On Some Integral Equations with Carleman Kernel (original) (raw)
Journal of Mathematics and Statistics, 2011
In this study, numerical solution for the Fredholm integral equation of the second kind with Cauchy singular kernel is presented. Approach: The Chebyshev polynomials of the second kind are used to approximate the unknown function. Results: Numerical results are given to show the accuracy of the present numerical solution. Conclusion: The present numerical solution to the Fredholm integral equation of the second kind with Cauchy kernel is accurate.
Algebraic Kernel Method for Solving Fredholm Integral Equations
International Frontier Science Letters, 2016
In this paper, we study the exact solution of linear Fredholm integral equations using some classical methods including degenerate kernel method and Fredholm determinants method. We propose an analytical method for solving such integral equations. This work has some goals related to suggested technique for solving Fredholm integral equations. The primary goal gives analytical solutions of such equations with minimum steps. Another goal is to compare the suggested method used in this study with classical methods. The final goal is that the propose method is an explicit formula that can be studied in detail for non-algebraic function kernels by using Taylor series expansion and for system of Fredholm integral equations.
Fredholm Integral Equations of the Second Kind (General Kernel
In Chap. 1, we conducted a thorough examination of the Fredholm integral equation of the second kind for an arbitrary complex parameter λ , assuming that the free term f (x) is complex-valued and continuous on the interval [a, b] and that the kernel K(x,t) is complex-valued, continuous, and separable on the square Q(a, b) = {(x,t) : [a, b] × [a, b]}. We stated the four Fredholm theorems and the Fredholm Alternative Theorem which provide for the construction of all possible solutions to the equation under these assumptions. A question naturally arises: What, if anything, can be proven if K(x,t) is a general kernel, i.e., an arbitrary kernel that is only assumed to be complex-valued and continuous? In this chapter, we will answer this question completely by proving that all of the Fredholm theorems continue to hold in this eventuality. In Sect. 2.1, we present several tools of the trade which are indispensible for the comprehension of the material in this chapter. In Sects. 2.2 and 2.3, we use these tools to show that the Fredholm integral equation of the second kind with a general kernel has a unique solution if the product of the parameter λ and the " size " of the kernel is small. In Sect. 2.4, we prove the validity of the Fredholm theorems for unrestricted λ and a general kernel. In Sect. 2.5, we show how to construct the resolvent kernel that appears in the solution to the integral equation recursively. In Sect. 2.6, we introduce numerical methods for producing an approximation to the solution of a Fredholm integral equation. These methods are necessary due to the inherent computational difficulties in constructing the resolvent kernel.
On the solution of Fredholm-Volterra integral equation with discontinuous kernel in time
2014
The existence of a unique solution of Fredholm-Volterra integral equation (F-VIE) under certain conditions, are discussed and proved. The Fredholm integral term (FIT) is considered in position with continuous kernel, while the Volterra integral term (VIT) in time with singular kernel. Using a numerical method, the F-VIE is transformed to a linear system of Volterra integral equations (LSVIEs). Then after using Toeplitz matrix method (TMM), we have a linear algebraic system (LAS). Finally, two applications are given, numerical results are obtained, and the error, in each case, is calculated .
A STUDY OF SOME EFFECTIVE TECHNIQUES FOR SOLVING VOLTERRA-FREDHOLM INTEGRAL EQUATIONS
Watam Press, 2019
In this paper, based on a strictly convex fuzzy number space and the Riemann integral of fuzzy-number-valued function which is taken value in the space, we propose iterative procedures based on Adomian Decomposition Method (ADM), Modified Adomian Decomposition Method (MADM) and Modified Variational Iteration Method (MVIM) to solve fuzzy Volterra-Fredholm integral equations of the second kind. That, a fuzzy Volterra-Fredholm integral equation has been converted to a system of Volterra-Fredholm integral equations in crisp case. The approximated methods using to find the approximate solution of this system and hence obtain an approximation for the fuzzy solution of the fuzzy Volterra-Fredholm integral equation. Moreover, we will prove the uniqueness of the solution and convergence of the proposed methods. Also, some numerical examples are included to demonstrate the validity and applicability of the proposed techniques.
On a discussion of Volterra–Fredholm integral equation with discontinuous kernel
Journal of the Egyptian Mathematical Society, 2020
The purpose of this paper is to establish the general solution of a Volterra-Fredholm integral equation with discontinuous kernel in a Banach space. Banach's fixed point theorem is used to prove the existence and uniqueness of the solution. By using separation of variables method, the problem is reduced to Volterra integral equations of the second kind with continuous kernel. Normality and continuity of the integral operator are also discussed.
The Use of Lavrentiev Regularization Method in Fredholm Integral Equations of the First Kind
2019
The Fredholm integral equations of the first kind are often considered as ill-posed problems. The conventional way of solving them is to first convert them into the Fredholm integral equations of the second kind by means of a regularization method. This is followed by applying some standard techniques that are available for solving Fredholm integral equations of the second kind. This combination of two methods usually has some significant drawbacks in the sense that it may not produce a solution or produces only one solution after tedious calculations. The aim of this study is to remove these impediments once and for all for separable kernels and provide a closed-form expression for obtaining one or infinitely many solutions using the Lavrentiev regularization method. MSC: 47A52 • 45B05
Numerical Solution of the Linear Fredholm Integral Equations of the Second Kind
2010
The theory of integral equation is one of the major topics of applied mathematics. The main purpose of this paper is to introduce a numerical method based on the interpolation for approximating the solution of the second kind linear Fredholm integral equation. In this case, the divided difierences method is applied. At last, two numerical examples are presented to show the accuracy of the proposed method.
The numerical solution of Fredholm integral equations of the second kind with singular kernels
Numerische Mathematik, 1972
A numerical method is given for integral equations with singular kernels. The method modifies the ideas of product integration contained in [3], and it is analyzed using the general schema of [! ]. The emphasis is on equations which were not amenable to the method in [3]; in addition, the method tries to keep computer running time to a minimum, while maintaining an adequate order of convergence. The method is illustrated extensively with an integral equation reformulation of boundary value problems for d uP (r*) u = 0; see [9].