On The Numerical Solution of Linear Fredholm-Volterra İntegro Differential Difference Equations With Piecewise İntervals

The numerical solution of a mixed linear integro delay differential-difference equation with piecewise interval is presented using the Chebyshev collocation method. The aim of this article is to present an efficient numerical procedure for solving a mixed linear integro delay differential difference equations. Our method depends mainly on a Chebyshev expansion approach. This method transforms a mixed linear integro delay differential-difference equations and the given conditions into a matrix equation which corresponds to a system of linear algebraic equation. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system Maple 10

Fibonacci Collocation Method for Solving High-Order Linear Fredholm Integro-Differential-Difference Equations

A new collocation method based on the Fibonacci polynomials is introduced for the approximate solution of high order-linear Fredholm integro-differential-difference equations with the mixed conditions. The proposed method is analyzed to show the convergence of the method. Some further numerical experiments are carried out to demonstrate the method

Doctor of Philosophy

dissertationThis dissertation consists of two parts that focus on two interrelated areas of Applied Mathematics. The first part explores fundamental properties and applications of functions with values in L-spaces. The second part is connected to Approximation Theory and dives deeper into the analysis of functions with values in specific classes of L-spaces (in particular, L-spaces of sets). In the first project devoted to the theory and numerical methods for the solution of integral equations, we explore linear Volterra and Fredholm integral equations for functions with values in L-spaces (which are generalizations of set-valued and fuzzy-valued functions). In this study, we prove the existence and uniqueness of the solution for such equations, suggest algorithms for finding approximate solutions, and study their convergence. The exploration of these equations is of great importance given the wide variety of their applications in biology (population modeling), physics (heat conduction), and engineering (feedback systems), among others. We extend the aforementioned results of existence and uniqueness to nonlinear equations. In addition, we study the dependence of solutions of such equations on variations in the data. In order to be able to better analyze the convergence of the suggested algorithms for the solutions of integral equations, we develop new results on the approximation of functions with values in L-spaces by adapted linear positive operators (Bernstein, Schoenberg, modified Schoenberg operators, and piecewise linear interpolation). The second project is devoted to problems of interpolation by generalized polynomials and splines for functions whose values lie in a specific L-space, namely a space of sets. Because the structure of such a space is richer than the structure of a general L-space, we have additional tools available (e.g., the support function of a set) which allow us to obtain deeper results for the approximation and interpolation of set-valued functions. We are working on defining various methods of approximation based on the support function of a set. Questions related to error estimates of the approximation of set-valued functions by those novel methods are also investigated

Numerical Approximate Methods for Solving Linear and Nonlinear Integral Equations

Integral equation has been one of the essential tools for various area of applied mathematics. In this work, we employed different numerical methods for solving both linear and nonlinear Fredholm integral equations. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations. Integral equations can be viewed as equations which are results of transformation of points in a given vector spaces of integrable functions by the use of certain specific integral operators to points in the same space. If, in particular, one is concerned with function spaces spanned by polynomials for which the kernel of the corresponding transforming integral operator is separable being comprised of polynomial functions only, then several approximate methods of solution of integral equations can be developed. This work, specially, deals with the development of different wavelet methods for solving integral and intgro-differential equations. Wavelets theory is a relatively new and emerging area in mathematical research. It has been applied in a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for waveform representations and segmentations, time frequency analysis, and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms. Wavelets can be separated into two distinct types, orthogonal and semi-orthogonal. The preliminary concept of integral equations and wavelets are first presented in Chapter 1. Classification of integral equations, construction of wavelets and multi-resolution analysis (MRA) have been briefly discussed and provided in this chapter. In Chapter 2, different wavelet methods are constructed and function approximation by these methods with convergence analysis have been presented. In Chapter 3, linear semi-orthogonal compactly supported B-spline wavelets together with their dual wavelets have been applied to approximate the solutions of Fredholm integral equations (both linear and nonlinear) of the second kind and their systems. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. Convergence analysis of B-spline method has been discussed in this chapter. Again, in Chapter 4, system of nonlinear Fredholm integral equations have been solved by using hybrid Legendre Block-Pulse functions and xiii Bernstein collocation method. In Chapter 5, two practical problems arising from chemical phenomenon, have been modeled as Fredholm- Hammerstein integral equations and solved numerically by different numerical techniques. First, COSMO-RS model has been solved by Bernstein collocation method, Haar wavelet method and Sinc collocation method. Second, Hammerstein integral equation arising from chemical reactor theory has been solved by B-spline wavelet method. Comparison of results have been demonstrated through illustrative examples. In Chapter 6, Legendre wavelet method and Bernoulli wavelet method have been developed to solve system of integro-differential equations. Legendre wavelets along with their operational matrices are developed to approximate the solutions of system of nonlinear Volterra integro-differential equations. Also, nonlinear Volterra weakly singular integro-differential equations system has been solved by Bernoulli wavelet method. The properties of these wavelets are used to reduce the system of integral equations to a system of algebraic equations which can be solved numerically by Newton's method. Rigorous convergence analysis has been done for these wavelet methods. Illustrative examples have been included to demonstrate the validity and applicability of the proposed techniques. In Chapter 7, we have solved the second order Lane-Emden type singular differential equation. First, the second order differential equation is transformed into integro-differential equation and then solved by Legendre multi-wavelet method and Chebyshev wavelet method. Convergence of these wavelet methods have been discussed in this chapter. In Chapter 8, we have developed a efficient collocation technique called Legendre spectral collocation method to solve the Fredholm integro-differential-difference equations with variable coefficients and system of two nonlinear integro-differential equations which arise in biological model. The proposed method is based on the Gauss-Legendre points with the basis functions of Lagrange polynomials. The present method reduces this model to a system of nonlinear algebraic equations and again this algebraic system has been solved numerically by Newton's method. The study of fuzzy integral equations and fuzzy differential equations is an emerging area of research for many authors. In Chapter 9, we have proposed some numerical techniques for solving fuzzy integral equations and fuzzy integro-differential equations. Fundamentals of fuzzy calculus have been discussed in this chapter. Nonlinear fuzzy Hammerstein integral equation has been solved by Bernstein polynomials and Legendre wavelets, and then compared with homotopy analysis method. We have solved nonlinear fuzzy Hammerstein Volterra integral equations with constant delay by Bernoulli wavelet method and then compared with B-spline wavelet method. Finally, fuzzy integro-differential equation has been solved by Legendre wavelet method and compared with homotopy analysis method. In fuzzy case, we have applied two-dimensional numerical methods which are discussed in chapter 2. Convergence analysis and error estimate have been also provided for Bernoulli wavelet method. xiv The study of fractional calculus, fractional differential equations and fractional integral equations has a great importance in the field of science and engineering. Most of the physical phenomenon can be best modeled by using fractional calculus. Applications of fractional differential equations and fractional integral equations create a wide area of research for many researchers. This motivates to work on fractional integral equations, which results in the form of Chapter 10. First, the preliminary definitions and theorems of fractional calculus have been presented in this chapter. The nonlinear fractional mixed Volterra-Fredholm integro-differential equations along with mixed boundary conditions have been solved by Legendre wavelet method. A numerical scheme has been developed by using Petrov-Galerkin method where the trial and test functions are Legendre wavelets basis functions. Also, this method has been applied to solve fractional Volterra integro-differential equations. Uniqueness and existence of the problem have been discussed and the error estimate of the proposed method has been presented in this work. Sinc Galerkin method is developed to approximate the solution of fractional Volterra-Fredholm integro-differential equations with weakly singular kernels. The proposed method is based on the Sinc function approximation. Uniqueness and existence of the problem have been discussed and the error analysis of the proposed method have been presented in this chapte

Abstract book

Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics

Nonlinear Systems

The editors of this book have incorporated contributions from a diverse group of leading researchers in the field of nonlinear systems. To enrich the scope of the content, this book contains a valuable selection of works on fractional differential equations.The book aims to provide an overview of the current knowledge on nonlinear systems and some aspects of fractional calculus. The main subject areas are divided into two theoretical and applied sections. Nonlinear systems are useful for researchers in mathematics, applied mathematics, and physics, as well as graduate students who are studying these systems with reference to their theory and application. This book is also an ideal complement to the specific literature on engineering, biology, health science, and other applied science areas. The opportunity given by IntechOpen to offer this book under the open access system contributes to disseminating the field of nonlinear systems to a wide range of researchers

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

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia