scholarly journals A Unified Formulation of Analytical and Numerical Methods for Solving Linear Fredholm Integral Equations

Algorithms ◽  
2021 ◽  
Vol 14 (10) ◽  
pp. 293
Author(s):  
Efthimios Providas
Keyword(s):  
Banach Space ◽  
Approximate Solution ◽  
Integral Equations ◽  
Kernel Methods ◽  
Projection Methods ◽  
Analytic Solutions ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Degenerate Kernel ◽  
Quadrature Methods

This article is concerned with the construction of approximate analytic solutions to linear Fredholm integral equations of the second kind with general continuous kernels. A unified treatment of some classes of analytical and numerical classical methods, such as the Direct Computational Method (DCM), the Degenerate Kernel Methods (DKM), the Quadrature Methods (QM) and the Projection Methods (PM), is proposed. The problem is formulated as an abstract equation in a Banach space and a solution formula is derived. Then, several approximating schemes are discussed. In all cases, the method yields an explicit, albeit approximate, solution. Several examples are solved to illustrate the performance of the technique.

scholarly journals Convergence of degenerate-kernel methods

1976 ◽  
Vol 19 (4) ◽  
pp. 422-431 ◽  
Author(s):  
Ian H. Sloan
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Integral Equations ◽  
Error Bounds ◽  
Kernel Methods ◽  
Fredholm Integral Equations ◽  
Degenerate Kernel ◽  
Kernel Approximation ◽  
Convergence Results

AbstractAdditional convergence results are given for the approximate solution in the space L2(a, b) of Fredholm integral equations of the second kind, y = f + Ky, by the degenerate-kernel methods of Sloan, Burn and Datyner. Convergence to the exact solution is provided for a class of these methods (including ‘method 2’), under suitable conditions on the kernel K, and error bounds are obtained. In every case the convergence is faster than that of the best approximate solution of the form yn = Σnan1u1, where u1, …, un are the appropriate functions used in the rank-n degenerate-kernel approximation. In addition, the error for method 2 is shown to be relatively unaffected if the integral equation has an eigenvalue near 1.

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.

Numerical study of stochastic Volterra–Fredholm integral equations by using second kind Chebyshev wavelets

2016 ◽  
Vol 24 (2) ◽  
Author(s):  
Fakhrodin Mohammadi ◽  
Parastoo Adhami
Keyword(s):  
Integral Equations ◽  
Numerical Study ◽  
Operational Matrix ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Wavelet Method ◽  
Chebyshev Wavelets ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Efficiency And Reliability

AbstractIn this paper, we present a computational method for solving stochastic Volterra–Fredholm integral equations which is based on the second kind Chebyshev wavelets and their stochastic operational matrix. Convergence and error analysis of the proposed method are investigated. Numerical results are compared with the block pulse functions method for some non-trivial examples. The obtained results reveal efficiency and reliability of the proposed wavelet method.

scholarly journals A Novel and Efficient Numerical Algorithm for Solving 2D Fredholm Integral Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
H. Bin Jebreen
Keyword(s):  
Approximate Solution ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Numerical Algorithm ◽  
Numerical Experiments ◽  
Operational Matrix ◽  
Fredholm Integral Equations ◽  
Scaling Functions ◽  
Algebraic Equations

A novel and efficient numerical method is developed based on interpolating scaling functions to solve 2D Fredholm integral equations (FIE). Using the operational matrix of integral for interpolating scaling functions, FIE reduces to a set of algebraic equations that one can obtain an approximate solution by solving this system. The convergence analysis is investigated, and some numerical experiments confirm the accuracy and validity of the method. To show the ability of the proposed method, we compare it with others.

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