764 research outputs found
The Dimensions of Field Theory : From Particles to Strings
This is an editorial summary of the contents of a Book comprising a set of
Articles by acknowledged experts dealing with the impact of Field Theory on
major areas of physics (from elementary particles through condensed matter to
strings), arranged subjectwise under six broad heads. The Book which emphasizes
the conceptual, logical and formal aspects of the state of the art in these
respective fields, carries a Foreword by Freeman Dyson, and is to be published
by the Indian National Science Academy on the occasion of the International
Mathematical Year 2000. The authors and full titles of all the Articles (33)
are listed sequentially (in the order of their first appearance in the
narration) under the bibliography at the end of this Summary, while a few of
the individual articles to appear in the Book are already available on the LANL
internet.Comment: LaTex file, 24 page
Some Applications of Homotopy Perturbation Method
In this thesis paper, I review the basic idea of Homotopy perturbation method (HPM), Modified Homotopy perturbation method (MHPM) and Homotopy perturbation transform method (HPTM). Then apply these on some higher order non-linear problems.Further, I tried to compare the results obtained from Modified homotopy perturbation method with HPM using the Sine-Gordon and fractional Klein-Gordon equation respectively. Homotopy perturbation transform method is the coupling of homotopy perturbation and Laplace transform method. Lastly, I applied the homotopy perturbation and homotopy perturbation transform method for solving linear and non-linear Schrödinger equation
Application of functional renormalization group approach to spin systems and long range models
During the last two decades, functional renormalization group (FRG) emerged as a versatile
class of methodologies which can be used in a variety of cases, ranging from statistical
mechanics models and interacting field theories to condensed matter strongly interacting
systems. The development of FRG techniques from one hand had a methodological side \u2013
aimed at testing different ways to handle with approximations of the functional equations \u2013
and from the other concurred to define FRG as a method to be seriously considered to face
new and poorly/not yet understood problems. Both directions, i.e. improve and test known
techniques and proposed refinements against well known problems and concretely deal with
non-testbed problems, have motivated the work done during the PhD.
This thesis aims at present, provide details and put in perspective results obtained during
my PhD in the last three years. Many of such results have been published, in a more
synthetic way, in peer review journals, others are going to appear online soon. Some others
will probably remain only in this thesis, but together with the published material will be
instrumental to delineate a path leading to a deeper understanding and a (hopefully) clearer
presentation of potentialities and issues of the FRG method
Wavelet Methods for the Solutions of Partial and Fractional Differential Equations Arising in Physical Problems
The subject of fractional calculus has gained considerable popularity and importance during the past three decades or so, mainly due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. It deals with derivatives and integrals of arbitrary orders. The fractional derivative has been occurring in many physical problems, such as frequency-dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PI D controller for the control of dynamical systems etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, control theory, neutron point kinetic model, anomalous diffusion, Brownian motion, signal and image processing, fluid dynamics and material science are well described by differential equations of fractional order. Generally, nonlinear partial differential equations of fractional order are difficult to solve. So for the last few decades, a great deal of attention has been directed towards the solution (both exact and numerical) of these problems. The aim of this dissertation is to present an extensive study of different wavelet methods for obtaining numerical solutions of mathematical problems occurring in disciplines of science and engineering. This present work also provides a comprehensive foundation of different wavelet methods comprising Haar wavelet method, Legendre wavelet method, Legendre multi-wavelet methods, Chebyshev wavelet method, Hermite wavelet method and Petrov-Galerkin method. The intension is to examine the accuracy of various wavelet methods and their efficiency for solving nonlinear fractional differential equations. With the widespread applications of wavelet methods for solving difficult problems in diverse fields of science and engineering such as wave propagation, data compression, image processing, pattern recognition, computer graphics and in medical technology, these methods have been implemented to develop accurate and fast algorithms for solving integral, differential and integro-differential equations, especially those whose solutions are highly localized in position and scale. The main feature of wavelets is its ability to convert the given differential and integral equations to a system of linear or nonlinear algebraic equations, which can be solved by numerical methods. Therefore, our main focus in the present work is to analyze the application of wavelet based transform methods for solving the problem of fractional order partial differential equations. The introductory concept of wavelet, wavelet transform and multi-resolution analysis (MRA) have been discussed in the preliminary chapter. The basic idea of various analytical and numerical methods viz. Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Homotopy Analysis Method (HAM), First Integral Method (FIM), Optimal Homotopy Asymptotic Method (OHAM), Haar Wavelet Method, Legendre Wavelet Method, Chebyshev Wavelet Method and Hermite Wavelet Method have been presented in chapter 1. In chapter 2, we have considered both analytical and numerical approach for solving some particular nonlinear partial differential equations like Burgers’ equation, modified Burgers’ equation, Huxley equation, Burgers-Huxley equation and modified KdV equation, which have a wide variety of applications in physical models. Variational Iteration Method and Haar wavelet Method are applied to obtain the analytical and numerical approximate solution of Huxley and Burgers-Huxley equations. Comparisons between analytical solution and numerical solution have been cited in tables and also graphically. The Haar wavelet method has also been applied to solve Burgers’, modified Burgers’, and modified KdV equations numerically. The results thus obtained are compared with exact solutions as well as solutions available in open literature. Error of collocation method has been presented in this chapter. Methods like Homotopy Perturbation Method (HPM) and Optimal Homotopy Asymptotic Method (OHAM) are very powerful and efficient techniques for solving nonlinear PDEs. Using these methods, many functional equations such as ordinary, partial differential equations and integral equations have been solved. We have implemented HPM and OHAM in chapter 3, in order to obtain the analytical approximate solutions of system of nonlinear partial differential equation viz. the Boussinesq-Burgers’ equations. Also, the Haar wavelet method has been applied to obtain the numerical solution of BoussinesqBurgers’ equations. Also, the convergence of HPM and OHAM has been discussed in this chapter. The mathematical modeling and simulation of systems and processes, based on the description of their properties in terms of fractional derivatives, naturally leads to differential equations of fractional order and the necessity to solve such equations. The mathematical preliminaries of fractional calculus, definitions and theorems have been presented in chapter 4. Next, in this chapter, the Haar wavelet method has been analyzed for solving fractional differential equations. The time-fractional Burgers-Fisher, generalized Fisher type equations, nonlinear time- and space-fractional Fokker-Planck equations have been solved by using two-dimensional Haar wavelet method. The obtained results are compared with the Optimal Homotopy Asymptotic Method (OHAM), the exact solutions and the results available in open literature. Comparison of obtained results with OHAM, Adomian Decomposition Method (ADM), VIM and Operational Tau Method (OTM) has been demonstrated in order to justify the accuracy and efficiency of the proposed schemes. The convergence of two-dimensional Haar wavelet technique has been provided at the end of this chapter. In chapter 5, the fractional differential equations such as KdV-Burger-Kuramoto (KBK) equation, seventh order KdV (sKdV) equation and Kaup-Kupershmidt (KK) equation have been solved by using two-dimensional Legendre wavelet and Legendre multi-wavelet methods. The main focus of this chapter is the application of two-dimensional Legendre wavelet technique for solving nonlinear fractional differential equations like timefractional KBK equation, time-fractional sKdV equation in order to demonstrate the efficiency and accuracy of the proposed wavelet method. Similarly in chapter 6, twodimensional Chebyshev wavelet method has been implemented to obtain the numerical solutions of the time-fractional Sawada-Kotera equation, fractional order Camassa-Holm equation and Riesz space-fractional sine-Gordon equations. The convergence analysis has been done for these wavelet methods. In chapter 7, the solitary wave solution of fractional modified Fornberg-Whitham equation has been attained by using first integral method and also the approximate solutions obtained by optimal homotopy asymptotic method (OHAM) are compared with the exact solutions acquired by first integral method. Also, the Hermite wavelet method has been implemented to obtain approximate solutions of fractional modified Fornberg-Whitham equation. The Hermite wavelet method is implemented to system of nonlinear fractional differential equations viz. the fractional Jaulent-Miodek equations. Convergence of this wavelet methods has been discussed in this chapter. Chapter 8 emphasizes on the application of Petrov-Galerkin method for solving the fractional differential equations such as the fractional KdV-Burgers’ (KdVB) equation and the fractional Sharma-TassoOlver equation with a view to exhibit the capabilities of this method in handling nonlinear equation. The main objective of this chapter is to establish the efficiency and accuracy of Petrov-Galerkin method in solving fractional differential equtaions numerically by implementing a linear hat function as the trial function and a quintic B-spline function as the test function. Various wavelet methods have been successfully employed to numerous partial and fractional differential equations in order to demonstrate the validity and accuracy of these procedures. Analyzing the numerical results, it can be concluded that the wavelet methods provide worthy numerical solutions for both classical and fractional order partial differential equations. Finally, it is worthwhile to mention that the proposed wavelet methods are promising and powerful methods for solving fractional differential equations in mathematical physics. This work also aimed at, to make this subject popular and acceptable to engineering and science community to appreciate the universe of wonderful mathematics, which is in between classical integer order differentiation and integration, which till now is not much acknowledged, and is hidden from scientists and engineers. Therefore, our goal is to encourage the reader to appreciate the beauty as well as the usefulness of these numerical wavelet based techniques in the study of nonlinear physical system
A novel analytical Aboodh residual power series method for solving linear and nonlinear time-fractional partial differential equations with variable coefficients
The goal of this research is to develop a novel analytic technique for obtaining the approximate and exact solutions of the Caputo time-fractional partial differential equations (PDEs) with variable coefficients. We call this technique as the Aboodh residual power series method (ARPSM), because it apply the Aboodh transform along with the residual power series method (RPSM). It is based on a new version of Taylor's series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates the computation of the fractional derivatives each time. As ARPSM just requires the idea of an infinite limit, we simply need a few computations to get the coefficients. This technique solves nonlinear problems without the He's polynomials and Adomian polynomials, so the small size of computation of this technique is the strength of the scheme, which is an advantage over the homotopy perturbation method and the Adomian decomposition method. The absolute and relative errors of five linear and non-linear problems are numerically examined to determine the efficacy and accuracy of ARPSM for time-fractional PDEs with variable coefficients. In addition, numerical results are also compared with other methods such as the RPSM and the natural transform decomposition method (NTDM). Some graphs are also plotted for various values of fractional orders. The results show that our technique is easy to use, accurate, and effective. Mathematica software is used to calculate the numerical and symbolic quantities in the paper
On new and improved semi-numerical techniques for solving nonlinear fluid flow problems.
Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.Most real world phenomena is modeled by ordinary and/or partial differential equations.
Most of these equations are highly nonlinear and exact solutions are not always possible.
Exact solutions always give a good account of the physical nature of the phenomena modeled.
However, existing analytical methods can only handle a limited range of these equations.
Semi-numerical and numerical methods give approximate solutions where exact solutions are
impossible to find. However, some common numerical methods give low accuracy and may lack
stability. In general, the character and qualitative behaviour of the solutions may not always
be fully revealed by numerical approximations, hence the need for improved semi-numerical
methods that are accurate, computational efficient and robust.
In this study we introduce innovative techniques for finding solutions of highly nonlinear
coupled boundary value problems. These techniques aim to combine the strengths of both
analytical and numerical methods to produce efficient hybrid algorithms. In this work, the
homotopy analysis method is blended with spectral methods to improve its accuracy. Spectral
methods are well known for their high levels of accuracy. The new spectral homotopy analysis
method is further improved by using a more accurate initial approximation to accelerate
convergence. Furthermore, a quasi-linearisation technique is introduced in which spectral
methods are used to solve the linearised equations. The new techniques were used to solve
mathematical models in fluid dynamics.
The thesis comprises of an introductory Chapter that gives an overview of common numerical
methods currently in use. In Chapter 2 we give an overview of the methods used in this
work. The methods are used in Chapter 3 to solve the nonlinear equation governing two-dimensional
squeezing flow of a viscous fluid between two approaching parallel plates and the
steady laminar flow of a third grade fluid with heat transfer through a flat channel. In Chapter
4 the methods were used to find solutions of the laminar heat transfer problem in a rotating
disk, the steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipation and
the classical von Kάrmάn equations for boundary layer flow induced by a rotating disk. In
Chapter 5 solutions of steady two-dimensional flow of a viscous incompressible fluid in a
rectangular domain bounded by two permeable surfaces and the MHD viscous flow problem
due to a shrinking sheet with a chemical reaction, were solved using the new methods
Computational and numerical analysis of differential equations using spectral based collocation method.
Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.In this thesis, we develop accurate and computationally efficient spectral collocation-based methods,
both modified and new, and apply them to solve differential equations. Spectral collocation-based
methods are the most commonly used methods for approximating smooth solutions of differential
equations defined over simple geometries. Procedurally, these methods entail transforming the gov
erning differential equation(s) into a system of linear algebraic equations that can be solved directly.
Owing to the complexity of expanding the numerical algorithms to higher dimensions, as reported
in the literature, researchers often transform their models to reduce the number of variables or
narrow them down to problems with fewer dimensions. Such a process is accomplished by making
a series of assumptions that limit the scope of the study. To address this deficiency, the present
study explores the development of numerical algorithms for solving ordinary and partial differential
equations defined over simple geometries. The solutions of the differential equations considered are
approximated using interpolating polynomials that satisfy the given differential equation at se
lected distinct collocation points preferably the Chebyshev-Gauss-Lobatto points. The size of the
computational domain is particularly emphasized as it plays a key role in determining the number
of grid points that are used; a feature that dictates the accuracy and the computational expense of
the spectral method. To solve differential equations defined on large computational domains much
effort is devoted to the development and application of new multidomain approaches, based on
decomposing large spatial domain(s) into a sequence of overlapping subintervals and a large time
interval into equal non-overlapping subintervals. The rigorous analysis of the numerical results con
firms the superiority of these multiple domain techniques in terms of accuracy and computational
efficiency over the single domain approach when applied to problems defined over large domains.
The structure of the thesis indicates a smooth sequence of constructing spectral collocation method
algorithms for problems across different dimensions. The process of switching between dimensions
is explained by presenting the work in chronological order from a simple one-dimensional problem
to more complex higher-dimensional problems. The preliminary chapter explores solutions of or
dinary differential equations. Subsequent chapters then build on solutions to partial differential
i
equations in order of increasing computational complexity. The transition between intermediate
dimensions is demonstrated and reinforced while highlighting the computational complexities in
volved. Discussions of the numerical methods terminate with development and application of a
new method namely; the trivariate spectral collocation method for solving two-dimensional initial
boundary value problems. Finally, the new error bound theorems on polynomial interpolation are
presented with rigorous proofs in each chapter to benchmark the adoption of the different numerical
algorithms. The numerical results of the study confirm that incorporating domain decomposition
techniques in spectral collocation methods work effectively for all dimensions, as we report highly
accurate results obtained in a computationally efficient manner for problems defined on large do
mains. The findings of this study thus lay a solid foundation to overcome major challenges that
numerical analysts might encounter
New Challenges Arising in Engineering Problems with Fractional and Integer Order
Mathematical models have been frequently studied in recent decades, in order to obtain the deeper properties of real-world problems. In particular, if these problems, such as finance, soliton theory and health problems, as well as problems arising in applied science and so on, affect humans from all over the world, studying such problems is inevitable. In this sense, the first step in understanding such problems is the mathematical forms. This comes from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research done involving fractional ordinary or partial differential equations and other relevant topics relating to integer order have attracted the attention of experts from all over the world. Various methods have been presented and developed to solve such models numerically and analytically. Extracted results are generally in the form of numerical solutions, analytical solutions, approximate solutions and periodic properties. With the help of newly developed computational systems, experts have investigated and modeled such problems. Moreover, their graphical simulations have also been presented in the literature. Their graphical simulations, such as 2D, 3D and contour figures, have also been investigated to obtain more and deeper properties of the real world problem
The Effect of Malaysia General Election on Financial Network: An Evidence from Shariah-Compliant Stocks on Bursa Malaysia
Instead of focusing the volatility of the market, the market participants should consider on how the general election affects the correlation between the stocks during 14th general election Malaysia. The 14th general election of Malaysia was held on 9th May 2018. This event has a great impact towards the stocks listed on Bursa Malaysia. Thus, this study investigates the effect of 14th general election Malaysia towards the correlation between stock in Bursa Malaysia specifically the shariah-compliant stock. In addition, this paper examines the changes in terms of network topology for the duration, sixth months before and after the general election. The minimum spanning tree was used to visualize the correlation between the stocks. Also, the centrality measure, namely degree, closeness and betweenness were computed to identify if any changes of stocks that plays a crucial role in the network for the duration of before and after 14th general election Malaysia
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