Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials

A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique

Cooperating Newton’s Method with Series Solution Method for Solving System of Linear Mixed Volterra-Fredholm Integral Equation of the Second Kind

In this paper, for the 1st time, we use Newton’s method with series solution method (SSM) for solving system of linear mixed Volterra-Fredholm integral equations of the second kind (SLMVFIE-2). In this work, we use a new technique for studying the numerical solutions for SLMVFIE-2 which is Newton’s method with SSM, first solving this system using SSM and after that cooperation Newton’s method with SSM, suggesting an algorithm for the technique. The new results are achieved and some new theorems have proved for convergence of the method, several numerical examples are tested for illustrating the usefulness of the technique; the numerical results are obtained and compared with the exact solution, computing the least square error, and running times which are criterion of discussion. For programming the technique, we use general Matlab program

Applications of Composite Convolution Operators

The Composite Convolution Operator is an operator which is obtained by composing Convolution operator with Composition operator. Volterra composite convolution operator is a composition of Volterra convolution operator and Composition operator. The Composite Convolution Operators and Composite Convolution Volterra operators have been defined by using the Expectation operator and Radon-Nikodym derivative. In this paper an attempt has been made to investigate applications of Composite Convolution Operators (CCO) in Integral Convolution Type Equations (ICTE). The study may explore a new technique to solve Fredholm Convolution type integral equations and Volterra Convolution type integral equations. Some methods for solving integral convolution type equations by using Composite Convolution Operators have also been studied. For integral convolution type equations, theorems on existence, uniqueness and estimates for solution have also been proved without any restriction for the parameter. In order to determine the solution by the method of successive approximations in this paper, I have made use of the concept of the Resolvent Kernel to obtain Neumann Series. The Banach Contraction Principle has also been used to obtain some results. The method of Variational Iteration has been applied to find out the approximate solution of integral equations by using Composite Convolution Operators. In this paper Numerical Methods have also been adopted for solution of these integral equations. Fourier transform has been used to solve Integral convolution type equations and Laplace transform has been applied to solve Volterra convolution type equations

Post-Processing Techniques and Wavelet Applications for Hammerstein Integral Equations

This dissertation is focused on the varieties of numerical solutions of nonlinear Hammerstein integral equations. In the first part of this dissertation, several acceleration techniques for post-processed solutions of the Hammerstein equation are discussed. The post-processing techniques are implemented based on interpolation and extrapolation. In this connection, we generalize the results in [29] and [28] to nonlinear integral equations of the Hammerstein type. Post-processed collocation solutions are shown to exhibit better accuracy. Moreover, an extrapolation technique for the Galerkin solution of Hammerstein equation is also obtained. This result appears new even in the setting of the linear Fredholm equation. In the second half of this dissertation, the wavelet-collocation technique of solving nonlinear Hammerstein integral equation is discussed. The main objective is to establish a fast wavelet-collocation method for Hammerstein equation by using a \u27linearization\u27 technique. The sparsity in the Jacobian matrix takes place in the fast wavelet-collocation method for Hammerstein equation with smooth as well as weakly singular kernels. A fast algorithm is based upon the block truncation strategy which was recently proposed in [10]. A multilevel augmentation method for the linearized Hammerstein equation is subsequently proposed which further accelerates the solution process while maintaining the order of convergence. Numerical examples are given throughout this dissertation

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

Improved Iterative Solution of Linear Fredholm Integral Equations of Second Kind via Inverse-Free Iterative Schemes

[EN] This work is devoted to Fredholm integral equations of second kind with non-separable kernels. Our strategy is to approximate the non-separable kernel by using an adequate Taylor's development. Then, we adapt an already known technique used for separable kernels to our case. First, we study the local convergence of the proposed iterative scheme, so we obtain a ball of starting points around the solution. Then, we complete the theoretical study with the semilocal convergence analysis, that allow us to obtain the domain of existence for the solution in terms of the starting point. In this case, the existence of a solution is deduced. Finally, we illustrate this study with some numerical experiments.This research was partially supported by a grant of the Spanish Ministerio de Ciencia, Innovacion y Universidades (Ref. PGC2018-095896-B-C21-C22).Gutiérrez, JM.; Hernández-Verón, MÁ.; Martínez Molada, E. (2020). Improved Iterative Solution of Linear Fredholm Integral Equations of Second Kind via Inverse-Free Iterative Schemes. Mathematics. 8(10):1-13. https://doi.org/10.3390/math8101747S113810Argyros, I. K. (1988). On a class of nonlinear integral equations arising in neutron transport. Aequationes Mathematicae, 36(1), 99-111. doi:10.1007/bf01837974Bruns, D. D., & Bailey, J. E. (1977). Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chemical Engineering Science, 32(3), 257-264. doi:10.1016/0009-2509(77)80203-0GANESH, M., & JOSHI, M. C. (1991). Numerical Solvability of Hammerstein Integral Equations of Mixed Type. IMA Journal of Numerical Analysis, 11(1), 21-31. doi:10.1093/imanum/11.1.21Anderson, B. D. O., & Kailath, T. (1971). Some Integral Equations with Nonsymmetric Separable Kernels. SIAM Journal on Applied Mathematics, 20(4), 659-669. doi:10.1137/0120065Ezquerro, J. A., & Hernández, M. A. (2004). A modification of the convergence conditions for Picard’s iteration. Computational & Applied Mathematics, 23(1). doi:10.1590/s0101-82052004000100003Amat, S., Ezquerro, J. A., & Hernández-Verón, M. A. (2013). Approximation of inverse operators by a new family of high-order iterative methods. Numerical Linear Algebra with Applications, 21(5), 629-644. doi:10.1002/nla.1917Barikbin, M. S., Vahidi, A. R., Damercheli, T., & Babolian, E. (2020). An iterative shifted Chebyshev method for nonlinear stochastic Itô–Volterra integral equations. Journal of Computational and Applied Mathematics, 378, 112912. doi:10.1016/j.cam.2020.112912Rabbani, M., Das, A., Hazarika, B., & Arab, R. (2020). Existence of solution for two dimensional nonlinear fractional integral equation by measure of noncompactness and iterative algorithm to solve it. Journal of Computational and Applied Mathematics, 370, 112654. doi:10.1016/j.cam.2019.11265

Fredholm factorization of Wiener-Hopf scalar and matrix kernels

A general theory to factorize the Wiener-Hopf (W-H) kernel using Fredholm Integral Equations (FIE) of the second kind is presented. This technique, hereafter called Fredholm factorization, factorizes the W-H kernel using simple numerical quadrature. W-H kernels can be either of scalar form or of matrix form with arbitrary dimensions. The kernel spectrum can be continuous (with branch points), discrete (with poles), or mixed (with branch points and poles). In order to validate the proposed method, rational matrix kernels in particular are studied since they admit exact closed form factorization. In the appendix a new analytical method to factorize rational matrix kernels is also described. The Fredholm factorization is discussed in detail, supplying several numerical tests. Physical aspects are also illustrated in the framework of scattering problems: in particular, diffraction problems. Mathematical proofs are reported in the pape

Exploring Periodic Orbit Expansions and Renormalisation with the Quantum Triangular Billiard

A study of the quantum triangular billiard requires consideration of a boundary value problem for the Green's function of the Laplacian on a trianglar domain. Our main result is a reformulation of this problem in terms of coupled non--singular integral equations. A non--singular formulation, via Fredholm's theory, guarantees uniqueness and provides a mathematically firm foundation for both numerical and analytic studies. We compare and contrast our reformulation, based on the exact solution for the wedge, with the standard singular integral equations using numerical discretisation techniques. We consider in detail the (integrable) equilateral triangle and the Pythagorean 3-4-5 triangle. Our non--singular formulation produces results which are well behaved mathematically. In contrast, while resolving the eigenvalues very well, the standard approach displays various behaviours demonstrating the need for some sort of ``renormalisation''. The non-singular formulation provides a mathematically firm basis for the generation and analysis of periodic orbit expansions. We discuss their convergence paying particular emphasis to the computational effort required in comparision with Einstein--Brillouin--Keller quantisation and the standard discretisation, which is analogous to the method of Bogomolny. We also discuss the generalisation of our technique to smooth, chaotic billiards.Comment: 50 pages LaTeX2e. Uses graphicx, amsmath, amsfonts, psfrag and subfigure. 17 figures. To appear Annals of Physics, southern sprin

Wiener-Hopf solution for impenetrable wedges at skew incidence

A new Wiener-Hopf approach for the solution of impenetrable wedges at skew incidence is presented. Mathematical aspects are described in a unified and consistent theory for angular region problems. Solutions are obtained using analytical and numerical-analytical approaches. Several numerical tests from the scientific literature validate the new technique, and new solutions for anisotropic surface impedance wedges are solved at skew incidence. The solutions are presented considering the geometrical and uniform theory of diffraction coefficients, total fields, and possible surface wave contribution