scholarly journals Approximate Solutions of Linear Fredholm Integral Equations System with Variable Coefficients

2013 ◽  
Vol 18 (1) ◽  
pp. 19-29
Author(s):  
Salih Yalçınbaş
Keyword(s):  
Integral Equations ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Fredholm Integral Equations

scholarly journals An Adaptive Approach to Solutions of Fredholm Integral Equations of the Second Kind

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Nebiye Korkmaz ◽  
Zekeriya Güney
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Iteration Method ◽  
Legendre Polynomials ◽  
Optimization Problem ◽  
Approximate Solutions ◽  
Basis Functions ◽  
Fredholm Integral Equations ◽  
Coarse Mesh ◽  
Fine Mesh

As an approach to approximate solutions of Fredholm integral equations of the second kind, adaptive hp-refinement is used firstly together with Galerkin method and with Sloan iteration method which is applied to Galerkin method solution. The linear hat functions and modified integrated Legendre polynomials are used as basis functions for the approximations. The most appropriate refinement is determined by an optimization problem given by Demkowicz, 2007. During the calculationsL2-projections of approximate solutions on four different meshes which could occur between coarse mesh and fine mesh are calculated. Depending on the error values, these procedures could be repeated consecutively or different meshes could be used in order to decrease the error values.

Application of Bernoulli wavelet method to solve a system of fuzzy integral equations

2018 ◽  
Vol 9 (1-2) ◽  
pp. 16-27 ◽  
Author(s):  
Mohamed Abdel- Latif Ramadan ◽  
Mohamed R. Ali
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Fuzzy Integral ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Numerical Examples ◽  
Bernoulli Wavelet

In this paper, an efficient numerical method to solve a system of linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM) is proposed. Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First we introduce properties of Bernoulli wavelets and Bernoulli polynomials, then we used it to transform the integral equations to the system of algebraic equations. The error estimates of the proposed method is given and compared by solving some numerical examples.

scholarly journals Solving of Linear Volterra-Fredholm Integral Equations via Modification of Block Pulse Functions

2021 ◽  
Vol 17 (1) ◽  
pp. 33
Author(s):  
Ayyubi Ahmad
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Triangular System ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Linear Algebraic Equations

A computational method based on modification of block pulse functions is proposed for solving numerically the linear Volterra-Fredholm integral equations. We obtain integration operational matrix of modification of block pulse functions on interval [0,T). A modification of block pulse functions and their integration operational matrix can be reduced to a linear upper triangular system. Then, the problem under study is transformed to a system of linear algebraic equations which can be used to obtain an approximate solution of  linear Volterra-Fredholm integral equations. Furthermore, the rate of convergence is  O(h) and error analysis of the proposed method are investigated. The results show that the approximate solutions have a good of efficiency and accuracy.

scholarly journals Asymptotic Solutions of Integral Equations with Convolution Kernels

1964 ◽  
Vol 14 (1) ◽  
pp. 5-19 ◽  
Author(s):  
V. Hutson
Keyword(s):  
Integral Equations ◽  
Fourier Transforms ◽  
Approximate Solutions ◽  
Asymptotic Solutions ◽  
Fredholm Integral Equations ◽  
Finite Range ◽  
End Points ◽  
Convolution Kernels ◽  
The Difference

SummaryThe equations considered are Fredholm integral equations of the second kind with regular kernels, whose argument depends only on the difference of the variables. Approximate solutions are sought for a given finite range of the eigenvalues, and for large values of the range of integration. Certain special conditions are imposed on the general form of the Fourier transforms of the kernel. Then it is shown that approximate solutions may be obtained in terms of the solutions of the corresponding (singular) Wiener-Hopf equations. Approximations to the eigenvalues are also found. It is shown that the eigenfunctions are unique, and that except possibly near the end points of the range, the solutions are of trigonometric type with the zeros of successive solutions interlacing.

A stochastic operational matrix method for numerical solutions of mixed stochastic Volterra–Fredholm integral equations

2019 ◽  
Vol 18 (02) ◽  
pp. 2050005
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Stochastic Integral ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Chebyshev Wavelets ◽  
Operational Matrix Method ◽  
Stochastic Operational Matrix ◽  
Stochastic Integral Equations

In this paper, we have studied space-time Brownian motion and its applications to mixed type stochastic integral equations. Approximate solutions of mixed stochastic integral equations have been obtained by using two-dimensional (2D) second kind Chebyshev wavelets (CWs). Furthermore, some examples have been presented to justify the efficiency of 2D second kind CWs.

scholarly journals Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations

2020 ◽  
Vol 4 (1) ◽  
pp. 9
Author(s):  
Atanaska Georgieva ◽  
Snezhana Hristova
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Homotopy Analysis Method ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Fuzzy Integral ◽  
Two Dimensional ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Fuzzy Solution

The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of this system are obtained by the help with HAM and hence an approximation for the fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integral equation is presented. The convergence of the proposed method is proved and the error estimate between the exact and the approximate solution is obtained. The validity and applicability of the proposed method is illustrated on a numerical example.

scholarly journals Iterated Petrov–Galerkin Method with Regular Pairs for Solving Fredholm Integral Equations of the Second Kind

2018 ◽  
Vol 23 (4) ◽  
pp. 73
Author(s):  
Silvia Alejandra Seminara ◽  
María Inés Troparevsky
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Correlation Matrix ◽  
Approximation Scheme ◽  
Approximate Solutions ◽  
Orthogonal Basis ◽  
Fredholm Integral Equations ◽  
Interesting Phenomenon ◽  
Weakly Singular ◽  
Selection Of

In this work we obtain approximate solutions for Fredholm integral equations of the second kind by means of Petrov–Galerkin method, choosing “regular pairs” of subspaces, Xn,Yn, which are simply characterized by the positive definitiveness of a correlation matrix. This choice guarantees the solvability and numerical stability of the approximation scheme in an easy way, and the selection of orthogonal basis for the subspaces make the calculations quite simple. Afterwards, we explore an interesting phenomenon called “superconvergence”, observed in the 1970s by Sloan: once the approximations un∈Xn to the solution of the operator equation u-Ku=g are obtained, the convergence can be notably improved by means of an iteration of the method, un*=g+Kun. We illustrate both procedures of approximation by means of two numerical examples: one for a continuous kernel, and the other for a weakly singular one.

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