Arising from an investigation in Hydrodynamics, the Korteweg-de Vries equation demonstrates existence of nonlinear waves that resume their profile after interaction. In this thesis, the classical equations governing wave motion are the starting point for the development of an analogue of the KdV that describes the evolution of a wave...
A nonlinear wave equation is developed, modeling the evolution in time of shallow water waves over a variable topography. As the usual assumptions of a perfect fluid and an irrotational flow are not made, the resulting model equation is dissipative due to the presence of a viscous boundary layer at...
A function translator is presented which was designed for
interactive programs which allow functions to be defined on-line. The
translator handles functions which are specified by a formula and
functions which are specified as the solution to a system of differential
equations.
Interval arithmetic is applied to the problem of obtaining
rigorous solutions to integral equations on a computer. The
integral equations considered are the linear Fredholm equation of
the second kind and the nonlinear Urysohn equation. Techniques are
presented which enable the computer to find an approximate
solution, prove the existence...
This paper is about the computation of the stresses on a rigid body from a knowledge
of the far field velocities in exterior Stokes and Oseen flows. The surface of the
body is assumed to be bounded and smooth, and the body is assumed to move with
constant velocity. We...
A numerical solution to Hodgkin and Huxley's partial differential
system for the propagated action potential is presented. In
addition a three dimensional demonstration of the absolute refractory
period is given. Lastly, theoretical evidence supporting
Rushton's hypothesis is presented.
In 1974 Davey and Stewartson used a multi-scale analysis to derive a coupled
system of nonlinear partial differential equations which describes the evolution of a
three dimensional wave packet in water of a finite depth. This system of equations
is the closest integrable two dimensional analog of the well-known one...
The thesis discusses stability of procedures based on linear
computing formulas for numerical integration of an ordinary first-order
differential equation. The theorems are proved: (1) If the
procedure is asymptotically stable it is stable for small positive step
size if the Lipschitz number is negative; (2) Relative stability always
exists...
The general theory of characteristics is reviewed for hyperbolic
partial differential equations of n independent variables. The
application of the theory of characteristics is made to unsteady, two-dimensional, rotational, inviscid flows; unsteady, two-dimensional,
irrotational, inviscid flows; and unsteady, axial symmetric, inviscid
flows. The characteristic surfaces and the compatibility relations
are...
A global solution is presented that accurately accounts for the
singular behavior at all irregular points. The linearized boundary
value problem in a semi-infinite strip in the physical plane is
transformed into a smooth unit disk by two successive conformal
mappings. The global solution results from a Fredholm integral
equation...
The main result of this paper is a proof of the existence of a solution
generated by a method for the variational assimilation of observational data
into the two-dimensional, incompressible Euler equations. The data are
assumed to be given by linear (measurement) functionals acting on the space
of functions representing...
In this dissertation, we investigate three numerical methods for inverting the Laplace transform. These methods are all based on the trapezoidal-type approximations to the Bromwich integral. The first method is the direct integration method: It is a straightforward application of the trapezoidal rule to the Bromwich integral, followed by convergence...
A numerical method based on the the method of characteristics for hyperbolic systems
of partial differential equations in four independent variables is developed and used
for solving time domain Maxwell's equations. The method uses the characteristic
hypersurfaces and the characteristic conditions to derive a set of independent equations
relating the...
A numerical technique to compute the time domain response of multiconductor lossy
uniform and nonuniform lines terminated in general nonlinear elements is presented. The
technique is based on the generalized method of characteristics. The method transforms the
original system of transmission line equations into a system of ordinary differential
equations....
This thesis is on the existence and uniqueness of weak solutions to the Navier-Stokes equations in R3 which govern the velocity of incompressible fluid with viscosity ν. The solution is obtained in the space of tempered distributions on R3 given an initial condition and forcing data which are dominated by...
Two new concepts have been explored in solving the neutron
diffusion equation in one and two dimensions. At the present time,
the diffusion equation is solved using source iterations. These
iterations are performed in a mathematical form which has a great
deal of physical significance. Specifically, the neutron production
term...
This thesis contains three manuscripts addressing the application of stochastic processes to the analysis and solution of partial differential equations (PDEs) in mathematical physics.
In the first manuscript, one dimensional diffusion and Burgers equation are considered. The Fourier transform of the solution to each PDE is represented as the expected...
Accurate modeling and simulation of wave propagation in dispersive dielectrics such as water, human tissue and sand, among others, has a variety of applications. For example in medical imaging, electromagnetic waves are used to interrogate human tissue in a non-invasive manner to detect anomalies that could be cancerous. In non-destructive...