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Partial Differential Equations

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Articles 211 - 240 of 265

Full-Text Articles in Numerical Analysis and Computation

Approximate Analytical Solutions For Fractional Space- And Time- Partial Differential Equations Using Homotopy Analysis Method, Subir, Das, R. Kumar, P. K. Gupta, Hossein Jafari Dec 2010

Approximate Analytical Solutions For Fractional Space- And Time- Partial Differential Equations Using Homotopy Analysis Method, Subir, Das, R. Kumar, P. K. Gupta, Hossein Jafari

Applications and Applied Mathematics: An International Journal (AAM)

This article presents the approximate analytical solutions of first order linear partial differential equations (PDEs) with fractional time- and space- derivatives. With the aid of initial values, the explicit solutions of the equations are solved making use of reliable algorithm like homotopy analysis method (HAM). The speed of convergence of the method is based on a rapidly convergent series with easily computable components. The fractional derivatives are described in Caputo sense. Numerical results show that the HAM is easy to implement and accurate when applied to space- time- fractional PDEs.

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Deformation Waves In Microstructured Materials: Theory And Numerics, Juri Engelbrecht, Arkadi Berezovski, Mihhail Berezovski Sep 2010

Deformation Waves In Microstructured Materials: Theory And Numerics, Juri Engelbrecht, Arkadi Berezovski, Mihhail Berezovski

Publications

A linear model of the microstructured continuum based on Mindlin theory is adopted which can be represented in the framework of the internal variable theory. Fully coupled systems of equations for macro-motion and microstructure evolution are represented in the form of conservation laws. A modification of wave propagation algorithm is used for numerical calculations. Results of direct numerical simulations of wave propagation in periodic medium are compared with similar results for the continuous media with the modelled microstructure. It is shown that the proper choice of material constants should be made to match the results obtained by both approaches

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A Note On He’S Parameter-Expansion Method Of Coupled Van Der Pol–Duffing Oscillators, N. H. Sweilam, M. M. Khader Aug 2010

A Note On He’S Parameter-Expansion Method Of Coupled Van Der Pol–Duffing Oscillators, N. H. Sweilam, M. M. Khader

Applications and Applied Mathematics: An International Journal (AAM)

This paper presents the analytical and approximate solutions of the coupled chaotic Van der Pol-Duffing systems, by using the He's parameter-expansion method (PEM). One iteration is sufficient to obtain a highly accurate solution, which is valid for the whole solution domain. From the obtained results, we can conclude that the suggest method, is of utter simplicity, and can be easily extended to all kinds of non-linear equations.

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Homotopy Perturbation Method And The Stagnation Point Flow, P. Donald Ariel Aug 2010

Homotopy Perturbation Method And The Stagnation Point Flow, P. Donald Ariel

Applications and Applied Mathematics: An International Journal (AAM)

The laminar steady flow of an incompressible, viscous fluid near a stagnation point has been computed using the homotopy perturbation method (HPM). Both the cases, (i) two-dimensional flow and (ii) axisymmetric flow, have been considered. A sequence of successive approximations has been obtained in the solution, and the convergence of the sequence is achieved by using the Padé approximants. It is found that there is a complete agreement between the results obtained by the HPM and the exact numerical solution.

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Elements Of Study On Dynamic Materials, Marine Rousseau, Gerard A. Maugin, Mihhail Berezovski Jul 2010

Elements Of Study On Dynamic Materials, Marine Rousseau, Gerard A. Maugin, Mihhail Berezovski

Publications

As a preliminary study to more complex situations of interest in small-scale technology, this paper envisages the elementary propagation properties of elastic waves in one-spatial dimension when some of the properties (mass density, elasticity) may vary suddenly in space or in time, the second case being of course more original. Combination of the two may be of even greater interest. Toward this goal, a critical examination of what happens to solutions at the crossing of pure space-like and time-like material discontinuities is given together with simple solutions for smooth transitions and numerical simulations in the discontinuous case. The effects on …

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High Order Finite Elements For Lagrangian Computational Fluid Dynamics, Truman Everett Ellis Apr 2010

High Order Finite Elements For Lagrangian Computational Fluid Dynamics, Truman Everett Ellis

Master's Theses

A general finite element method is presented to solve the Euler equations in a Lagrangian reference frame. This FEM framework allows for separate arbitrarily high order representation of kinematic and thermodynamic variables. An accompanying hydrodynamics code written in Matlab is presented as a test-bed to experiment with various basis function choices. A wide range of basis function pairs are postulated and a few choices are developed further, including the bi-quadratic Q2-Q1d and Q2-Q2d elements. These are compared with a corresponding pair of low order bi-linear elements, traditional Q1-Q0 and sub-zonal pressure Q1-Q1d. Several test problems are considered including static convergence …

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Morphological Evolution Of Single-Crystal Ultrathin Solid Films, Mikhail Khenner Mar 2010

Morphological Evolution Of Single-Crystal Ultrathin Solid Films, Mikhail Khenner

Mathematics Faculty Publications

An introduction to mathematical modeling of ultrathin solid films and the role of such modeling in nanotechnologies: Educational presentation for senior physics majors

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Morphological Evolution Of Single-Crystal Ultrathin Solid Films, Mikhail Khenner Mar 2010

Morphological Evolution Of Single-Crystal Ultrathin Solid Films, Mikhail Khenner

Mathematics Faculty Publications

An introduction to mathematical modeling of ultrathin solid films and the role of such modeling in nanotechnologies: Educational/Research presentation for senior physics majors

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Oscillatory And Monotonic Modes Of Long-Wave Marangoni Convection In A Thin Film, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev Jan 2010

Oscillatory And Monotonic Modes Of Long-Wave Marangoni Convection In A Thin Film, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev

Mathematics Faculty Publications

We study long-wave Marangoni convection in a layer heated from below. Using the scaling k=O Bi, where k is the wave number and Bi is the Biot number, we derive a set of amplitude equations. Analysis of this set shows presence of monotonic and oscillatory modes of instability. Oscillatory mode has not been previously found for such direction of heating. Studies of weakly nonlinear dynamics demonstrate that stable steady and oscillatory patterns can be found near the stability threshold.

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Oscillatory And Monotonic Modes Of Long-Wave Marangoni Convection In A Thin Film, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev Jan 2010

Oscillatory And Monotonic Modes Of Long-Wave Marangoni Convection In A Thin Film, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev

Mathematics Faculty Publications

We study long-wave Marangoni convection in a layer heated from below. Using the scaling k=O Bi, where k is the wave number and Bi is the Biot number, we derive a set of amplitude equations. Analysis of this set shows presence of monotonic and oscillatory modes of instability. Oscillatory mode has not been previously found for such direction of heating. Studies of weakly nonlinear dynamics demonstrate that stable steady and oscillatory patterns can be found near the stability threshold.

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Temporal Scales For Transport Patterns In The Gulf Of Finland, Bert Viikmae, Tarmo Soomere, Mikk Viidebaum, Mihhail Berezovski Jan 2010

Temporal Scales For Transport Patterns In The Gulf Of Finland, Bert Viikmae, Tarmo Soomere, Mikk Viidebaum, Mihhail Berezovski

Publications

The basic time scales for current-induced net transport of surface water and associated time scales of reaching the nearshore in the Gulf of Finland, the Baltic Sea, are analysed based on Lagrangian trajectories of water particles reconstructed from three-dimensional velocity fields by the Rossby Centre circulation model for 1987–1991. The number of particles reaching the nearshore exhibits substantial temporal variability whereas the rate of leaving the gulf is almost steady. It is recommended to use an about 3 grid cells wide nearshore area as a substitute to the coastal zone and about 10–15 day long trajectories for calculations of the …

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Waves In Materials With Microstructure: Numerical Simulation, Mihhail Berezovski, Arkadi Berezovski, Juri Engelbrecht Jan 2010

Waves In Materials With Microstructure: Numerical Simulation, Mihhail Berezovski, Arkadi Berezovski, Juri Engelbrecht

Publications

Results of numerical experiments are presented in order to compare direct numerical calculations of wave propagation in a laminate with prescribed properties and corresponding results obtained for an effective medium with the microstructure modelling. These numerical experiments allowed us to analyse the advantages and weaknesses of the microstructure model.

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Parallel And Distributed Simulation Of Parabolic And Telegraphic Equations., Ewedafe Simon Uzezi Jan 2010

Parallel And Distributed Simulation Of Parabolic And Telegraphic Equations., Ewedafe Simon Uzezi

Student Works (2010-2019)

In this thesis, a parallel implementation of explicit/implicit parallel algorithms such as the stationary iterative methods and the class of iterating alternating methods which includes: Alternating Direction Implicit (ADI), Iterative Alternating Direction Explicit (IADE), for D’Yakonov (IADE-DY), Double sweep Mitchell and Fairweather (MF-DS) and Alternating Group Explicit (AGE) method for solving 1-Dimensional (1-D), 2-Dimensional (2-D) Parabolic (special examples including 1-D, 2-D Bio-Heat Equation) and 1-D, 2-D and 3-D Telegraphic Equations on a distributed environment of Message Passing Interface (MPI) and Parallel Virtual Machine (PVM) platform is presented. To correlate the communication activity with computation, we counted events between significant MPI/PVM …

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Thermocapillary Effects In Driven Dewetting And Self-Assembly Of Pulsed Laser-Irradiated Metallic Films, Mikhail Khenner Dec 2009

Thermocapillary Effects In Driven Dewetting And Self-Assembly Of Pulsed Laser-Irradiated Metallic Films, Mikhail Khenner

Mathematics Faculty Publications

A mathematical model for the evolution of pulsed laser-irradiated, molten metallic films has been developed using the lubrication theory. The heat transfer problem that incorporates the absorbed heat from a single laser beam or the interfering laser beams is solved analytically. Using this temperature field, we derive the 3D long-wave evolution PDE for the film height. To get insights into dynamics of dewetting, we study the 2D version of the evolution equation by means of a linear stability analysis and by numerical simulations. The stabilizing and destabilizing effects of various system parameters, such as the reflectivity, the peak laser beam …

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Thermocapillary Effects In Driven Dewetting And Self-Assembly Of Pulsed Laser-Irradiated Metallic Films, Mikhail Khenner Dec 2009

Thermocapillary Effects In Driven Dewetting And Self-Assembly Of Pulsed Laser-Irradiated Metallic Films, Mikhail Khenner

Mathematics Faculty Publications

A mathematical model for the evolution of pulsed laser-irradiated, molten metallic films has been developed using the lubrication theory. The heat transfer problem that incorporates the absorbed heat from a single laser beam or the interfering laser beams is solved analytically. Using this temperature field, we derive the 3D long-wave evolution PDE for the film height. To get insights into dynamics of dewetting, we study the 2D version of the evolution equation by means of a linear stability analysis and by numerical simulations. The stabilizing and destabilizing effects of various system parameters, such as the reflectivity, the peak laser beam …

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Thermocapillary Effects In Driven Dewetting And Self-Assembly Of Pulsed Laser-Irradiated Metallic Films, Mikhail Khenner Dec 2009

Thermocapillary Effects In Driven Dewetting And Self-Assembly Of Pulsed Laser-Irradiated Metallic Films, Mikhail Khenner

Mathematics Faculty Publications

A mathematical model for the evolution of pulsed laser-irradiated, molten metallic films has been developed using the lubrication theory. The heat transfer problem that incorporates the absorbed heat from a single laser beam or the interfering laser beams is solved analytically. Using this temperature field, we derive the 3D long-wave evolution PDE for the film height. To get insights into dynamics of dewetting, we study the 2D version of the evolution equation by means of a linear stability analysis and by numerical simulations. The stabilizing and destabilizing effects of various system parameters, such as the reflectivity, the peak laser beam …

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Higher Order Difference Schemes For Heat Equation, Jianzhong Wang Dec 2009

Higher Order Difference Schemes For Heat Equation, Jianzhong Wang

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we construct the explicit difference schemes for the heat equation with arbitrary high orders. We also show the validity of the new schemes by numerical simulations.

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Waves In Inhomogeneous Solids, Arkadi Berezovski, Mihhail Berezovski, Juri Engelbrecht Aug 2009

Waves In Inhomogeneous Solids, Arkadi Berezovski, Mihhail Berezovski, Juri Engelbrecht

Publications

The paper aims at presenting a numerical technique used in simulating the propagation of waves in inhomogeneous elastic solids. The basic governing equations are solved by means of a finite-volume scheme that is faithful, accurate, and conservative. Furthermore, this scheme is compatible with thermodynamics through the identification of the notions of numerical fluxes (a notion from numerics) and of excess quantities (a notion from irreversible thermodynamics). A selection of one-dimensional wave propagation problems is presented, the simulation of which exploits the designed numerical scheme. This selection of exemplary problems includes (i) waves in periodic media for weakly nonlinear waves with …

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Thermocapillary Effects In Driven Dewetting And Self-Assembly Of Pulsed Laser-Irradiated Metallic Films, Agegnehu Atena, Mikhail Khenner Jan 2009

Thermocapillary Effects In Driven Dewetting And Self-Assembly Of Pulsed Laser-Irradiated Metallic Films, Agegnehu Atena, Mikhail Khenner

Mathematics Faculty Publications

In this paper the lubrication-type dynamical model is developed of a molten, pulsed laser-irradiated metallic film. The heat transfer problem that incorporates the absorbed heat from a single beam or interfering beams is solved analytically. Using this temperature field, we derive the 3D long-wave evolution PDE for the film height. To get insights into dynamics of dewetting, we study the 2D version of the evolution equation by means of a linear stability analysis and by numerical simulations. The stabilizing and destabilizing effects of various system parameters, such as the peak laser beam intensity, the film optical thickness, the Biot and …

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Thermocapillary Effects In Driven Dewetting And Self-Assembly Of Pulsed Laser-Irradiated Metallic Films, Agegnehu Atena, Mikhail Khenner Jan 2009

Thermocapillary Effects In Driven Dewetting And Self-Assembly Of Pulsed Laser-Irradiated Metallic Films, Agegnehu Atena, Mikhail Khenner

Mathematics Faculty Publications

In this paper the lubrication-type dynamical model is developed of a molten, pulsed laser-irradiated metallic film. The heat transfer problem that incorporates the absorbed heat from a single beam or interfering beams is solved analytically. Using this temperature field, we derive the 3D long-wave evolution PDE for the film height. To get insights into dynamics of dewetting, we study the 2D version of the evolution equation by means of a linear stability analysis and by numerical simulations. The stabilizing and destabilizing effects of various system parameters, such as the peak laser beam intensity, the film optical thickness, the Biot and …

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Singular Superposition/Boundary Element Method For Reconstruction Of Multi-Dimensional Heat Flux Distributions With Application To Film Cooling Holes, Mahmood Silieti, Eduardo Divo, Alain J. Kassab Jan 2009

Singular Superposition/Boundary Element Method For Reconstruction Of Multi-Dimensional Heat Flux Distributions With Application To Film Cooling Holes, Mahmood Silieti, Eduardo Divo, Alain J. Kassab

Publications

A hybrid singularity superposition/boundary element-based inverse problem method for the reconstruction of multi-dimensional heat flux distributions is developed. Cauchy conditions are imposed at exposed surfaces that are readily reached for measurements while convective boundary conditions are unknown at surfaces that are not amenable to measurements such as the walls of the cooling holes. The purpose of the inverse analysis is to determine the heat flux distribution along cooling hole surfaces. This is accomplished in an iterative process by distributing a set of singularities (sinks) inside the physical boundaries of the cooling hole (usually along cooling hole centerline) with a given …

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Determining The Shape Of A Resistor Grid, Esther Chiew, Vincent Selhorst-Jones Dec 2008

Determining The Shape Of A Resistor Grid, Esther Chiew, Vincent Selhorst-Jones

Mathematical Sciences Technical Reports (MSTR)

Impedance imaging has received a lot of attention in the past two decades, as a means for non-destructively imaging the interior of a conductive object. One injects a known electrical current pattern into an object at the exterior boundary, then measures the induced potential (voltage) on some portion of the boundary. The goal is to recover information about the interior conductivity of the object, which (we hope) influences the voltages we measure. Of course one can also use multiple input currents and measured voltages. A variation on this problem is that of "boundary identification," in which some portion of the …

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Morphologies And Kinetics Of A Dewetting Ultrathin Solid Film, Mikhail Khenner Jan 2008

Morphologies And Kinetics Of A Dewetting Ultrathin Solid Film, Mikhail Khenner

Mathematics Faculty Publications

The surface evolution model based on geometric partial differential equation is used to numerically study the kinetics of dewetting and dynamic morphologies for the localized pinhole defect in the surface of the ultrathin solid film with the strongly anisotropic surface energy. Depending on parameters such as the initial depth and width of the pinole, the strength of the attractive substrate potential and the strength of the surface energy anisotropy, the pinhole may either extend to the substrate and thus rupture the film, or evolve to the quasiequilibrium shape while the rest of the film surface undergoes phase separation into a …

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Enhanced Stability Of A Dewetting Thin Liquid Film In A Single-Frequency Vibration Field, Mikhail Khenner Jan 2008

Enhanced Stability Of A Dewetting Thin Liquid Film In A Single-Frequency Vibration Field, Mikhail Khenner

Mathematics Faculty Publications

Dynamics of a thin dewetting liquid film on a vertically oscillating substrate is considered. We assume moderate vibration frequency and large (compared to the mean film thickness) vibration amplitude. Using the lubrication approximation and the averaging method, we formulate the coupled sets of equations governing the pulsatile and the averaged fluid flows in the film, and then derive the nonlinear amplitude equation for the averaged film thickness. We show that there exists a window in the frequency-amplitude domain where the parametric and shear-flow instabilities of the pulsatile flow do not emerge. As a consequence, in this window the averaged description …

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Enhanced Stability Of A Dewetting Thin Liquid Film In A Single-Frequency Vibration Field, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev Jan 2008

Enhanced Stability Of A Dewetting Thin Liquid Film In A Single-Frequency Vibration Field, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev

Mathematics Faculty Publications

Dynamics of a thin dewetting liquid film on a vertically oscillating substrate is considered. We assume moderate vibration frequency and large (compared to the mean film thickness) vibration amplitude. Using the lubrication approximation and the averaging method, we formulate the coupled sets of equations governing the pulsatile and the averaged fluid flows in the film, and then derive the nonlinear amplitude equation for the averaged film thickness. We show that there exists a window in the frequency-amplitude domain where the parametric and shear-flow instabilities of the pulsatile flow do not emerge. As a consequence, in this window the averaged description …

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Enhanced Stability Of A Dewetting Thin Liquid Film In A Single-Frequency Vibration Field, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev Jan 2008

Enhanced Stability Of A Dewetting Thin Liquid Film In A Single-Frequency Vibration Field, Sergey Shklyaev, Mikhail Khenner, Alexei Alabuzhev

Mathematics Faculty Publications

Dynamics of a thin dewetting liquid film on a vertically oscillating substrate is considered. We assume moderate vibration frequency and large (compared to the mean film thickness) vibration amplitude. Using the lubrication approximation and the averaging method, we formulate the coupled sets of equations governing the pulsatile and the averaged fluid flows in the film, and then derive the nonlinear amplitude equation for the averaged film thickness. We show that there exists a window in the frequency-amplitude domain where the parametric and shear-flow instabilities of the pulsatile flow do not emerge. As a consequence, in this window the averaged description …

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Morphologies And Kinetics Of A Dewetting Ultrathin Solid Film, Mikhail Khenner Jan 2008

Morphologies And Kinetics Of A Dewetting Ultrathin Solid Film, Mikhail Khenner

Mathematics Faculty Publications

The surface evolution model based on geometric partial differential equation is used to numerically study the kinetics of dewetting and dynamic morphologies for the localized pinhole defect in the surface of the ultrathin solid film with the strongly anisotropic surface energy. Depending on parameters such as the initial depth and width of the pinole, the strength of the attractive substrate potential and the strength of the surface energy anisotropy, the pinhole may either extend to the substrate and thus rupture the film, or evolve to the quasiequilibrium shape while the rest of the film surface undergoes phase separation into a …

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Numerical Simulation Of Waves And Fronts In Inhomogeneous Solids, A. Berezovski, M. Berezovski, J. Engelbrecht, G. A. Maugin Jun 2007

Numerical Simulation Of Waves And Fronts In Inhomogeneous Solids, A. Berezovski, M. Berezovski, J. Engelbrecht, G. A. Maugin

Publications

Dynamic response of inhomogeneous materials exhibits new effects, which often do not exist in homogeneous media. It is quite natural that most of studies of wave and front propagation in inhomogeneous materials are associated with numerical simulations. To develop a numerical algorithm and to perform the numerical simulations of moving fronts we need to formulate a kinetic law of progress relating the driving force and the velocity of the discontinuity. The velocity of discontinuity is determined by means of the non-equilibrium jump relations at the front. The obtained numerical method generalizes the wave-propagation algorithm to the case of moving discontinuities …

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Parameter Estimation In Nonlinear Coupled Advection-Diffusion Equation, Robert R. Ferdinand Jun 2007

Parameter Estimation In Nonlinear Coupled Advection-Diffusion Equation, Robert R. Ferdinand

Applications and Applied Mathematics: An International Journal (AAM)

In this paper a coupled system of two nonlinear advection-diffusion equations is presented. Such systems of equations have been used in mathematical literature to describe the dynamics of contaminant present in groundwater flowing through cracks in a porous rock matrix and getting absorbed into it. An inverse method procedure that approximates infinite-dimensional model parameters is described and convergence results for the parameter approximants are proved. This is finally followed by a computational experiment to compare theoretical and numerical results to verify accuracy of the mathematics analysis presented.

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Utilizing Thermal Testing For Recovering, James Preciado, Thomas Werne May 2007

Utilizing Thermal Testing For Recovering, James Preciado, Thomas Werne

Mathematical Sciences Technical Reports (MSTR)

Given a two-dimensional region that contains one or more circular voids, we develop mathematical methods to locate the center and radius of the voids based on thermal boundary data. These methods can be readily applied in the field of non-destructive evaluation.

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