantum harmonic oscillators can be handled effectively using the proposed approach, and their behavior can then be better understood. The effectiveness of the method is validated by a number of numerical comparisons.

Many studies have employed a variety of techniques to further investigate topological space, particularly the notion of bitopological spaces, due to the significance of topological space in data processing as well as certain implementations. Numerous extended topological structures have been laid out subsequently. Of those abstractions, functions in topology was one of which was most noteworthy. In order to assist in this trend, we focused our research on the idea of open and closed sets, which is one of the strongest techniques available to present scientists for the study of computer graphics and digital topology.New functions, pairwise ω−closed functions, which are strictly weaker than pairwise closed functions, will be introduced in this study. By applying the P˝−space definition, whose is a P−space modification. Additionally, we establish different projection and product theories pertaining to pairwise Lindelöf of and pairwise paracompact spaces utilizing P˝−spaces. We analyze images and inverse images that have been chosen topological attributes for every one of these functions. In the final analysis, we explore several counterexamples that correspond to the offered definitions and theorems.

The Columbia space shuttle catastrophe in 2003 served as the inspiration for this paper’s improved mathematical model, which includes a nonlinear damping Neumann boundary condition. By creating and examining a modified heat equation with piecewise nonlinear source terms and damping Neumann boundary conditions, the study seeks to investigate the incident’s heat transport dynamics. To ensure that the problem is well-posed, we provide strong mathematical arguments for the existence of solutions both locally and globally. In addition, we use numerical simulations to show how the nonlinear boundary conditions affect heat dissipation and to confirm the theoretical results. The findings advance our knowledge of thermal modeling in aircraft applications and offer greater insights into heat propagation under such conditions.

This study introduces a novel higher-order implicit correction method derived from the Euler-Maclaurin formula to enhance the approximation of initial value problems. The proposed method surpasses the Runge-Kutta approach in accuracy, stability, and convergence. An error bound is established to demonstrate its theoretical reliability. To validate its effectiveness, numerical experiments are conducted, showcasing its superior performance compared to conventional methods. The results consistently confirm that the proposed method outperforms the Runge-Kutta method across various practical applications.