scholarly journals An Ulm-Type Inverse-Free Iterative Scheme for Fredholm Integral Equations of Second Kind

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1957
Author(s):  
José M. Gutiérrez ◽  
Miguel Á. Hernández-Verón
Keyword(s):  
Integral Equations ◽  
Order Of Convergence ◽  
Iterative Scheme ◽  
Semilocal Convergence ◽  
Fredholm Integral Equations ◽  
Linear Integral ◽  
Ulm’S Method ◽  
Theoretical Results ◽  
Quadratic Order ◽  
Linear Integral Equations

In this paper, we present an iterative method based on the well-known Ulm’s method to numerically solve Fredholm integral equations of the second kind. We support our strategy in the symmetry between two well-known problems in Numerical Analysis: the solution of linear integral equations and the approximation of inverse operators. In this way, we obtain a two-folded algorithm that allows us to approximate, with quadratic order of convergence, the solution of the integral equation as well as the inverses at the solution of the derivative of the operator related to the problem. We have studied the semilocal convergence of the method and we have obtained the expression of the method in a particular case, given by some adequate initial choices. The theoretical results are illustrated with two applications to integral equations, given by symmetric non-separable kernels.

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

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1747
Author(s):  
José Manuel Gutiérrez ◽  
Miguel Ángel Hernández-Verón ◽  
Eulalia Martínez
Keyword(s):  
Integral Equations ◽  
Theoretical Study ◽  
Iterative Scheme ◽  
Semilocal Convergence ◽  
Iterative Solution ◽  
Fredholm Integral Equations ◽  
Existence Of A Solution ◽  
Separable Kernel ◽  
Starting Point ◽  
Domain Of Existence

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.

scholarly journals Solving Linear Volterra – Fredholm Integral Equation of the Second Type Using Linear Programming Method

2020 ◽  
Vol 17 (1(Suppl.)) ◽  
pp. 0342
Author(s):  
Muna Mansoor Mustafaf
Keyword(s):  
Linear Programming ◽  
Integral Equations ◽  
New Technique ◽  
Fredholm Integral Equations ◽  
Quadrature Methods ◽  
Special Cases ◽  
A New Technique ◽  
Linear Integral ◽  
Romberg Integration ◽  
Linear Integral Equations

In this paper, a new technique is offered for solving three types of linear integral equations of the 2nd kind including Volterra-Fredholm integral equations (LVFIE) (as a general case), Volterra integral equations (LVIE) and Fredholm integral equations (LFIE) (as special cases). The new technique depends on approximating the solution to a polynomial of degree  and therefore reducing the problem to a linear programming problem(LPP), which will be solved to find the approximate solution of LVFIE. Moreover, quadrature methods including trapezoidal rule (TR), Simpson 1/3 rule (SR), Boole rule (BR), and Romberg integration formula (RI) are used to approximate the integrals that exist in LVFIE. Also, a comparison between those methods is produced. Finally, for more explanation, an algorithm is proposed and applied for testing examples to illustrate the effectiveness of the new technique.

scholarly journals On the numerical solutions for nonlinear Volterra-Fredholm integral equations

2022 ◽  
Vol 40 ◽  
pp. 1-11
Author(s):  
Parviz Darania ◽  
Saeed Pishbin
Keyword(s):  
Nonlinear System ◽  
Integral Equations ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Coefficient Matrix ◽  
Collocation Methods ◽  
Fredholm Integral Equations ◽  
Stability Estimates ◽  
Lower Triangular ◽  
Theoretical Results

In this note, we study a class of multistep collocation methods for the numerical integration of nonlinear Volterra-Fredholm Integral Equations (V-FIEs). The derived method is characterized by a lower triangular or diagonal coefficient matrix of the nonlinear system for the computation of the stages which, as it is known, can beexploited to get an efficient implementation. Convergence analysis and linear stability estimates are investigated. Finally numerical experiments are given, which confirm our theoretical results.

scholarly journals On Solvability Conditions for a Certain Conjugation Problem

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 234
Author(s):  
Vladimir Vasilyev ◽  
Nikolai Eberlein
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Integral Equations ◽  
Mellin Transform ◽  
Algebraic Equations ◽  
Conjugation Problem ◽  
Solvability Conditions ◽  
Linear Algebraic Equations ◽  
Linear Integral ◽  
Linear Integral Equations

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

On the location of the Weyl circles

1981 ◽  
Vol 88 (3-4) ◽  
pp. 345-356 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Integral Equations ◽  
Complex Plane ◽  
Riccati Equations ◽  
Sturm Liouville ◽  
Linear Integral ◽  
Linear Integral Equations

SynopsisThe paper deals with explicit estimates concerning certain circles in the complex plane which were associated with Sturm–Liouville problems by H. Weyl. By the use of Riccati equations instead of linear integral equations, improvements are obtained for results of Everitt and Halvorsen concerning the behaviour of the Titchmarsh–Weyl m-coefficient.

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