Applied Mathematics Commons^™

Superconvergence Of The Iterated Galerkin Methods For Hammerstein Equations, Hideaki Kaneko, Yuesheng Xu

Mathematics & Statistics Faculty Publications

In this paper, the well-known iterated Galerkin method and iterated Galerkin-Kantorovich regularization method for approximating the solution of Fredholm integral equations of the second kind are generalized to Hammerstein equations with smooth and weakly singular kernels. The order of convergence of the Galerkin method and those of superconvergence of the iterated methods are analyzed. Numerical examples are presented to illustrate the superconvergence of the iterated Galerkin approximation for Hammerstein equations with weakly singular kernels. © 1996, Society for Industrial and Applied Mathematics

Parallel Newton-Krylov-Schwarz Solvers For The Full Potential Flow Equation, Jie Zhang

Computer Science Theses & Dissertations

Newton-Krylov-Schwarz methods are increasingly applied in Computational Fluid Dynamics (CFD). We develop a parallel analysis code based on this method for the full potential flow model. The full potential model consists of a single nonlinear second-order partial differential equation of mixed type (elliptic/hyperbolic), which we solve as a steady boundary-value problem.

We use a nine-point finite-difference stencil to discretize the equation. A Newtonlike linearization and correction method is used to solve the resulting set of nonlinear algebraic equations. To solve the inner linear equations, we employ a Krylov space method. Preconditioners are used to improve the convergence rate. In order …

A Simple Mathematical-Model And Alternative Paradigm For Certain Chemotherapeutic Regimens, J. A. Adam, J. C. Panetta

Mathematics & Statistics Faculty Publications

A simplified two-compartment model for cell-specific chemotherapy is analysed by reformulating the governing system of differential equations as a Schrodinger equation in time. With the choice of an exponentially decaying function representing the effects of chemotherapy on cycling tumor cells, the potential function V(t) is a Morse-type potential, well known in the quantum mechanical literature; and the solutions are obtainable in terms of confluent hypergeometric functions (or the related Whittaker functions). Because the chemotherapy is administered periodically, the potential V(t) is periodic also, and use is made of existing theory (Floquet theory) as applied to scattering by periodic potentials in …

A Two-Dimensional Model For Watershed Runoff And Pollutant Transport, Laura Jean Howard

Civil & Environmental Engineering Theses & Dissertations

A two-dimensional mathematical model is developed for the quantitative and qualitative analysis of watershed runoff for a given rainfall event. The runoff is routed through a receiving channel. The model is designed for use on a personal computer. Full documentation is included in the Appendix. The model is capable of accommodating downstream boundary conditions, temporally varied effective rainfall conditions, and spatially varied watershed conditions. Output takes the form of flow depths, discharges, and concentrations. The model was tested for quantity and quality using existing experimental data, and the results are presented herein.

Mathematical Models Of Chemotherapy, John Carl Panetta

Mathematics & Statistics Theses & Dissertations

Several mathematical models are developed to describe the effects of chemotherapy on both cancerous and normal tissue. Each model is defined by either a single homogeneous equation or a system of heterogeneous equations which describe the states of the normal and/or cancer cells. Periodic terms are added to model the effects of the chemotherapy. What we obtain are regions, in parameter space (dose and period), of acceptable drug regimens.

The models take into account various aspects of chemotherapy. These include, interactions between the cancer and normal tissue, cell specific chemotherapeutic drug, the use of non-constant parameters to aid in modeling …

Wing-Section Optimization For Supersonic Viscous Flow, Cem Cihan Item

Mechanical & Aerospace Engineering Theses & Dissertations

The recent interest in the High Speed Civil Transport (HSCT) has resulted in renewed research studies of optimized supersonic cruise transport configurations. Incorporation of flow viscosity effects in the design process of such a supersonic wing is currently under investigation. This may lead to more accurate problem formulations and, in tum, greater aerodynamic efficiency than can be obtained by the traditional, inviscid, linear theories. In this context, for a design code to be a candidate for a complex optimization problem, such as three-dimensional viscous supersonic wing design, it should be validated using simpler building-block shapes.

To optimize the shape of …

An Optimal Path Cover Algorithm For Cographs, R. Lin, S. Olariu

Computer Science Faculty Publications

The class of cographs, or complement-reducible graphs, arises naturally in many different areas of applied mathematics and computer science. In this paper, we present an optimal algorithm for determining a minimum path cover for a cograph G. In case G has a Hamiltonian path (cycle) our algorithm exhibits the path (cycle) as well.

Linear Time Optimization Algorithms For P4-Sparse Graphs, Beverly Jamison, Stephan Olariu

Computer Science Faculty Publications

Quite often, real-life applications suggest the study of graphs that feature some local density properties. In particular, graphs that are unlikely to have more than a few chordless paths of length three appear in a number of contexts. A graph G is P₄-sparse if no set of five vertices in G induces more than one chordless path of length three. P₄-sparse graphs generalize both the class of cographs and the class of P₄-reducible graphs. It has been shown that P₄-sparse graphs can be recognized in time linear in the size of the …

Fixed Points Of Generalized Contractive Multi-Valued Mappings, Peter Z. Daffer, Hideaki Kaneko

Mathematics & Statistics Faculty Publications

In a recent paper N. Mizoguchi and W. Takahashi gave a positive answer to the conjecture of S. Reich concerning the existence of fixed points of multi-valued mappings that satisfy a certain contractive condition. In this paper, we provide an alternative and somewhat more straightforward proof for the theorem of Mizoguchi and Takahashi. Also the problems associated with fixed points of weakly contractive multi-valued mappings are studied. Finally, we make a few comments that improve other results from their paper (J. Math. Anal. Appl. 141 (1989), 177-188).

A Logistic Model Of Periodic Chemotherapy, J. C. Panetta

Mathematics & Statistics Faculty Publications

A logistic differential equation with a time-varying periodic parameter is used to model the growth of cells, in particular cancer cells, in the presences of chemotherapeutic drugs. The chemotherapeutic effects are modeled by a periodic parameter that modifies the growth rate of the cell tissue. A negative growth rate represents the detrimental effects of the drugs. A simple criterion is obtained for the behavior of the chemotherapy.

Error Estimates And Lipschitz Constants For Best Approximation In Continuous Function Spaces, M. Bartelt, W. Li

Mathematics & Statistics Faculty Publications

We use a structural characterization of the metric projection PG(f), from the continuous function space to its one-dimensional subspace G, to derive a lower bound of the Hausdorff strong unicity constant (or weak sharp minimum constant) for PG and then show this lower bound can be attained. Then the exact value of Lipschitz constant for PG is computed. The process is a quantitative analysis based on the Gâteaux derivative of PG, a representation of local Lipschitz constants, the equivalence of local and global Lipschitz constants for lower semicontinuous mappings, and construction …

Implementation Of A Multiblock Sensitivity Analysis Method In Numerical Aerodynamic Shape Optimization, James Matthew Lacasse

Mechanical & Aerospace Engineering Theses & Dissertations

A multiblock sensitivity analysis method is applied in a numerical aerodynamic shape optimization technique. The Sensitivity Analysis Domain Decomposition (SADD) scheme which is implemented in this study was developed to reduce the computer memory requirements resulting from the aerodynamic sensitivity analysis equations, Discrete sensitivity analysis offers the ability to compute quasi-analytical derivatives in a more efficient manner than traditional finite-difference methods, which tend to be computationally expensive and prone to inaccuracies.

The direct optimization procedure couples CFD analysis based on the two-dimensional thin-layer Navier-Stokes equations with a gradient-based numerical optimization technique. The linking mechanism is the sensitivity equation derived from …

A Direct Numerical Simulation Of Vortex Breakdown Induced Tail Buffet, Steven James Massey

Mechanical & Aerospace Engineering Theses & Dissertations

A simulation of tail buffet is presented for a delta wing-vertical tail configuration. Flow conditions are chosen such that the wing primary-vortex cores experience vortex breakdown and the resulting turbulent wake flow impinges on the vertical tail. The dimensions and material properties of the vertical tail are chosen such that the deflections are large enough to insure interaction with the flow, and the natural frequencies are high enough to facilitate a practical computational solution. This multidisciplinary problem is solved sequentially for the fiuid flow, the elastic deformations and the grid displacements. The fluid flow is simulated by time accurately solving …

Continuity Of Metric Projection, Pólya Algorithm, Strict Best Approximation, And Tubularity Of Convex Sets, Robert Huotari, Wu Li

Mathematics & Statistics Faculty Publications

The notion of tubularity of a convex subset, K, of l∞ (n) was originally introduced to study the convergence of the Pólya algorithm. It is shown in the present paper that this geometric condition provides a characterization of thosed closed convex sets onto which the set-valued metric projection is continuous. In the development of this result, Rice′s strict best approximation is characterized in three new ways, and is shown, assuming tubularity of K, to be a continuous selection. The class of sets on which the Pólya algorithm is known to converge is enlarged to include …

Some Triple Sine Series, G. Kerr, G. Melrose, J. Tweed

Mathematics & Statistics Faculty Publications

Two types of triple sine series are investigated. They are reduced to singular integral equations with kernels involving elliptic functions. Closed form solutions are obtained.

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 …

Quasi-Brittle Graphs, A New Class Of Perfectly Orderable Graphs, Stephan Olariu

Computer Science Faculty Publications

A graph G is quasi-brittle if every induced subgraph H of G contains a vertex which is incident to no edge extending symmetrically to a chordless path with three edges in either Hor its complement H¯. The quasi-brittle graphs turn out to be a natural generalization of the well-known class of brittle graphs. We propose to show that the quasi-brittle graphs are perfectly orderable in the sense of Chvátal: there exists a linear order < on their set of vertices such that no induced path with vertices a, b, c, d and edges ab, bc, cd has a < b and d < c.

Optimal Greedy Algorithms For Indifference Graphs, Peter J. Looges, Stephan Olariu

Computer Science Faculty Publications

A fundamental problem in social sciences and management is understanding and predicting decisions made by individuals, various groups, or the society as a whole. In this context, one important concept is the notion of indifference. We characterize the class of indifference graphs, that is, graphs which arise in the process of quantifying indifference relations. In particular, we show that these graphs are characterized by the existence of a special ordering of their vertices. As it turns out, this ordering leads naturally to optimal greedy algorithms for a number of computational problems, including coloring, finding a shortest path between two vertices, …

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.

Hierarchical Flux-Based Thermal-Structural Finite Element Analysis Method, Sandra P. Polesky

Mechanical & Aerospace Engineering Theses & Dissertations

A hierarchical flux-based finite element method is developed for both one- and two-dimensional thermal-structural analyses. Derivation of the finite element equations is presented. The resulting finite element matrices associated with the flux-based formulation are evaluated in closed-form. The hierarchical finite elements include additional degrees of freedom in the approximation of the element variable distributions by the use of nodeless variables. The nodeless variables offer increased solution accuracy without the need for defining actual nodes and rediscretizing the finite element model. Thermal and structural responses obtained using the hierarchical flux-based method are compared with results obtained from a conventional linear finite …

The Morphology Of Convex Polygons, Stephan Olariu

Computer Science Faculty Publications

A simple polygon P is said to be unimodal if for every vertex of P, the Euclidian distance function to the other vertices of P is unimodal. The study of unimodal polygons has emerged as a fruitful area of computational and discrete geometry. We study unimodality properties of a number of special convex polygons from the morphological point of view. In particular, we establish a hierarchy among three classes of convex polygons in terms of their unimodality properties.