Advances in Applied Mathematics and Mechanics AAMM Volume 3, Number 5

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Volume 3, Number 5

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Hydrodynamic Regimes, Knudsen Layer, Numerical Schemes: Definition of Boundary Fluxes

Christophe Besse, Saja Borghol, Thierry Goudon, Ingrid Lacroix-Violet & Jean-Paul Dudon

10.4208/aamm.10-m1041

Adv. Appl. Math. Mech., 3 (2011), pp. 519-561.

Preview Full PDF 2270 26015 Abstract

Abstract

We propose a numerical solution to incorporate in the simulation of a system of conservation laws boundary conditions that come from a microscopic modeling in the small mean free path regime. The typical example we discuss is the derivation of the Euler system from the BGK equation. The boundary condition relies on the analysis of boundary layers formation that accounts from the fact that the incoming kinetic flux might be far from the thermodynamic equilibrium.

Perturbations of Multipliers of Systems of Periodic Ordinary Differential Equations

Leonid Berezansky, Michael Gil’ & Liora Troib

10.4208/aamm.09-m0992

Adv. Appl. Math. Mech., 3 (2011), pp. 562-571.

Preview Full PDF 2298 25847 Abstract

Abstract

The paper deals with periodic systems of ordinary differential equations (ODEs). A new approach to the investigation of variations of multipliers under perturbations is suggested. It enables us to establish explicit conditions for the stability and instability of perturbed systems.

A Boundary Meshless Method for Solving Heat Transfer Problems Using the Fourier Transform

A. Tadeu, C. S. Chen, J. Antόnio & Nuno Simões

10.4208/aamm.10-m1039

Adv. Appl. Math. Mech., 3 (2011), pp. 572-585.

Preview Full PDF 2271 26091 Abstract

Abstract

Fourier transform is applied to remove the time-dependent variable in the diffusion equation. Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation, which is solved by the method of fundamental solutions and the method of particular solutions. The particular solution of Helmholtz equation is available as shown in [4, 15]. The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm. Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response. Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.

Dependence Analysis of the Solutions on the Parameters of Fractional Delay Differential Equations

Shuiping Yang, Aiguo Xiao & Xinyuan Pan

10.4208/aamm.10-m1045

Adv. Appl. Math. Mech., 3 (2011), pp. 586-597.

Preview Full PDF 2272 25885 Abstract

Abstract

In this paper, we investigate the dependence of the solutions on the parameters (order, initial function, right-hand function) of fractional delay differential equations (FDDEs) with the Caputo fractional derivative. Some results including an estimate of the solutions of FDDEs are given respectively. Theoretical results are verified by some numerical examples.

The Lower Bounds of Eigenvalues by the Wilson Element in Any Dimension

Youai Li

10.4208/aamm.10-m1046

Adv. Appl. Math. Mech., 3 (2011), pp. 598-610.

Preview Full PDF 2339 25976 Abstract

Abstract

In this paper, we analyze the Wilson element method of the eigenvalue problem in arbitrary dimensions by combining a new technique recently developed in [10] and the a posteriori error result. We prove that the discrete eigenvalues are smaller than the exact ones.

The Modified Ghost Fluid Method Applied to Fluid-Elastic Structure Interaction

Tiegang Liu, A. W. Chowdhury & Boo Cheong Khoo

10.4208/aamm.10-m1054

Adv. Appl. Math. Mech., 3 (2011), pp. 611-632.

Preview Full PDF 2211 25644 Abstract

Abstract

In this work, the modified ghost fluid method is developed to deal with 2D compressible fluid interacting with elastic solid in an Euler-Lagrange coupled system. In applying the modified Ghost Fluid Method to treat the fluid-elastic solid coupling, the Navier equations for elastic solid are cast into a system similar to the Euler equations but in Lagrangian coordinates. Furthermore, to take into account the influence of material deformation and nonlinear wave interaction at the interface, an Euler-Lagrange Riemann problem is constructed and solved approximately along the normal direction of the interface to predict the interfacial status and then define the ghost fluid and ghost solid states. Numerical tests are presented to verify the resultant method.

Flow of Newtonian Fluid in Non-Uniform Tubes with Application to Renal Flow: A Numerical Study

P. Muthu & Tesfahun Berhane

10.4208/aamm.10-m1064

Adv. Appl. Math. Mech., 3 (2011), pp. 633-648.

Preview Full PDF 2327 26036 Abstract

Abstract

In this paper, a numerical method employing a finite difference technique is used for an investigation of viscous, incompressible fluid flow in a tube with absorbing wall and slowly varying cross-section. The effect of fluid absorption through permeable wall is accounted by prescribing flux as a function of axial distance. The method is not restricted by the parameters in the problem such as wave number, permeability parameter, amplitude ratio and Reynolds number. The effects of these parameters on the radial velocity and mean pressure drop are studied and the results are presented graphically. Comparison is also made between the results obtained by perturbation method of solution and present approach.

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