Novel efficient collocation method for Sturm–Liouville problems with nonlocal integral boundary conditions

A time delay exists between driver input and vehicle braking state response during the working process of the anti-lock braking system (ABS), and the braking performance of vehicles will be further reduced due to the delay of controllers. This paper investigates a systematic method of stability analysis for time delay ABS, and the analysis focuses on the stability and critical delay algorithm of ABS with delay time. Firstly, the dynamic structure and modelling process of ABS are briefly introduced, and PD control algorithm is adopted to improve the control performance. Then, dynamic models of ABS with time delay are derived, and the full delay stability interval and critical time delay algorithm of ABS are deduced by using the generalized Sturm criterion method. Finally, the validity of the critical delay algorithm by the proposed method and the stability and accuracy of ABS with time delay, different road conditions, vehicle speeds and control parameters are illustrated by numerical simulations, and the results show that the critical time delay algorithm of ABS can be verified under different conditions.

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“…The generalised Sturm method 20,21 indicates that the discriminant factor d 6 should be larger than zero, which can be shown as…”

Section: Time-delay Stability Analysis Of Absmentioning

confidence: 99%

Stability analysis of the anti-lock braking system with time delay

Gao

Wang

et al. 2021

Proceedings of the Institution of Mechanical Engineers, Part I:

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“…Bernstein-collocation method (BCM) is one of the familiar techniques to solve linear and nonlinear problems, which was studied by many researchers, for example, El-Gamel and Abd El-Hady, 44,45 Isler and Dascioglu, 46 Yüzbasi, 47 and others. [48][49][50][51][52] B-spline functions have important properties.…”

Section: Introductionmentioning

confidence: 99%

On the solution of MHD Jeffery–Hamel problem involving flow between two nonparallel plates with a blood flow application

El‐Shenawy,

El‐Gamel,

Abd El‐Hady

2024

Heat Trans

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The Jeffery–Hamel flow phenomenon appears in a variety of real‐world applications involving the flow of two nonparallel plates. BY using a similarity transformation derived from the equation of continuity, partial differential equations determining flow characteristics are translated into nonlinear ordinary differential equations. The problem involves the flow of a specific type of fluid, namely, an incompressible and electrically conducting fluid, between two nonparallel plates. The flow is assumed to be steady, two‐dimensional, and subject to certain boundary conditions. Specifically, the plates are impermeable, and the fluid adheres to a no‐slip condition, resulting in zero fluid velocity at the plates' surfaces. Moreover, the problem incorporates the effects of magnetic fields and pressure fluctuations, making it highly applicable to scenarios, such as blood flow through arteries in the human body, which can be modeled as a special case of the magnetohydrodynamic (MHD) Jeffery–Hamel problem referred to as the (MHD) blood pressure equation. This work compares two numerical approaches for solving the MHDs Jeffery–Hamel problem: B‐spline and Bernstein polynomial collocation. The given approaches are used to discretize and transform the equation into a system of algebraic equations. Matrix algebra techniques are then used to solve the resultant system. A complete error analysis and convergence rates for different grid sizes are derived for both methods and are used to compare the accuracy and efficiency of the two approaches. Both approaches produce correct solutions, according to the numerical findings, although the Bernstein polynomial collocation method is more efficient and accurate than the B‐spline collocation.

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Jacobi polynomials and the numerical solution of ray tracing through the crystalline lens

Abd El-Hady,

El-shenawy

2024

Opt Quant Electron

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The human eye is a fascinating optical system, with the crystalline lens playing a significant role in focusing light onto the retina of the eye. The ray tracing through the crystalline lens problem is a challenging problem in optics. In this paper, the case of a non-homogeneous optical medium is investigated, and the ray equation is numerically solved to get the ray paths. The governing equation is an ODE with a fractional derivative given in the Caputo sense. A novel numerical scheme is based on the Jacobi polynomial collocation technique to tackle this problem. A fast and accurate Broyden’s Quasi-Newton algorithm is applied to solve the nonlinear system of equations obtained from the collocation process. Numerical results are stated in detail to show the efficiency of our technique and are compared with other analytical and numerical methods using tables and illustrated figures, which will be useful to corroborate the clinical and physical data. Ray tracing through the crystalline lens is not only fascinating from a scientific perspective but also has practical implications across various domains, and the proposed scheme is considered a promising and practically reliable method to address such types of applications.

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Novel efficient collocation method for Sturm–Liouville problems with nonlocal integral boundary conditions

Cited by 8 publications

References 38 publications

Stability analysis of the anti-lock braking system with time delay

Stability analysis of the anti-lock braking system with time delay

On the solution of MHD Jeffery–Hamel problem involving flow between two nonparallel plates with a blood flow application

Jacobi polynomials and the numerical solution of ray tracing through the crystalline lens

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