Reza Mollapourasl

NUMERICAL SOLUTION OF FREDHOLM INTEGRAL EQUATION OF THE FIRST KIND WITH COLLOCATION METHOD AND ESTIMATION OF ERROR BOUND

Fredholm integral equation of the first kind is one of the inverse problems that arise in many engineering fields such as image processing and electromagnetic. Most inverse problems are ill-posed, so that solving discretized system of such problems with large condition number has a lot of difficulties. In this paper with using wavelet basis and collocation method the integral equation reduce to a linear system of equation. Then for solving the system we use the CG method. Furthermore, we get an estimation of error bound for this method.

Convergence of numerical solution of the Fredholm integral equation of the first kind with degenerate kernel

Fredholm integral equation of the first kind is one of the ill posed problems since in the operator form of integral equation the integral operator does not have bounded inverse. In this article we consider integral equation of the first kind with degenerate kernel which has bounded inverse. We use collocation method by wavelet families for numerical solving of the equation. Then we prove the convergence for the numerical method. We use conjugate gradiant method for solving the system of linear equations after discretizing the integral equation. For showing efficiency of the method some examples are used.

Numerical solution of Volterra type integral equation of the first kind with wavelet basis

Volterra type integral equations are appeared in many engineering fields, so that, we select Volterra integral equation of the first kind and wavelets as basis functions to estimate a solution for this kind of equations. In this procedure, we use collocation method as a projection method to convert integral equation to the system of linear equations. Finally, some numerical examples indicate the accuracy of this method.

Fixed point method for solving nonlinear quadratic Volterra integral equations

We consider a paper of Banas and Sadarangani (2008) [11] which deals with monotonicity properties of the superposition operator and their applications. An application of the monotonicity properties is to study the solvability of a quadratic Volterra integral equation. In this paper, we prepare an efficient numerical technique based on the fixed point method and quadrature rules to approximate a solution for quadratic Volterra integral equation. Then convergence of numerical scheme is proved by some theorems and some numerical examples are given to show applicability and accuracy of the numerical method and guarantee the theoretical results.

Computational projection methods for solving Fredholm integral equation

In this paper first, Fredholm integral equation of the first kind is introduced. Then we decide to solve this kind of equations by projection methods such as Galerkin and collocation methods. Then we use Haar Wavelet as a base function to estimate the solution of the integral equation of the first kind. Finally by some numerical examples the efficiency of these methods are discussed.

On the solution of a nonlinear integral equation on the basis of a fixed point technique and cubic B-spline scaling functions

In this paper we apply the fixed point method to solve some nonlinear functional Volterra integral equations which appear in many physical, chemical, and biological problems. In each iteration of this method, cubic semi-orthogonal compactly supported B-spline wavelets are used as basis functions to approximate the solution. Also, the convergence of this numerical method is investigated and some examples are presented to show the accuracy and convergence of the method.

Convergence analysis of Sinc-collocation methods for nonlinear Fredholm integral equations with a weakly singular kernel

In this paper, the efficient methods are proposed for solving nonlinear Fredholm integral equations with a weakly singular kernel. By using Sinc approximation with the single exponential (SE) and double exponential (DE) transformations, the methods are developed. Sinc approximation has considerable advantages. Approximation by Sinc functions handles singularities in the problem and such approximations yield rapidly convergent schemes for solving the problem. Furthermore, convergence of proposed methods is discussed by preparing the theorems which show exponential convergence and guarantee the applicability of those. Finally, some numerical examples are presented to illustrate efficiency and accuracy of the numerical schemes.

An efficient numerical scheme for a nonlinear integro-differential equations with an integral boundary condition

Nonlinear functional integro-differential equations with an integral boundary condition is appeared in chemical engineering, underground water flow and population dynamics phenomena and other field of physics and mathematical chemistry. So, this paper presents a powerful numerical approach based on an iterative technique and Sinc quadrature to estimate a solution for this equation. Then convergence of this technique is discussed by preparing a theorem which guarantees the applicability of that. Finally, some numerical examples are given to confirm efficiency and accuracy of the numerical scheme.

Convergence of sinc approximation for Fredholm integral equation with degenerate kernel

Purpose – The purpose of this paper is to discuss a numerical method for solving Fredholm integral equations of the first kind with degenerate kernels and convergence of this numerical method.Design/methodology/approach – By using sinc collocation method in strip, the authors try to estimate a numerical solution for this kind of integral equation.Findings – Some numerical results support the accuracy and efficiency of the stated method.Originality/value – The paper presents a method for solving first kind integral equations which are ill‐posed.

On solution of functional integral equation of fractional order

The aim of this paper is to investigate existence and stability of the solution of the functional integral equations of fractional order arising in physics, mechanics and chemical reactions. These equations are considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval R + . The main tools used in our considerations are the concept of a measure of noncompactness and the classical Schauder fixed point theorem. Also, the numerical method is employed successfully for solving these functional integral equations of fractional order.