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Articles 1 - 17 of 17
Full-Text Articles in Physical Sciences and Mathematics
Lost At Sea: Introduction To Numerical Methods Through Navigation, R. Corban Harwood
Lost At Sea: Introduction To Numerical Methods Through Navigation, R. Corban Harwood
Faculty Publications - Department of Mathematics
Excerpt: "The ship, El Perdido, was damaged during a storm which knocked out its main and backup power generators. Before the backup generator failed, Captain Miguel Gomez sent a distress call and the crew have been able to keep El Perdido a oat, but the ship is adrift in the Pacific Ocean off the coast of California. Thankfully, a US Coast Guard rescue operation is underway after receiving the distress call. The Coast Guard has El Perdido's last known position and has mapped out the surface water velocities in this area as slope fields for longitude (x) and latitude (y), …
Eigenvalue Dependence Of Numerical Oscillations In Parabolic Partial Differential Equations, R. Corban Harwood
Eigenvalue Dependence Of Numerical Oscillations In Parabolic Partial Differential Equations, R. Corban Harwood
Faculty Publications - Department of Mathematics
This paper investigates oscillation-free stability conditions of numerical methods for linear parabolic partial differential equations with some example extrapolations to nonlinear equations. Not clearly understood, numerical oscillations can create infeasible results. Since oscillation-free behavior is not ensured by stability conditions, a more precise condition would be useful for accurate solutions. Using Von Neumann and spectral analyses, we find and explore oscillation-free conditions for several finite difference schemes. Further relationships between oscillatory behavior and eigenvalues is supported with numerical evidence and proof. Also, evidence suggests that the oscillation-free stability condition for a consistent linearization may be sufficient to provide oscillation-free stability …
Steady And Stable: Numerical Investigations Of Nonlinear Partial Differential Equations, R. Corban Harwood
Steady And Stable: Numerical Investigations Of Nonlinear Partial Differential Equations, R. Corban Harwood
Faculty Publications - Department of Mathematics
Excerpt: "Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation."
Simulating The Spread Of The Common Cold, R. Corban Harwood
Simulating The Spread Of The Common Cold, R. Corban Harwood
Faculty Publications - Department of Mathematics
This modeling scenario guides students to simulate and investigate the spread of the common cold in a residence hall. An example floor plan is given, but the reader is encouraged to use a more relevant example. In groups, students run repeated simulations, collect data, derive a differential equation model, solve that equation, estimate parameter values by hand and through regression, visually evaluate the consistency of the model with their data, and present their results to the class.
1-65-S-Algal Blooms: Algal Blooms Threatening Lake Chapala, R. Corban Harwood
1-65-S-Algal Blooms: Algal Blooms Threatening Lake Chapala, R. Corban Harwood
Faculty Publications - Department of Mathematics
This modeling scenario investigates the massive algal blooms that struck Lake Chapala, Mexico, starting in 1994. After reading a summary of articles written on the incidents, students are guided through the process of creating a first order differential equation from a verbal model of the factors and analyze the nonautonomous ODE using direction field, parameter evaluation, and exact solution computation to fully describe the population behavior. Students are expected to be familiar with the separable method and direction fields. Students will learn building and improving a model from qualitative descriptions, nondimensionalization, evaluating parameters, and how to use DFIELD software to …
Logistics Of Mathematical Modeling-Focused Projects, R. Corban Harwood
Logistics Of Mathematical Modeling-Focused Projects, R. Corban Harwood
Faculty Publications - Department of Mathematics
Projects provide tangible connections to course content and can motivate students to learn at a deeper level. This article focuses on the implementation of projects in both lower and upper division math courses which develop and analyze mathematical models of a problem based upon known data and real-life situations. Logistical pitfalls and insights are highlighted as well as several key implementation resources. Student feedback demonstrate a positive correlation between the use of projects and an enhanced understanding of the course topics when the impact of logistics is reduced. Best practices learned over the years are given along with example project …
An Elementary Proof Of Dodgson's Condensation Method For Calculating Determinants, R. Corban Harwood, Mitch Main, Micah Donor
An Elementary Proof Of Dodgson's Condensation Method For Calculating Determinants, R. Corban Harwood, Mitch Main, Micah Donor
Faculty Publications - Department of Mathematics
In 1866, Charles Ludwidge Dodgson published a paper concerning a method for evaluating determinants called the condensation method. His paper documented a new method to calculate determinants that was based on Jacobi's Theorem. The condensation method is presented and proven here, and is demonstrated by a series of examples. The condensation method can be applied to a number of situations, including calculating eigenvalues, solving a system of linear equations, and even determining the different energy levels of a molecular system. The method is much more efficient than cofactor expansions, particularly for large matrices; for a 5 x 5 matrix, the …
Oscillation-Free Method For Semilinear Diffusion Equations Under Noisy Initial Conditions, R. Corban Harwood, Likun Zhang, V. S. Manoranjan
Oscillation-Free Method For Semilinear Diffusion Equations Under Noisy Initial Conditions, R. Corban Harwood, Likun Zhang, V. S. Manoranjan
Faculty Publications - Department of Mathematics
Noise in initial conditions from measurement errors can create unwanted oscillations which propagate in numerical solutions. We present a technique of prohibiting such oscillation errors when solving initial-boundary-value problems of semilinear diffusion equations. Symmetric Strang splitting is applied to the equation for solving the linear diffusion and nonlinear remainder separately. An oscillation-free scheme is developed for overcoming any oscillatory behavior when numerically solving the linear diffusion portion. To demonstrate the ills of stable oscillations, we compare our method using a weighted implicit Euler scheme to the Crank-Nicolson method. The oscillation-free feature and stability of our method are analyzed through a …
Algae Population Self-Replenishment, R. Corban Harwood
Algae Population Self-Replenishment, R. Corban Harwood
Faculty Publications - Department of Mathematics
This modeling scenario investigates the massive algal blooms that struck Lake Chapala, Mexico, in 1994. After reading a summary of news articles on the incident, students create an ODE system model from a verbal description of the factors, visualize this system using an executable Java applet (PPLANE) to predict overall behavior, and then analyze the nonlinear system using the Jacobian matrix, eigenvalues, phase plane, and feasibility conditions on parameters to fully describe the system behavior. Students are expected to be familiar with systems of differential equations, equilibria, jacobian matrices, and eigenvalues. Students will learn modeling from qualitative descriptions, nondimensionalization, applying …
Two Integration Of Faith And Mathematics Projects For Freshmen Mathematics Majors, Nicholas J. Willis
Two Integration Of Faith And Mathematics Projects For Freshmen Mathematics Majors, Nicholas J. Willis
Faculty Publications - Department of Mathematics
Two projects will be presented that integrate faith and Mathematics in a freshman Introduction to Proofs class at George Fox University. The first project asks students to look at the life of a Christian Mathematician. The focus of this project is to show students that many great mathematicians also had immense faith. The second project asks students to take a close look at their own life. How do they plan to live a life of Christian faith in their chosen profession? Both projects are designed to encourage students to look at their careers in Mathematics as a vocation.
Singular Points Of Reducible Sextic Curves, David A. Weinberg, Nicholas J. Willis
Singular Points Of Reducible Sextic Curves, David A. Weinberg, Nicholas J. Willis
Faculty Publications - Department of Mathematics
No abstract provided.
Operator Splitting Method And Applications For Semilinear Parabolic Partial Differential Equations, R. Corban Harwood
Operator Splitting Method And Applications For Semilinear Parabolic Partial Differential Equations, R. Corban Harwood
Faculty Publications - Department of Mathematics
This dissertation presents a redefined operator splitting method used in solving semilinear parabolic partial differential equations. As one such form, the reaction-diffusion equation is highly prevalent in mathematical modeling. Besides being physically meaningful as a separation of two distinct physical processes in this equation, operator splitting simplifies the solution method in several ways. The super-linear speed-up of computations is a rewarding simplification as it presents great benefits for large-scale systems. In solving these semilinear equations, we will develop a condition for oscillation-free methods, a condition independent of the usual stability condition. This numerical consideration is important to fully embody our …
Lead-Acid Battery Model Under Discharge With A Fast Splitting Method, R. Corban Harwood, Valipuram S. Manoranjan, Dean B. Edwards
Lead-Acid Battery Model Under Discharge With A Fast Splitting Method, R. Corban Harwood, Valipuram S. Manoranjan, Dean B. Edwards
Faculty Publications - Department of Mathematics
A mathematical model of a valve-regulated lead-acid battery under discharge is presented as simplified from a standard electrodynamics model. This nonlinear reaction–diffusion model of a battery cell is solved using an operator splitting method to quickly and accurately simulate sulfuric acid concentration. This splitting method incorporates one-sided approximation schemes to preserve continuity over material interfaces encompassing discontinuous parameters. Numerical results are compared with measured data by calculating battery voltage from modeled acid concentration as derived from the Nernst equation.
Singular Points Of Real Sextic Curves I, David A. Weinberg, Nicholas J. Willis
Singular Points Of Real Sextic Curves I, David A. Weinberg, Nicholas J. Willis
Faculty Publications - Department of Mathematics
A complete classification of the individual types of singular points is given for irreducible real sextic curves. This classification is derived by using the computer algebra system Maple. There are 191 types of singular points for real irreducible sextic curves. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. A significant portion of the proof consists of a sequence of large symbolic computations that can be done nicely using Maple.
Singular Points Of Real Quartic And Quintic Curves, David A. Weinberg, Nicholas J. Willis
Singular Points Of Real Quartic And Quintic Curves, David A. Weinberg, Nicholas J. Willis
Faculty Publications - Department of Mathematics
There are thirteen types of singular points for irreducible real quartic curves and seventeen types of singular points for reducible real quartic curves. This classification is originally due to D. A. Gudkov. There are nine types of singular points for irreducible complex quartic curves and ten types of singular points for reducible complex quartic curves. There are 42 types of real singular points for irreducible real quintic curves and 49 types of real singular points for irreducible real quintic curves. The classification of real singular points for irreducible real quintic curves is originally due to Golubina and Tai. There are …
An Evolutionary Method For The Minimum Toll Booth Problem: The Methodology, Lihui Bai, Matthew T. Stamps, R. Corban Harwood, Christopher J. Kollmann
An Evolutionary Method For The Minimum Toll Booth Problem: The Methodology, Lihui Bai, Matthew T. Stamps, R. Corban Harwood, Christopher J. Kollmann
Faculty Publications - Department of Mathematics
This paper considers the minimum toll booth problem (MINTB) for determining a tolling strategy in a transportation network that requires the least number of toll locations, and simultaneously causes the most efficient use of the network. The paper develops a methodology for using the genetic algorithm to solve MINTB and presents the algorithm GAMINTB. The proposed method is tested and validated through a computational study with six example networks. Additional numerical test discovers some interesting properties for the proposed method, and provides guidelines for further application of the GAMINTB.
Predictive Control Of A Munition Using Low-Speed Linear Theory, Nathan Slegers
Predictive Control Of A Munition Using Low-Speed Linear Theory, Nathan Slegers
Faculty Publications - Biomedical, Mechanical, and Civil Engineering
"Modified linear theory provides reasonable impact predictions at high speeds. However, for typical small UAS mission speeds, less than 20-m/s impact errors were substantial due to large angles of attack and pitch rates. Low-speed linear theory was developed by including higher-order terms involving w and q that modified linear theory neglects. As a result, the angle of attack, pitch, and yaw predictions are significantly improved, leading to accurate impact predictions even at very low speeds. A predictive control scheme was developed to reduce dispersion using control surfaces near the tail. The predictive controller uses low-speed linear theory to rapidly predict …