High-Order Accurate Adaptive Kernel Compression Time-Stepping Schemes for Fractional Differential Equations

We present a goal-oriented a posteriori error estimator for finite element approximations of a class of homogenization problems. As a rule, homogenization problems are defined through the coupling of a macroscopic solution and the solution of auxiliary problems. In this work we assume that the homogenized problem is known and that it depends on a finite number of auxiliary problems. The accuracy in the goal functional depends therefore on the discretization error of the macroscopic and the auxiliary solutions. We show that it is possible to compute the error contributions of all solution components separately and use this information to balance the different discretization errors. Additionally, we steer a local mesh refinement for both the macroscopic problem and the auxiliary problems. The high efficiency of this approach is shown by numerical examples. These include the upscaling of a periodic diffusion tensor, the case of a Stokes flow over a porous bed, and the homogenization of a fuel cell model which includes the flow in a gas channel over a porous substrate coupled with a multispecies nonlinear transport equation.     

High-Order Accurate Local Schemes for Fractional Differential Equations

High-order methods inspired by the multi-step Adams methods are proposed for systems of fractional differential equations. The schemes are based on an expansion in a weighted \(L^2\) space. To obtain the schemes this expansion is terminated after \(P+1\) terms. We study the local truncation error and its behavior with respect to the step-size h and P. Building on this analysis, we develop an error indicator based on the Milne device. Methods with fixed and variable step-size are tested numerically on a number of problems, including problems with known solutions, and a fractional version on the Van der Pol equation.     

Growth factor dependence of progression through G1 and S phases of adult rat hepatocytes in vitro. Evidence of a mitogen restriction point in mid-late G1

Several hepatocyte mitogens have been identified, but the signals triggering the G0/G1 transition and cell cycle progression of hepatocytes remain unknown. Using hepatocyte primary cultures, we investigated the role of epidermal growth factor/pyruvate during the entry into and progression through the G1 phase and analyzed the expression of cell cycle markers. We show that the G0/G1 transition occurs during hepatocyte isolation as evidenced by the expression of early genes such as c-fos, c-jun, and c-myc. In culture, hepatocytes progress through G1 regardless of growth factor stimulation until a restriction point (R point) in mid-late G1 beyond which they cannot complete the cell cycle without mitogenic stimulation. Changes in cell cycle gene expression were associated with progression in G1; the cyclin E mRNA level is low early in G1 but increases at the G1/S boundary, while the protein is constantly detected during cell cycle but undergoes a change of electrophoretic mobility in mid-late G1 after the R point. In addition, a drastic induction of cyclin D1 mRNA and protein, and to a lesser extent of cyclin D2 mRNA, takes place in mitogen-stimulated cells after the R point. In contrast, cyclin D3 mRNA appears early in G1, remains constant in stimulated cells, but accumulates in unstimulated arrested cells, paralleling the cyclin-dependent kinase 4 mRNA expression. These results characterize the different steps of G1 phase in hepatocytes.     

Estimation of vehicle sideslip, tire force and wheel cornering stiffness

This paper presents a process for the estimation of tire–road forces, vehicle sideslip angle and wheel cornering stiffness. The method uses measurements (yaw rate, longitudinal/lateral accelerations, steering angle and angular wheel velocities) only from sensors which can be integrated or have already been integrated in modern cars. The estimation process is based on two blocks in series: the first block contains a sliding-mode observer whose principal role is to calculate tire–road forces, while in the second block an extended Kalman filter estimates sideslip angle and cornering stiffness. More specifically, this study proposes an adaptive tire-force model that takes variations in road friction into account. The paper also presents a study of convergence for the sliding-mode observer. The estimation process was applied and compared to real experimental data, in particular wheel force measurements. The vehicle mass is assumed to be known. Experimental results show the accuracy and potential of the estimation process.