THE INITIAL VALUE PROBLEM FOR THE SEQUENCE OF GENERALIZED KORTEWEG-DE VRIES EQUEATIONS
THE INITIAL VALUE PROBLEM FOR THE SEQUENCE OF GENERALIZED KORTEWEG-DE VRIES EQUEATIONS
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Date
1987-11
Authors
SALIU, A. YUSUF
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Abstract
Interest in nonlinear dispersive wave equations
focussed recently on the simplest model equation of
this type, namely,
u.t = uux + ux xx,'
where subscripts denote partial differentiations.
Kbrteweg and de-Vries first derived the equation in
their study of long water waves in a (relatively shallow)
channel. Recently, this equation has been derived in
plasma physics and in studies of anharmonic (nonlinear)
lattices. Existence and uniqueness of solutions of the
above equation for appropriate initial and boundary
conditions have recently been proved by Sjoberg.
This dissertation focusses on the initial value
problem for the sequence of generalized Korteweg de-Vries
equations, namely,
We have shown that for each m, solutions of the
generalized Korteweg-de-Vries equations exist for all
time and are uniquely determined by arbitrary initial
values. The entire work has been divided into four
chapters. The first chapter covers the introduction,
back-ground and definitions of some basic terms. In
this chapter, we have also formulated, more precisely,
our main results, in a theorem.
In chapter II, we stated and proved lemmas relating
the various norms. Also in this chapter, an priori
bound was obtained using the relationships among the
norms.
Chapter III concentrates on the proof of the
existence of a global solution to our initial value
problem. Here, we proved the first part of the theorem
formulated in chapter I.
The last chapter covers the proof of uniqueness of
solution of the initial value problem started in chapter
III. We have shown in this chapter that given an
appropriate intial value the solution can be determined
uniquely for all time.
Notation: Theorems, lemmas, definitions and
equations are numbered decimally within the section
and the number of the chapter is prefixed. Equations
are indicated by a parenthesis :- (1.3.2) and a
definition by a bracket [1,3.2]. A theorem or lemma
is indicated without parenthesis or bracket :-
1.3.2. A single number in a bracket like [3] refers
to the bibliography.
Description
A THESIS SUBMITTED TO THE POST-GRADUATE
SCHOOL, AHMADU BELLO UNIVERSITY ZARIA, IN
PARTIAL FULFILMENT OF THE REQUIREMENTS FOR
THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICS
DEPARTMENT OF MATHEMATICS
FACULTY OF SCIENCE
AHMADU BELLO UNIVERSITY
ZARIA, (NIGERIA)
NOV. 1987
Keywords
INITIAL,, VALUE,, PROBLEM,, SEQUENCE,, GENERALIZED,, KORTEWEG-DE VRIES, EQUEATIONS.