DSpace Collection:
http://localhost:8081/xmlui/handle/123456789/33
2022-08-16T01:56:25ZFOURIER SERIES APPROXIMATION BY LINEAR OPERATORS IN ๐ณ๐-NORM
http://localhost:8081/xmlui/handle/123456789/15065
Title: FOURIER SERIES APPROXIMATION BY LINEAR OPERATORS IN ๐ณ๐-NORM
Authors: Arti
Abstract: In this thesis, we study the degree of approximation of functions belonging to certain
function classes through trigonometric Fourier series using summability methods.
We divide the thesis into six chapters.
Chapter one is an introductory part of the thesis which deals with the upbringing
of approximation theory, basic de nitions and some notations which are used
throughout the thesis. Literature survey and the objective of the work done is also
given in this chapter.
Chapter two is about the approximation of 2 -periodic functions in the weighted
Lipschitz class W(Lp; (t)) (p 1) by almost summability means of their Fourier
series. We also obtain a result on the approximation of conjugate functions through
almost matrix means of their conjugate Fourier series, which in turn improves some
of the previous results. The deviation is measured in the corresponding weighted
norm. We also discuss some corollaries derived from our main results.
Chapter three deals with the approximation of functions by using -method
of summability of conjugate Fourier series. Here we obtained a degree of approximation
of the conjugate function e f, conjugate to a 2 -periodic function f in the
generalized H older space H ;p (0 < 1; p 1) through Borel means of the
conjugate Fourier series. Our result improves some of the previous result.
In the fourth chapter, we obtain an estimate for the degree of approximation of
functions belonging to the generalized Zygmund space Z!
p (p 1) through product
means of Fourier series, which generalizes and improves some of the previous results.
The results are obtain in terms of the moduli of continuity. We also derive some
corollaries from our theorems.
In the fth chapter, we obtain a quantitative estimate of Young's theorem (well
known in the classical Fourier analysis) by using matrix means which generalizes the
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result obtained by Mazhar and Budaiwi [76].
In the sixth chapter, we study the degree of approximation of 2 -periodic functions
of two variables, de ned on T2 := [๔ ; ] [๔ ; ] and belonging to certain
Lipschitz classes, by means of almost Euler summability of their Fourier series. The
degree of approximation so obtained depends on the modulus of continuity associated
with the functions. We also derive some corollaries from our theorems for the
functions of Zygmund classes.2019-04-01T00:00:00ZON WELL-POSEDNESS FOR STOCHASTIC BURGERSTYPE EQUATIONS
http://localhost:8081/xmlui/handle/123456789/15064
Title: ON WELL-POSEDNESS FOR STOCHASTIC BURGERSTYPE EQUATIONS
Authors: Kumar, Vivek
Abstract: In this thesis, different classes of generalized stochastic Burgers-type equations are studied. In
particular, we focus on stochastic Burgers equation and its different types of generalizations.
Here, we have mainly discussed three types of generalized equations: first equation considers
the polynomial types nonlinearity in place of quadratic nonlinearity, second one is equipped with
the fractional differential operator or mixed fractional differential operator in place of Laplacian
operator and last one mainly uses of different types of stochastic noises. The main aspects of
discussion is to show the existence and uniqueness of the solution to the these equations.
The very first goal is to study of the existence of weak solutions of the one-dimensional generalized
stochastic Burgers equation with polynomial nonlinearity perturbed by space time white
noise with Dirichlet boundary conditions and a๔Hรถlder continuous coefficient in the noise term
with a 2
1
2 ;1
. The existence result is established by solving an equivalent martingale problem.
The second aim is to investigate the global existence and uniqueness of solutions to the
one-dimensional generalized stochastic Burgers equation containing a nonlinearity of polynomial
type and perturbed by cylindrical Volterra process having Dirichlet boundary conditions.
In addition, we are also interested to prove that there exists an invariant measure for the same
equation with the quadratic nonlinearity.
As a third task, we investigate the existence and uniqueness of solutions to the fractional
Burgers-type nonlinear stochastic partial differential equation driven by cylindrical fractional
Brownian motion in Hรถlder spaces. The existence proof relies on a finite dimensional Galerkin
approximation. Moreover, the rate of convergence of the Galerkin approximation as well as
fully discretization of solution are also obtained.
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ii
Finally, we address a class of stochastic nonlinear partial differential equation of Burgerstype
driven by pseudo differential operator (D+Da) where Da ๔(๔D)
a2
with a 2 (0;2) and
which is perturbed by the fractional Brownian sheet. The existence and uniqueness of an Lpvalued
(local) solution is established for the initial boundary valued problem to this equation2019-01-01T00:00:00ZEXISTENCE OF SOLUTIONS AND APPROXIMATE CONTROLLABILITY OF SOME EVOLUTION EQUATIONS
http://localhost:8081/xmlui/handle/123456789/15063
Title: EXISTENCE OF SOLUTIONS AND APPROXIMATE CONTROLLABILITY OF SOME EVOLUTION EQUATIONS
Authors: Meraj, Arshi
Abstract: The work presented in this thesis deals with the investigation of the existence of
mild solutions and approximate controllability of some fractional and integer order
di erential and integro-di erential equations. To prove our results, we will use semigroup
theory, evolution system, xed point theorems, fractional calculus, measure of
noncompactness, basic theory of functional analysis, and monotone iterative technique.
The present work consists of the following eight chapters.
Chapter 1 contains a brief introduction to the problems which are discussed
in the consecutive chapters and provides a motivational background to study the
problems which are discussed in this thesis. Further, it contains a review of relevant
literature and an outline of the thesis.
Chapter 2 contains some basic concepts of fractional calculus, functional analysis,
semigroup theory and measure of noncompactness that will be required in the
subsequent chapters.
Chapter 3 concerns with the study of a fractional nonlocal neutral integrodi
erential equation having
ux type integral boundary conditions. The existence
and uniqueness results are proved by using Banach and Leray-Schauder nonlinear
alternative xed point theorems.
Chapter 4 contains fractional integro-di erential equations having non-instantaneous
impulses. The existence result is obtained by the help of xed point theorem and
iii
noncompact semigroup.
Chapter 5 consists of fractional nonlocal semilinear integro-di erential equations
having impulsive conditions for which the impulses are not instantaneous. The
approximate controllability is proved via semigroup theory, Kuratowski measure of
noncmpactness and -set contractive xed point technique, without imposing the
condition of Lipschitz continuity on nonlinear term as well as the condition of compactness
on impulsive functions and nonlocal function.
Chapter 6 contains deformable fractional di erential equations. The results
of existence and approximate controllability are obtained via semigroup theory,
Schauder and Banach xed point technique.
Chapter 7 considers non-autonomous semilinear di erential equations having
nonlocal conditions. The existence and uniqueness are obtained via monotone iterative
method with the upper and lower solutions in an ordered complete norm space,
using evolution system and measure of noncompactness.
Chapter 8 extends the results of chapter 7 for non-autonomous integro-di erential
equations having nonlocal conditions.
The relevant references are appended at the end.2019-02-01T00:00:00ZVARIANTS OF GREY WOLF OPTIMIZER AND SINE COSINE ALGORITHM FOR GLOBAL OPTIMIZATION AND THEIR APPLICATIONS
http://localhost:8081/xmlui/handle/123456789/15062
Title: VARIANTS OF GREY WOLF OPTIMIZER AND SINE COSINE ALGORITHM FOR GLOBAL OPTIMIZATION AND THEIR APPLICATIONS
Authors: Gupta, Shubham
Abstract: Grey Wolf Optimizer (GWO) and Sine Cosine Algorithm (SCA) are recently developed
population based metaheuristic algorithms to solve global optimization problems. The GWO is
inspired by the social and leadership behaviour of grey wolves, and the SCA is designed from the
inspiration of sine and cosine trigonometric functions. Although these algorithms are recently
developed, their effectiveness and advantages are demonstrated in various real world applications
like feature selection, thresholding, multi objective optimization, load dispatch problem in
electrical engineering, clustering and training of neural network etc.
The aim of this PhD Thesis is to propose some modified variants of the classical GWO and
classical SCA which are more effective and reliable in terms of search strategy and solution
accuracy of the optimization problems. To achieve these objectives, in the Thesis, First a modified
variant of classical GWO called RW-GWO is introduced which improves the exploration as well
as exploitation ability of the wolves in a grey wolf pack by introducing two different strategies. In
the first strategy, a new search equation based on random walk search mechanism is introduced
for the leading hunters, and in second, a greedy selection is applied at the end of each iteration
corresponding to each wolf between its current and previous state. The random walk search
strategy focuses on enhancing the exploration and exploitation ability of leading guidance and
greedy selection preserves the discovered promising areas of the search space. The performance
of the RW-GWO algorithm is analyzed and compared with classical GWO on IEEE CEC 2014
benchmark set of unconstrained optimization problems. The numerical results of these test
problems demonstrate the superior search ability of proposed algorithm as compared to classical
GWO.
Next, another variant of the classical GWO called Memory-based Grey Wolf Optimizer (mGWO)
is introduced. The mGWO algorithm utilizes the personal best history of individual wolves to
enhance the collaborative strength of grey wolf pack through modified encircling and hunting
mechanism. The mGWO also integrates the personal best guidance during the search to share the
available best knowledge regarding the search space among the individual search agents. Hence,
the leading and personal best guidance together perform the search process in the mGWO. The
evaluation of the proposed mGWO is performed on IEEE CEC 2014 benchmark set of
unconstrained problems. The numerical results of these test problems demonstrate the better search
ability of proposed algorithm as compare to the classical GWO in all the category of optimization
problems such as unimodal, multimodal, composite and hybrid functions. The comparison
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between the RW-GWO and mGWO concludes that RW-GWO can be preferred for the unimodal
and composite problems and for the multimodal and hybrid problems mGWO can be preferred.
To improve the search accuracy of candidate solutions, a new variant of classical SCA called m-
SCA is proposed in the Thesis which is based on opposition-based learning and modified position
update mechanism. The opposition-based learning is used to generate the opposite candidate
solutions so that the stagnation at local optima can be avoided. The jumping rate which allows the
algorithm to perform the opposition-based learning phase in the algorithm is fixed to a low value
to keep the balance between exploration and exploitation. The search equation of classical SCA is
modified based on the cognitive component to reduce the inefficient diversity of search agents and
to maintain the balance between exploration and exploitation during the search. The performance
of the m-SCA is analyzed and compared with classical SCA on unconstrained benchmark
problems given in IEEE CEC 2014. The analysis of the results demonstrates the superior search
ability of the m-SCA as compared to classical SCA on all category of problems such as unimodal,
multimodal, composite and hybrid benchmark problems.
Next, another modified variant of classical SCA called ISCA is introduced which enhances the
performance of the classical SCA based on the personal best history of candidate solutions,
crossover operator and modified position update mechanism. In the ISCA, the greedy selection is
also employed for each candidate solution between its current and previous state to avoid its
divergence from discovered promising search areas. The performance evaluation of the proposed
algorithm is performed on IEEE CEC 2014 benchmark suite of unconstrained optimization
problems. The numerical results of these test problems demonstrate the superior search ability of
proposed algorithm as compared to the classical SCA in all category of benchmark optimization
problems. The comparison between the m-SCA and ISCA concludes that ISCA can be preferred
for the unimodal, multimodal and hybrid problems and for the composite problems both the
algorithms are very competitive to each other.
Further, the performance of classical versions of GWO and SCA, and their proposed variants
called RW-GWO, mGWO, m-SCA and ISCA is evaluated on constrained optimization problems.
The constrained versions of these algorithms are designed by introducing a simple constraint
handling mechanism based on the constraint violation. The constrained benchmark problems given
in IEEE CEC 2006 are used for experimentation. The analysis on these problems demonstrate the
better search ability of the mGWO algorithm than the classical GWO and RW-GWO algorithms.
Similarly, the proposed ISCA algorithm shows their better search ability to solve constrained
optimization problems as compared to the classical SCA and m-SCA.
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In order to analyze the applicability of the classical GWO, classical SCA and their proposed
variants, an unconstrained and nonlinear optimization problem which arises in the field of image
processing is selected. The problem is defined to determine the optimal thresholds for image
segmentation in grey images. To find the optimum thresholds for an image, Otsuโs between-class
variance criterion is employed as the fitness function. Nine benchmark images are used for
experimentation and several statistical measures are used for comparison. The analysis of results
ensure that the proposed improved variant RW-GWO and mGWO perform better than classical
GWO, classical SCA, m-SCA and ISCA algorithms.
Next, the classical GWO, classical SCA and their proposed variants called RW-GWO, mGWO,
m-SCA and ISCA are implemented on another real-life application which is unconstrained in
nature and arises in the field of electrical engineering. The objective of this problem is to determine
the optimal setting for the proper coordination of overcurrent relays. The IEEE 3, 4, 6, and 14-bus
systems are used for experimentation and validation. The comparison of results demonstrate the
better search efficiency and solution accuracy of the proposed RW-GWO algorithm than all other
variants of GWO and SCA and their classical versions in finding the optimal setting for overcurrent
relays.
Finally, the Thesis is concluded with the limitations and scope of the proposed algorithms. Later
it suggests future scope and some new directions of research in this area.2019-07-01T00:00:00Z