Linear Fredholm Integral Equations Research Articles

On solving an n×n system of nonlinear Volterra integral equations by the Newton-Kantorovich method

We consider an nxn system of nonlinear integral equations of Volterra type (nonlinear VIEs) arising from an economic model. By applying the Newton-Kantorovich method to the nonlinear VIEs we linearize them into linear Volterra type integral equations (linear VIEs). Uniqueness of the solution of the system is shown. An idea has been proposed to find the approximate solution by transforming the system of linear VIEs into a system of linear Fredholm integral equations by using sub-collocation points. Then the backward Newton interpolation formula is used to find the approximate solution at the collocation points. Each iteration is solved by the Nystrom type Gauss-Legendre quadrature formula (QF). It is found that by increasing the number of collocation points of QF with fewer iterations, a high accurate approximate solution can be obtained. Finally, an illustrative example is demonstrated to validate the accuracy of the method.

One dimensional nonlinear integral operator with Newton–Kantorovich method

The Newton–Kantorovich method (NKM) is widely used to find approximate solutions for nonlinear problems that occur in many fields of applied mathematics. This method linearizes the problems and then attempts to solve the linear problems by generating a sequence of functions. In this study, we have applied NKM to Volterra-type nonlinear integral equations then the method of Nystrom type Gauss–Legendre quadrature formula (QF) was used to find the approximate solution of a linear Fredholm integral equation. New concept of determining the solution based on subcollocation points is proposed. The existence and uniqueness of the approximated method are proven. In addition, the convergence rate is established in Banach space. Finally illustrative examples are provided to validate the accuracy of the presented method.

On reconstruction of thermalphysic characteristics of functionally graded hollow cylinder

An inverse coefficient problem of thermal conductivity for a functionally graded hollow cylinder is considered. After applying the Laplace transform, the direct thermal conductivity problem is solved by using two methods: (1) based on a reduction to the Fredholm integral equation of the 2nd kind; (2) by means of the Galerkin method. A comparison of the direct problem solving techniques is provided. The nonlinear inverse problem is solved on the basis of an iterative process; at every step of the latter the linear Fredholm integral equation of the 1st kind is solved. Results of the computational experiments on a reconstruction of variation laws of thermal conductivity and specific volumetric heat capacity are presented.

Moving least square for systems of integral equations

This paper aims at developing a meshless approximation based on the Moving Least Square (MLS), in addition to its application for solving a system of linear Fredholm integral equations of the second kind. For the MLS, nodal points are used to approximate the unknown functions. These points can be selected as regular or random from the domain under study. The method is a meshless one, and since it uses a local shape function in the vicinity of each nodal point which is chosen from the support points, it does not depend on the geometry of the domain. In this method, the unknown function is considered as a vector of functions of its kind. An error analysis has also been provided for this new method. A simple and efficient application of this method has also demonstrated through several numerical examples.

Collocation method for linear and nonlinear Fredholm and Volterra integral equations

A collocation procedure is developed for the linear and nonlinear Fredholm and Volterra integral equations, using the globally defined B-spline and auxiliary basis functions. The solution is collocated by cubic B-spline and then the integral equation is approximated by the 5-points Gauss–Turán quadrature formula with respect to the Legendre weight function. Combination of these two approaches is the main idea of this paper to reduce the cost of computation and complexity. The error analysis of proposed numerical method is studied theoretically. Numerical results are given to illustrate the efficiency of the proposed method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.

Using Collocation Fuzzy Wavelet like Operator to Approximate the Solution of Fuzzy Fredholm Integral Equation

In this paper, first we introduce a numerical method for approximating the solution of linear Fuzzy Fredholm Integral Equations of the second kind (FFIE-2). This method is based on the fuzzy wavelet like operator. Then, we discuss and investigate the convergence and error analysis of the proposed method. Finally, to show the accuracy of the proposed method, we present two numerical examples.

NUMERICAL METHODS FOR SOLVING FUZZY FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND

In this article some numerical methods, namely: the Taylor expan- sion method and the Trapezoidal method, have been investigated and imple- mented to solve a fuzzy Fredholm integral equation of the second kind. Conse- quently, we convert a linear fuzzy Fredholm integral equation of the second kind into a linear system of integral equations of the second kind in crisp case. To demonstrate the credibility of these numerical schemes we consider a numerical test example. The numerical results show to be in a close agreement with the exact solution.

A Direct Analytic Method for Solution of Linear Fredholm Integral and Integro-differential Equations of the Second Kind

Abstract: In this paper, a direct analytic method is given for the solution of the linear Fredholm integral and integro-differential equations of the second kind, which is based on the span of the known function, under the action of the operator defined by the kernel. The necessary conditions for using this method are so weak that extends its applicability. The solved xamples show the strength of this method.

Numerical solution of two-dimensional fuzzy Fredholm integral equations of the second kind using triangular functions

The main purpose of this paper is to approximate the solution of linear two-dimensional fuzzy Fredholm integral equations of the second kind (2D-FFIE-2). We use fuzzy two-dimensional triangular functions (2D-TFs) to reduce the 2D–FFIE–2 to a system of linear Fredholm integral equations of the second kind with three variables in crisp. More over, we prove the convergence of the method. Finally we illustrate this method with some numerical examples to demonstrate the validity and applicability of the technique.

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

A Novel Numerical Method of Two-Dimensional Fredholm Integral Equations of the Second Kind

A novel numerical method is developed for solving two-dimensional linear Fredholm integral equations of the second kind by integral mean value theorem. In the proposed algorithm, each element of the generated discrete matrix is not required to calculate integrals, and the approximate integral operator is convergent according to collectively compact theory. Convergence and error analyses of the approximate solution are provided. In addition, an algorithm is given. The reliability and efficiency of the proposed method will be illustrated by comparison with some numerical results.

Numerical solution of linear Fredholm integral equations via two-dimensional modification of hat functions

In this work we approximate the solution of the linear Fredholm integral equations, by means of a new two-dimensional modification of hat functions (2D-MHFs) and a new operational matrix of integration. By this idea, the basic equations will be changed into the associated systems of algebraic equations. Also, an error analysis is provided under several mild conditions. The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.

A new iterative method for solving linear Fredholm integral equations using the least squares method

In this paper, a new iterative method is proposed for solving linear integral equations. This method is based on the LSQR method, an algorithm for sparse linear equations and sparse least squares, reducing the solution of linear integral equations to the solution of a bidiagonal linear system of algebraic equations. A simple recurrence formula is presented for generating the sequence of approximate solutions. Some theoretical properties and error analysis of the new method are discussed. Although the new method can be used for solving the ill-posed first kind integral equations independently, combining of the new method with the method of regularization is presented to solve this kind of integral equations. Also the perturbing effect of the first kind integral equations is analyzed. Some properties and convergence theorem are proposed. Finally, some numerical examples are presented to show the efficiency of the new method.

A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations

This paper suggests a novel multi-parametric homotopy method for systems of Fredholm integral equations. This modified method contains three convergence-control parameters to form an improved homotopy. We present efficient error estimation for the approximate solution. The results of present method are compared with the Adomian decomposition method (ADM), the homotopy perturbation method (HPM) and standard homotopy analysis method results that provides confirmation for the validity of proposed approach. Some test examples are given to clarify the efficiency and high accuracy of the present method. The results reveal that the present method is very effective and convenient.

A computational method based on hybrid of Bernstein and block-pulse functions for solving linear fuzzy Fredholm integral equations system

This paper deals with the solutions of linear fuzzy Fredholm integral equation systems by using a combination of Bernstein and block-pulse functions on the interval [0, 1), that is called hybrid functions. Moreover, the existence of the solution and convergence of the proposed method is proved. Finally, illustrative examples are included in order to demonstrate the accuracy and the convergence of this method.

Half-Sweep Arithmetic Mean Method with High-Order Newton-Cotes Quadrature Schemes to Solve Linear Second Kind Fredholm Equations

The main purpose of this paper is to examine the effectiveness of the Half-Sweep Arithmetic Mean (HSAM) method in solving the dense linear systems generated from the discretization of the linear Fredholm integral equations of the second kind. In addition, the applications of the various orders of closed Newton-Cotes quadrature discretization schemes will be investigate in order to form linear systems. Furthermore, the basic formulation and implementation for the proposed method are also presented. Some illustrative examples are given to point out the efficiency of the proposed method.

Taylor matrix method for solving linear two-dimensional Fredholm integral equations with Piecewise Intervals

This paper contributes an efficient numerical approach, which produces an approximate polynomial solution, for solving linear two-dimensional Fredholm integral equations with piecewise intervals. The main technique is based upon approximating the unknown function (i.e., expected solution) and known functions (including four variable kernel functions and two variable source functions) in terms of their Taylor expansions in a new matrix form. With the aid of these approximations, we convert the main problems into their associated matrix equations. The matrix equations correspond to systems of linear equations with the unknown Taylor coefficients. Moreover, under several mild conditions, error analysis of the proposed method is provided. Numerical examples are included to demonstrate the validity and applicability of the method and the comparisons are made with the existing results.

Solving second kind linear Fredholm integral equations via Quarter-Sweep SOR iterative method

In this paper, we consider the numerical solutions of linear Fredholm integral equations of the second kind. The Quarter-Sweep Successive Over-Relaxation (QSSOR) iterative method is applied to solve linear systems generated from discretization of the second kind linear Fredholm integral equations using quadrature method. In addition, the formulation and implementation of the proposed method to solve the problem are also presented. Numerical tests and comparisons with other existing methods are given to illustrate the effectiveness of the proposed method.

Using the Monte Carlo method to solve integral equations using a modified control variate

One of the numerical methods for solving definite integrals is Monte Carlo (MC) randomized simulation. In this paper we intend to apply a Monte Carlo variance reduction technique to solve the linear Fredholm integral equations, which is based on a modified control variate. Taylor expansion to each subinterval of the objective function is regarded as the modified control variate. And thus we solve the linear Fredholm integral equations of the second kind more accurately. One of the main advantage of the proposed method is reducing the problem caused by the linear system of equations.

Application of Semiorthogonal B-Spline Wavelets for the Solutions of Linear Second Kind Fredholm Integral Equations

In this paper, the linear semiorthogonal compactly supported B-spline wavelets together with their dual wavelets have been applied to approximate the solutions of Fredholm integral equations of the second kind. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of in tegral equations to some algebraic equations. The method is computationally attractive, and application of it has been demonstrated thr ough illustrative examples.