DSpace Collection: ThesesTheseshttp://ir.library.ui.edu.ng/handle/123456789/4132021-09-26T08:49:47Z2021-09-26T08:49:47ZSTUDIES IN THE THEORY OF MULTIPLIERS WITH APPLICATIONS TO SEMIGROUPS OF OPERATORSADEWOYE, T. O.http://ir.library.ui.edu.ng/handle/123456789/47522019-09-17T13:36:59Z1974-08-01T00:00:00ZTitle: STUDIES IN THE THEORY OF MULTIPLIERS WITH APPLICATIONS TO SEMIGROUPS OF OPERATORS
Authors: ADEWOYE, T. O.
Abstract: This thesis divides naturally into two parts. The first half is a study of the general theory of multipliers, while the second half deals with applications of the theory of multipliers to the theory of semigroups of operators defined on a Banach spice. We study the multiplier problem for an abstract Hilbert space li, and generalise to H certain important results established for La(G)-multipliers (Larsen [12], Hewitt and Ross [8]). A significant result of this study is the Identification of certain projection operators on L2(G) which are, in several respects. Like the translation operators on Ls (G). We also discuss the restricted multiplier problem for the Banach algebra L1(G) of all absolutely integrable complex-valued functions defined on a compact group G, and we obtain results which are analogous to those obtained by Brainerd and Edwards [1] for L1 (G), where G is a locally compact abelian group. In connection with, semigroups of operators, we discuss, in the context of various Banach spaces, the representation of the multipliers which arise semigroups of operators on these Banach spaces. In this respect, we extend the results of Hille and Phillips [10] proved for the circle group (and generalised to compact abelian groups by Olubummo A. and Babalola V.A. [13]) to certain Banach spaces which are not even function spaces. All these results put together provide a good link between the theory of multipliers for a Banach space and the theory of semigroups of operators on the Banach space.
Description: A THESIS IN THE DEPARTMENT OF MATHEMATICS SUBMITTED TO THE FACULTY OF SCIENCE IN PARTIAL FULFIMENT OF THE REQUIREMENTS FOR THE DEGREE OF D0CT0R OF PHILOSOPHY OF THE UNIVERSITY OF IBADAN1974-08-01T00:00:00ZMELLIN TRANSFORM METHOD FOR THE VALUATION OF AMERICAN POWER PUT OPTIONFadugba, S.E.http://ir.library.ui.edu.ng/handle/123456789/40562019-01-22T11:35:13Z2017-05-01T00:00:00ZTitle: MELLIN TRANSFORM METHOD FOR THE VALUATION OF AMERICAN POWER PUT OPTION
Authors: Fadugba, S.E.
Abstract: An
p (Sn
t ; t) =
1
2 i Z c+i1
ci1
K!+1
!(! + 1)
e
1
2n2 2(!2+ 1! 2)(Tt)(Sn
t )!d!
+
rK
2 i Z c+i1
ci1 Z T
t
(Sn
t )! ( Sn
y )!
!
e
1
2n2 2(!2+ 1 ! 2)(yt)dyd!
q
2 i Z c+i1
ci1 Z T
t
(Sn
t )! ( Sn
y )!+1
! + 1
e
1
2n2 2(!2+ 1 ! 2)(yt)dyd!
Description: A Thesis in the Department of Mathematics, Submitted to the Faculty of Science in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY of the UNIVERSITY OF IBADAN2017-05-01T00:00:00ZGALERKIN APPROXIMATION OF A NONLINEAR PARABOLIC INTERFACE PROBLEM ON FINITE AND SPECTRAL ELEMENTSADEWOLE, M. O.http://ir.library.ui.edu.ng/handle/123456789/40542019-01-25T07:59:07Z2017-05-01T00:00:00ZTitle: GALERKIN APPROXIMATION OF A NONLINEAR PARABOLIC INTERFACE PROBLEM ON FINITE AND SPECTRAL ELEMENTS
Authors: ADEWOLE, M. O.
Abstract: Nonlinear parabolic interface problems are frequently encountered in the modelling
of physical processes which involved two or more materials with di erent properties.
Research had focused largely on solving linear parabolic interface problems
with the use of Finite Element Method (FEM). However, Spectral Element Method
(SEM) for approximating nonlinear parabolic interface problems is scarce in literature.
This work was therefore designed to give a theoretical framework for the
convergence rates of nite and spectral element solutions of a nonlinear parabolic
interface problem under certain regularity assumptions on the input data.
A nonlinear parabolic interface problem of the form
ut r (a(x; u)ru) = f(x; u) in
(0; T]
with initial and boundary conditions
u(x; 0) = u0(x) ; u(x; t) = 0 on @
[0; T]
and interface conditions
[u] = 0; a(x; u)
@u
@n
= g(x; t)
was considered on a convex polygonal domain
2 R2 with the assumption that
the unknown function u(x; t) is of low regularity across the interface, where f :
R ! R, a :
R ! R are given functions and g : [0; T] ! H2() \ H1=2()
is the interface function. Galerkin weak formulation was used and the solution
domain was discretised into quasi-uniform triangular elements after which the
unknown function was approximated by piecewise linear functions on the nite
elements. The time discretisation was based on Backward Di erence Schemes
(BDS). The implementation of this was based on predictor-corrector method due
to the presence of nonlinear terms. A four-step linearised FEM-BDS was proposed
and analysed to ease the computational stress and improve on the accuracy of the
ii
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time-discretisation. On spectral elements, the formulation was based on Legendre
polynomials evaluated at Gauss-Lobatto-Legendre points. The integrals involved
were evaluated by numerical quadrature. The linear theories of interface and noninterface
problems as well as Sobolev imbedding inequalities were used to obtain
the a priori and the error estimates. Other tools used to obtain the error estimates
were approximation properties of linear interpolation operators and projection
operators.
The a priori estimates of the weak solution were obtained with low regularity
assumption on the solution across the interface, and almost optimal convergence
rates of O h 1 + 1
j log hj 1=2 and O h2 1 + 1
j log hj in the L2(0; T;H1(
)) and
L2(0; T; L2(
)) norms respectively were established for the spatially discrete scheme.
Almost optimal convergence rates of O k + h 1 + 1
j log hj and
O k + h2 1 + 1
j log hj in the L2(0; T;H1(
)) and L2(0; T; L2(
)) norms were obtained
for the fully discrete scheme based on the backward Euler scheme, respectively
for small mesh size h and time step k. Similar error estimates were obtained
for two-step implicit scheme and four-step linearised FEM-BDS. The solution by
SEM was found to converge spectrally in the L2(0; T; L2(
))-norm as the degree
of the Legendre polynomial increases.
Convergence rates of almost optimal order in the L2(0; T;H1(
)) and
L2(0; T; L2(
)) norms for nite element approximation of a nonlinear parabolic
interface problem were established when the integrals involved were evaluated by
numerical quadratur
Description: A Thesis in the Department of Mathematics, Submitted to the Faculty of Sciences in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY of the UNIVERSITY OF IBADAN2017-05-01T00:00:00Z