Error bounds for maximum entropy approximate solutions to Fredholm integral equations

1991 ◽  
Vol 32 (12) ◽  
pp. 3312-3314 ◽  
Author(s):  
S. Kopeć
Keyword(s):  
Maximum Entropy ◽  
Integral Equations ◽  
Error Bounds ◽  
Approximate Solutions ◽  
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.

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