Ordinary Differential Equations and Applied Dynamics | Open Access Articles

Gradient Estimates And The Fundamental Solution For Higher-Order Elliptic Systems With Lower-Order Terms, Michael J. Duffy Jr. Jul 2021

Gradient Estimates And The Fundamental Solution For Higher-Order Elliptic Systems With Lower-Order Terms, Michael J. Duffy Jr.

Graduate Theses and Dissertations

Here we generalize the higher-order divergence-form elliptic differential equations studied by Barton in [4] by the inclusion of certain lower-order terms. The methods used here compare to those used in [4], with the addition of further Sobolev-type estimates to handle included lower-order terms. In section 3 we derive a Caccioppoli inequality in which we bound the L2 norm of the mth order gradient, in terms of the L2 norm of the solution. In section 5 we adapt some of the ideas from [9] to derive Lp bounds on gradients of solutions as a substitute for a reverse Holder inequality. Finally …

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Lecture 07: Nonlinear Preconditioning Methods And Applications, Xiao-Chuan Cai Apr 2021

Lecture 07: Nonlinear Preconditioning Methods And Applications, Xiao-Chuan Cai

Mathematical Sciences Spring Lecture Series

We consider solving system of nonlinear algebraic equations arising from the discretization of partial differential equations. Inexact Newton is a popular technique for such problems. When the nonlinearities in the system are well-balanced, Newton's method works well, but when a small number of nonlinear functions in the system are much more nonlinear than the others, Newton may converge slowly or even stagnate. In such a situation, we introduce some nonlinear preconditioners to balance the nonlinearities in the system. The preconditioners are often constructed using a combination of some domain decomposition methods and nonlinear elimination methods. For the nonlinearly preconditioned problem, …

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Design Of Orbital Maneuvers With Aeroassisted Cubesatellites, Stephanie Clark May 2012

Design Of Orbital Maneuvers With Aeroassisted Cubesatellites, Stephanie Clark

Graduate Theses and Dissertations

Recent advances within the field of cube satellite technology has allowed for the possible development of a maneuver that utilizes a satellite's Low Earth Orbit (LEO) and increased atmospheric density to effectively use lift and drag to implement a noncoplanar orbital maneuver. Noncoplanar maneuvers typically require large quantities of propellant due to the large delta-v that is required. However, similar maneuvers using perturbing forces require little or no propellant to create the delta-v required. This research reported here studied on the effects of lift on orbital changes, those of noncoplanar types in particular, for small satellites without orbital maneuvering thrusters. …

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An Analysis Of Unsolvable Linear Partial Differential Equations Of Order One, Laura J. Fields Jan 2001

An Analysis Of Unsolvable Linear Partial Differential Equations Of Order One, Laura J. Fields

Inquiry: The University of Arkansas Undergraduate Research Journal

It is difficult to underestimate the importance of differential equations in understanding the physical world. These equations, involving not just simple variables like temperature, speed or mass, but also the derivatives, i.e. the rate of change of these variables, are found in nearly every branch of science. Until the mid 20th century, all such equations were thought to be solvable. This was based on the discovery by Leonard Euler that certain differential equations, called ordinary differential equations (ODEs), are indeed always solvable. While ODEs deal with simple conditions, under which some quantity changes with some other quantity and its derivatives, …

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Moment-Generating Functions And Laplace Transforms, James F. Miller Jan 1951