Mathematical Modelling and Analysis

Modelling for a new mechanical passive damper and its mathematical analysis

2024-10-11T18:28:29+03:00

To improve ride comfort, an oil damper should restrain its damping force in a high-velocity range. A lot of dampers with such properties were developed. For example, the one controlling the flux of oil by a leaf valve is widely adapted and reasonable. However, it is difficult to represent its dynamics with a simple mathematical model, and the cost of a computational fluid dynamics is too expensive. To overcome the disadvantages, the first author in [15] developed the other mechanical oil damper with sub-pistons instead of the leaf valve, which enabled us to propose a simple mathematical model with linear ordinary differential equations, thanks to the simple mechanism to control the oil flow. In this article, we give a more detailed mathematical model for the damper taking the dynamic pressure resistance into account, which is represented by nonlinear ordinary differential equations. In addition, a numerical scheme for the model is also proposed and its mathematical analysis such as the validity of the numerical solutions is shown.

A class of explicit second derivative general linear methods for non-stiff ODEs

2024-10-11T18:28:29+03:00

In this paper, we construct explicit second derivative general linear methods (SGLMs) with quadratic stability and a large region of absolute stability for the numerical solution of non-stiff ODEs. The methods are constructed in two different cases: SGLMs with p = q = r = s and SGLMs with p = q and r = s = 2 in which p, q, r and s are respectively the order, stage order, the number of external stages and the number of internal stages. Examples of the methods up to order five are given. The efficiency of the constructed methods is illustrated by applying them to some well-known non-stiff problems and comparing the obtained results with those of general linear methods of the same order and stage order.

Micropolar fluid-thin elastic structure interaction: variational analysis

2024-10-11T18:28:28+03:00

We consider the non-stationary flow of a micropolar fluid in a thin channel with an impervious wall and an elastic stiff wall, motivated by applications to blood flows through arteries. We assume that the elastic wall is composed of several layers with different elastic characteristics and that the domains occupied by the two media are infinite in one direction and the problem is periodic in the same direction. We provide a complete variational analysis of the two dimensional interaction between the micropolar fluid and the stratified elastic layer. For a suitable data regularity, we prove the existence, the uniqueness and the regularity of the solution to the variational problem associated to the physical system. Increasing the data regularity, we prove that the fluid pressure is unique, we obtain additional regularity for all the unknown functions and we show that the solution to the variational problem is solution for the physical system, as well.

A new approach for Solving a nonlinear system of second-order BVPs

2024-10-11T18:28:29+03:00

In this paper, we introduce a new approach based on the Reproducing Kernel Method (RKM) for solving a nonlinear system of second-order Boundary Value Problems (BVPs) without the Gram-Schmidt orthogonalization process. What motivates us to use the RKM without the Gram-Schmidt orthogonalization process is its easy implementation, elimination of the Gram-Schmidt process, fewer calculations, and high accuracy. Finally, the compatibility of numerical results and theorems demonstrates that the Present method is effective.

The BKM criterion to the 3D double-diffusive magneto convection systems involving planar components

2024-10-11T18:28:28+03:00

In this paper, we investigate the BKM type blowup criterion applied to 3D double-diffusive magneto convection systems. Specifically, we demonstrate that a unique local strong solution does not experience blow-up at time T, given that ). To prove this, we employ the logarithmic Sobolev inequality in the Besov spaces with negative indices and a well-known commutator estimate established by Kato and Ponce. This result is the further improvement and extension of the previous works by O (2021) and Wu (2023).

Averaged reaction for nonlinear boundary conditions on a grill-type Winkler foundation

2024-11-22T18:29:09+02:00

We consider a homogenization problem for the elasticity operator posed in a bounded domain of the half-space, a part of its boundary being in contact with the plane. This surface is traction-free out of “small regions”, where we impose nonlinear Winkler-Robin boundary conditions containing “large reaction parameters”. Non-periodical distribution of these regions is allowed provided that they have the same area. We show the convergence of solutions towards those of the homogenized problems depending on the relations between the parameters distance, sizes, and reaction.

A two-derivative time integrator for the Cahn-Hilliard equation

2024-11-22T18:29:08+02:00

This paper presents a two-derivative energy-stable method for the Cahn-Hilliard equation. We use a fully implicit time discretization with the addition of two stabilization terms to maintain the energy stability. As far as we know, this is the first time an energy-stable multiderivative method has been developed for phase-field models. We present numerical results of the novel method to support our mathematical analysis. In addition, we perform numerical experiments of two multiderivative predictor-corrector methods of fourth and sixth-order accuracy, and we show numerically that all the methods are energy stable.