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

Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations

Due to the ability of function representation, hybrid functions and wavelets have a special position in research. In this thesis, we state elementary definitions, then we introduce hybrid functions and some wavelets such as Haar, Daubechies, Cheby- shev, sine-cosine and linear Legendre multi wavelets. The construction of most wavelets are based on stepwise functions and the comparison between two categories of wavelets will become easier if we have a common construction of them. The properties of the Floor function are used to and a function which is one on the interval [0; 1) and zero elsewhere. The suitable dilation and translation parameters lead us to get similar function corresponding to the interval [a; b). These functions and their combinations enable us to represent the stepwise functions as a function of floor function. We have applied this method on Haar wavelet, Sine-Cosine wavelet, Block - Pulse functions and Hybrid Fourier Block-Pulse functions to get the new representations of these functions. The main advantage of the wavelet technique for solving a problem is its ability to transform complex problems into a system of algebraic equations. We use the Legendre multi-wavelets on the interval [0; 1) to solve the linear integro-differential and Fredholm integral equations of the second kind. We also use collocation points and linear legendre multi wavelets to solve an integro-differential equation which describes the charged particle motion for certain configurations of oscillating magnetic fields. Illustrative examples are included to reveal the sufficiency of the technique. In linear integro-differential equations and Fredholm integral equations of the second kind cases, comparisons are done with CAS wavelets and differential transformation methods and it shows that the accuracy of these results are higher than them. Homotopy Analysis Method (HAM) is an analytic technique to solve the linear and nonlinear equations which can be used to obtain the numerical solution too. We extend the application of homotopy analysis method for solving Linear integro- differential equations and Fredholm and Volterra integral equations. We provide some numerical examples to demonstrate the validity and applicability of the technique. Numerical results showed the advantage of the HAM over the HPM, SCW, LLMW and CAS wavelets methods. For future studies, some problems are proposed at the end of this thesis

Multistage Homotopy Analysis Method for Solving Nonlinear Integral Equations

In this paper, we present an efficient modification of the homotopy analysis method (HAM) that will facilitate the calculations. We then conduct a comparative study between the new modification and the homotopy analysis method. This modification of the homotopy analysis method is applied to nonlinear integral equations and mixed Volterra-Fredholm integral equations, which yields a series solution with accelerated convergence. Numerical illustrations are investigated to show the features of the technique. The modified method accelerates the rapid convergence of the series solution and reduces the size of work

Hybrid functions approach to solve a class of Fredholm and Volterra integro-differential equations

In this paper, we use a numerical method that involves hybrid and block-pulse functions to approximate solutions of systems of a class of Fredholm and Volterra integro-differential equations. The key point is to derive a new approximation for the derivatives of the solutions and then reduce the integro-differential equation to a system of algebraic equations that can be solved using classical methods. Some numerical examples are dedicated for showing efficiency and validity of the method that we introduce

a collocation method via the quasi affine biorthogonal systems for solving weakly singular type of volterra fredholm integral equations

Abstract Tight framelet system is a recently developed tool in applied mathematics. Framelets, due to their nature, are widely used in the area of image manipulation, data compression, numerical analysis, engineering mathematical problems such as inverse problems, visco-elasticity or creep problems, and many more. In this manuscript we provide a numerical solution of important weakly singular type of Volterra - Fredholm integral equations WSVFIEs using the collocation type quasi-affine biorthogonal method. We present a new computational method based on special B-spline tight framelets and use it to introduce our numerical scheme. The method provides a robust solution for the given WSVFIE by using the resulting matrices based on these biorthogonal wavelet. We demonstrate the validity and accuracy of the proposed method by some numerical examples

Numerical Solution of Linear Volterra-Fredholm Integro-Differential Equations Using Lagrange Polynomials

In this paper, we introduce a numerical method for solving linear Volterra-Fredholm integro-differential Equations (LVFIDE’s) of the first order. To solve these equations, we consider the polynomial approximation from original Lagrange polynomial approximation, barycentric Lagrange polynomial approximation, and modified Lagrange polynomial approximation. Finally, some examples are included to improve the validity and applicability of the techniques. Keywords: Linear Volterra-Fredholm integro-differential equation, Original Lagrange polynomial, Barycentric Lagrange Polynomial, Modified Lagrange polynomial

Collocation Orthonormal Berntein Polynomials method for Solving Integral Equations

In this paper, we use a combination of Orthonormal Bernstein functions on the interval  for degree ,and 6 to produce anew approach implementing Bernstein Operational matrix of derivative as a method for the numerical solution of linear Fredholm integral equations of the second kind and Volterra integral equations. The method converges rapidly to the exact solution and gives very accurate results even by low value of m. Illustrative examples are included to demonstrate the validity and efficiency of the technique and convergence of method to the exact solution. Keywords: Bernstein polynomials, Operational Matrix of Derivative, Linear Fredholm Integral Equations of the Second  Kind and Volterra Integral Equations