Applied Mathematics Commons^™

Gauss-Type Quadratures For Weakly Singular Integrals And Their Application To Fredholm Integral Equations Of The Second Kind, Hideaki Kaneko, Yuesheng Xu

Mathematics & Statistics Faculty Publications

In this paper we establish Gauss-type quadrature formulas for weakly singular integrals. An application of the quadrature scheme is given to obtain numerical solutions of the weakly singular Fredholm integral equation of the second kind. We call this method a discrete product-integration method since the weights involved in the standard product-integration method are computed numerically.

A Fast Numerical Solution Of Scattering By A Cylinder: Spectral Method For The Boundary Integral Equations, Fang Q. Hu

Mathematics & Statistics Faculty Publications

It is known that the exact analytic solutions of wave scattering by a circular cylinder, when they exist, are not in a closed form but in infinite series which converge slowly for high frequency waves. In this paper, a fast numerical solution is presented for the scattering problem in which the boundary integral equations, reformulated from the Helmholtz equation, are solved using a Fourier spectral method. It is shown that the special geometry considered here allows the implementation of the spectral method to be simple and very efficient. The present method differs from previous approaches in that the singularities of …

Periodic And Homoclinic Orbits In A Toy Climate Model, M. Toner, A. D. Kirwan Jr.

Mathematics & Statistics Faculty Publications

A two dimensional system of autonomous nonlinear ordinary differential equations models glacier growth and temperature changes on an idealized planet. We apply standard perturbative techniques from dynamical systems theory to study small amplitude periodic orbits about a constant equilibrium. The equations are put in cononical form and the local phase space topology is examined. Maximum and minimum periods of oscillation are obtained and related to the radius of the orbit. An adjacent equilibrium is shown to have saddle character and the inflowing and outflowing manifolds of this saddle are studied using numerical integration. The inflowing manifolds show the region of …

A Numerical Study Of Wave Propagation In A Confined Mixing Layer By Eigenfunction Expansions, Fang Q. Hu

Mathematics & Statistics Faculty Publications

It is well known that the growth rate of instability waves of a two-dimensional free shear layer is reduced greatly at supersonic convective Mach numbers. In previous works, it has been shown that new wave modes exist when the shear layers are bounded by a channel due to the coupling effect between the acoustic wave modes and the motion of the mixing layer. The present work studies the simultaneous propagation of multiple stability waves using numerical simulation. It is shown here that the coexistence of two wave modes in the flow field can lead to an oscillatory growth of disturbance …

Temporal Model Of An Optically Pumped Co-Doped Solid State Laser, T. G. Wangler, J. J. Swetits, A. M. Buoncristiani

Mathematics & Statistics Faculty Publications

Currently, research is being conducted on the optical properties of materials associated with the development of solid-state lasers in the 2 micron region. In support of this effort, a mathematical model describing the energy transfer in a holmium laser sensitized with thulium is developed. In this paper, we establish some qualitative properties of the solution of the model, such as non-negativity, boundedness, and integrability. A local stability analysis is then performed from which conditions for asymptotic stability are obtained. Finally, we report on our numerical analysis of the system and how it compares with experimental results.

The Scattering Potential For A Polytrope Of Degree-5, J. A. Adam

Mathematics & Statistics Faculty Publications

By regarding the study of radial and non-redial stellar oscillations as a problem in potential scattering theory, a standard form of the radial Schrödinger equation can be derived. After establishing some preliminary results of astrophysical interest, an analytic expression for the potential is derived for a truncated (i.e., finite radius) polytrope (or class of self-gravitating compressible spheres) of degree n = 5. Properties of the potential are discussed.

Induced Mach Wave-Flame Interactions In Laminar Supersonic Fuel Jets, F. Q. Hu, T. L. Jackson, D. G. Lasseigne, C. E. Grosch

Mathematics & Statistics Faculty Publications

A model problem is proposed to investigate the steady response of a reacting, compressible laminar jet to Mach waves generated by wavy walls in a channel of finite width. The model consists of a two-dimensional jet of fuel emerging into a stream of oxidizer which are allowed to mix and react in the presence of the Mach waves. The governing equations are taken to be the steady parabolized Navier-Stokes equations which are solved numerically. The kinetics is assumed to be a one-step, irreversible reaction of the Arrhenius type. Two important questions on the Mach wave-flame interactions are discussed: (i) how …

The Dynamics Of Growth-Factor-Modified Immune-Response To Cancer Growth: One-Dimensional Models, J. A. Adam

Mathematics & Statistics Faculty Publications

By characterizing the effect of tumor growth factors as deviations from normal logistic-type growth rates, the spatio-temporal dynamics for a one-dimensional model of cancer growth incorporating immune response are studied. The growth rates considered are classified respectively as normal, activated, inhibited and delay activated. The homogeneous steady states are defined by relative extrema of a ''free energy'' function V(x) for each of the above four cases. This function is of particular importance in studying the coexistence of tumoral and cancer-free steady states, and in identifying the nature (progressive or regressive) of travelling wave solutions to the nonlinear partial differential equation …

Erratum: "Temperature And Suction Effects On The Instability Of An Infinite Swept Attachment Line" [Physics Of Fluids A 4, 2008 (1992)], D. G. Lasseigne, T. L. Jackson, F. Q. Hu

Mathematics & Statistics Faculty Publications

Erratum to:

Lasseigne, D. G., Jackson, T. L., & Hu, F. Q. (1992). Temperature and suction effects on the instability of an infinite swept attachment line. Physics of Fluids A: Fluid Dynamics, 4(9), 2008-2012. doi:10.1063/1.858370

The Sharp Lipschitz-Constants For Feasible And Optimal-Solutions Of A Perturbed Linear Program, Wu Li

Mathematics & Statistics Faculty Publications

The purpose of this paper is to derive the sharp Lipschitz constants for the feasible solutions and optimal solutions of a linear program with respect to right-hand-side perturbations. The Lipschitz constants are given in terms of pseudoinverses of submatrices of the matrices involved and are proven to be sharp.

Scattering Parameters For An Epstein Profile In A Half-Space, J. A. Adam

Mathematics & Statistics Faculty Publications

The reflection of waves for a stratified medium in (-∞, ∞) can be studied by transforming the hypergeometric differential equation into a wave equation. In the context of an astrophysical problem [1], the corresponding analysis is carried out for an Epstein profile in (0, ∞). This profile represents a scattering potential in quantum mechanical terminology: its properties are briefly discussed.

Temperature And Suction Effects On The Instability Of An Infinite Swept Attachment Line, D. G. Lasseigne, T. L. Jackson, F. Q. Hu

Mathematics & Statistics Faculty Publications

It is known that the incompressible, infinite swept attachment line flow is unstable to streamwise disturbances that originate in the boundary layer when the cross-flow exceeds a critical magnitude. Furthermore, a small degree of suction at the surface has a significant stabilizing influence while a small degree of blowing has a considerable destabilizing influence. This paper investigates the stabilizing and destabilizing effects of, respectively, cooling or heating the plate and the competing or enhancing effects of suction or blowing. A nonorthogonal flow with respect to the attachment line is also considered by adding a component of shear to the mean …

Numerical Solutions For Weakly Singular Hammerstein Equations And Their Superconvergence, Hideaki Kaneko, Richard D. Noren, Yuesheng Xu

Mathematics & Statistics Faculty Publications

In the recent paper [7], it was shown that the solutions of weakly singular Hammerstein equations satisfy certain regularity properties. Using this result, the optimal convergence rate of a standard piecewise polynomial collocation method and that of the recently proposed collocationtype method of Kumar and Sloan [10] are obtained. Superconvergence of both of these methods are also presented. In the final section, we discuss briefly a standard productintegration method for weakly singular Hammerstein equations and indicate its superconvergence property. © 1992 Rocky Mountain Mathematics Consortium.

Solution Uniqueness And Stability Criteria For A Model Of Growth Factor Production, J. A. Adam

Mathematics & Statistics Faculty Publications

Uniqueness and stability criteria are established for the steady states of a nonlinear model of growth factor production. A specific expression for the nonlinearity is chosen, containing three parameters which can be adjusted to fit a specific biological context, but much of the analysis applies to a general class of source terms that exhibit the same qualitative behavior.

Nonlinear-Interaction Of A Detonation Vorticity Wave, D. G. Lasseigne, T. L. Jackson, M. Y. Hussaini

Mathematics & Statistics Faculty Publications

The interaction of an oblique, overdriven detonation wave with a vorticity disturbance is investigated by a direct two-dimensional numerical simulation using a multidomain, finite-difference solution of the compressible Euler equations. The results are compared to those of linear theory, which predict that the effect of exothermicity on the interaction is relatively small except possibly near a critical angle where linear theory no longer holds. It is found that the steady-state computational results whenever obtained in this study agree with the results of linear theory. However, for cases with incident angle near the critical angle, moderate disturbance amplitudes, and/or sudden transient …

Self-Activation And Inhibition: A Simple Nonlinear Model, J. A. Adam

Mathematics & Statistics Faculty Publications

Self-activation and self-inhibition of cell number density (or growth factor concentration) due to a spatially localized source are studied. Both the time-independent and time-dependent models are examined, and the linear stability of the resulting three steady states of the former is discussed.

A Duality Approach To Best Uniform Convex Approximation, S. E. Weinstein, Yuesheng Xu

Mathematics & Statistics Faculty Publications

Let C[a, b] be the space of continuous functions on [a, b] endowed with the uniform norm llƒll _∞ = sup{ Iƒ(x)1 :x∈ [a, b]}. Let K be the set of convex functions defined on [a, b]. A function g* ∈ K is said to be a best uniform convex approximation to ƒ ∈ C[a, b] if ∥ƒ - g*∥ _∞ = inf { ∥ƒ - g∥_∞ : g ∈ K}.

Parametric Instability Of Supersonic Shear Layers Induced By Periodic Mach Waves, Fang Q. Hu, Christopher K. W. Tam

Mathematics & Statistics Faculty Publications

It is suggested that parametric instability can be induced in a confined supersonic shear layer by the use of a periodic Mach wave system generated by a wavy wall. The existence of such an instability solution is demonstrated computationally by solving the Floquet system of equations. The solution is constructed by means of a Fourier-Chebyshev expansion. Numerical convergence is assured by using a very large number of Fourier and Chebyshev basis functions. The computed growth rate of the induced flow instability is found to vary linearly with the amplitude of the mach waves when the amplitude is not excessively large. …

Stability Of A Viscoelastic Burgers Flow, D. Glenn Lasseigne, W. E. Olmstead

Mathematics & Statistics Faculty Publications

The system of equations proposed by Burgers to model turbulent flow in a channel is extended to include viscoelastic affects. The stability and bifurcation properties are examined in the neighborhood of the critical Reynolds number. For highly elastic fluids, the bifurcated state is periodic with a shift in frequency.

A Characterization Of The Solution Of A Fredholm Integral Equation With L∞ Forcing Term, Hideaki Kaneko, Richard Noren, Yuesheng Xu

Mathematics & Statistics Faculty Publications

In this paper we investigate the regularity properties of the Fredholm equation (Formula Presented) . The kernel is the product of the smooth function k and the singular function gα (Formula Presented). The forcing function f is in L∞. We obtain a decomposition of the solution as the sum of two functions—one with a discontinuity reflecting that of the forcing function—and the other a regular function. Our results extend those of C. Schneider [6], who assumes a condition that is stronger than f ∈ C[a, b] ∩ C^m(a,b) (for some integer m). © 1990 …

Best Quasi-Convex Uniform Approximation, S. E. Weinstein, Yuesheng Xu

Mathematics & Statistics Faculty Publications

No abstract provided.

Uniform L1 Behavior In Classes Of Integrodifferential Equations With Convex Kernels, Richard Noren

Mathematics & Statistics Faculty Publications

No abstract provided.

Ignition Of A Combustible Solid With Reactant Consumption, D. Glenn Lasseigne, W. E. Olmstead

Mathematics & Statistics Faculty Publications

The effects of excessive reactant consumption on the ignition of a combustible solid are introduced through a revised scaling of the heat release constant. Large activation energy asymptotics then yields a new one-parameter integral equation governing the temperature evolution near ignition. Analysis of the integral equation reveals a critical value of the parameter which distinguishes between the cases of ignition and nonignition. © 1987 Society for Industrial and Applied Mathematics

On The Existence Of Periodic And Eventually Periodic Solutions Of A Fluid Dynamic Forced Harmonic Oscillator, Charlie H. Cooke

Mathematics & Statistics Faculty Publications

For certain flow regimes, the nonlinear differential equation Y¨=F(Y)−G, Y≥0, G>0 and constant, models qualitatively the behaviour of a forced, fluid dynamic, harmonic oscillator which has been a popular department store attraction. The device consists of a ball oscillating suspended in the vertical jet from a household fan. From the postulated form of the model, we determine sets of attraction and exploit symmetry properties of the system to show that all solutions are either initially periodic, with the ball never striking the fan, or else eventually approach a periodic limit cycle, after a sufficient number of bounces away from …

Sufficiency Of A Numerical Downstream Continuation, Charlie H. Cooke

Mathematics & Statistics Faculty Publications

(First paragraph) Customarily one does not impose n-th order boundary conditions on the solution of initial/boundary value problems whose characterizing partial differential equations are also n-th order. However, conjecture that such problems are not well-posed, or that a solution might not exist, is not always justified [l]. Perhaps a physically more natural example is provided by problems of computational fluid dynamics. Here boundary conditions which correctly should be applied at an infinite distance downstream from the region of interest are for computational convenience often applied at a finite location [2]. Results of numerical experimentation on viscous flows governed by …

Complementary Extremum Principles, J. Swetits, C. Rogers

Mathematics & Statistics Faculty Publications

Important complementary extremum principles are generated without recourse to general variational theory. The results are illustrated by an application to a class of boundary value problems in Magnetohydrodynamics.

Solar Magnetoatmospheric Waves-A Simplified Mathematical Treatment, John A. Adam

Mathematics & Statistics Faculty Publications

The inhomogeneous wave equation for a special class of magnetoatmospheric waves is formally solved, and the principle of stationary phase used to provide information on the group velocity properties of such waves. General results are presented concerning the associated mechanical energy flux. The basic problem considered is relevant to waves initiated by sudden events in the solar atmosphere.