James T. Sandefur

Existence and Uniqueness of Solutions of Second Order Nonlinear Differential Equations

A factoring technique is used to prove existence, uniqueness, and continuation properties for solutions to a class of second order semilinear differential equations in a Banach space. These results are then used to derive local and global existence results for a large class of partial differential equations. Among the examples considered are the semilinear versions of the wave equation (possibly damped or strongly damped), the telegraph equation, and the equation of motion for a vibrating plate.Contrary to most techniques, this method does not require commutativity of the operators. An example of this is also given.

Higher order abstract Cauchy problems

Abstract The purpose of this paper is to show the existence and uniqueness of a solution to several types of higher order abstract Cauchy problems in a Banach space. Semigroup methods and spectral theory are used to arrive at these results.

An abstract d'Alembert formula

he abstract d’Alembert formula expresses solutions of the factored linear equation

ASYMPTOTIC EQUIPARTITION OF ENERGY FOR DIFFERENTIAL EQUATIONS IN HILBERT SPACE

\prod_{j = 1}^n {({d / {dt - A_j }})u(t) = 0}

Abstract Equipartition of Energy Theorems

as

Using Self-Similarity to Find Length, Area, and Dimension

u(t) = \sum_{j = 1}^n {u_j (t)}

How Recursion Supports Algebraic Understanding

, where

The amount of non-uniqueness for factored equations with Euler-Poisson-Darboux factors

({d / {dt - A_j }})u_j (t) = 0

Discrete Dynamical Systems: Theory and Applications

for

Generating and using examples in the proving process

j = 1, \cdots ,n