Numerical solution of the linear two-dimensional Fredholm integral equations of the second kind via two-dimensional triangular orthogonal functions

Mirzaee, Farshid; Piroozfar, Sima

doi:10.1016/j.jksus.2010.04.007

Cited by 21 publications

(30 citation statements)

References 10 publications

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“…Tables 3-7 show that the MMC method is more accurate than the traditional MC method. Tables 3-5 show comparisons between the obtained results in the works [10,21,23], traditional MC and our proposed method. From these comparisons, one can see that the MMC method is more accurate than the methods introduced in [10,21,23] and MC as well.…”

Section: Discussionmentioning

confidence: 99%

“…For this purpose, the absolute error has been calculated using MMC and traditional MC methods. Also, in a column, the calculated error in the work [21], using the two-dimensional triangular orthogonal functions for numerical solution, is shown. Figure 1 is presented to show the error for the unknown function uðx; yÞ between the exact solution of Equation (17) and numerical results that were obtained using the combined algorithm of the present work.…”

Section: Example 42mentioning

confidence: 99%

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Combined probabilistic algorithm for solving high dimensional problems

2014

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The present study establishes an accurate and efficient algorithm based on Monte Carlo (MC) simulation for solving high dimensional linear systems of algebraic equations (LSAEs) and two-dimensional Fredholm integral equations of the second kind (FIESK). This new combined numerical-probabilistic algorithm is based on Jacobi over-relaxation method and MC simulation in conjunction with the iterative refinement technique to find the unique solution of the large sparse LSAEs. It has an excellent accuracy, low cost and simple structure. Theoretical results are established to justify the convergence of the algorithm. To confirm the accuracy and efficiency of the present work, the proposed algorithm is used for solving 1000 £ 1000 and 2000 £ 2000 LSAEs. Furthermore, the algorithm is coupled with Galerkin's method to illustrate the power and effectiveness of the proposed algorithm for solving twodimensional FIESK.

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Section: Discussionmentioning

confidence: 99%

Section: Example 42mentioning

confidence: 99%

Combined probabilistic algorithm for solving high dimensional problems

2014

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“…Regarding the Fredholm integral equation, we obtain analogous results. Let v 0 (s, t) ∶=h(s, t) ∈ C 1 (Ω) and define inductively as in (22) the function J r−1 . Then,…”

Section: The Numerical Methods Using the Usual Schauder Bases In C(ω 2 )mentioning

confidence: 99%

“…Example Now, we consider the following 2‐dimensional Fredholm integral equation included in Mirzaee and Piroozfar:

h false(s, t false) = \frac{1}{{false(1 + s + t false)}^{2}} - \frac{s}{6 false(8 + t false)} + \int_{0}^{1} \int_{0}^{1} \frac{s}{false(8 + t false) false(1 + x + y false)} h false(x, y false) d x d y, false(s, t false) \in false[0, 1 false] \times false[0, 1 false],

where

h false(s, t false) = \frac{1}{{false(1 + s + t false)}^{2}}

is the exact solution. The numerical results are collected in Table .…”

Section: Numerical Examplesmentioning

confidence: 99%

“…For instance, a reduced differential transform method, a method using radial basis functions, a numerical method based on Haar wavelet, 2‐dimensional orthogonal triangular functions, and the 2‐dimensional rationalized Haar can be used to approximate the solution, in approximation method based on a polynomial system . There are furthermore a spectral meshless radial point interpolation method, a method based on developing the 2‐dimensional differential transform, a discrete Galerkin and iterated Galerkin method, an integral mean‐value method, a piecewise interpolating polynomial technique (a 1‐, 2‐, and 3‐dimensional modification of the hat functions method, respectively), a 2‐dimensional triangular orthogonal functions method, a numerical scheme based on the moving least squares method, an operational matrix and 2‐dimensional block pulse functions methods, a method using Legendre polynomials, an iterative numerical method of successive approximations, and He variational method …”

Section: Introductionmentioning

confidence: 99%

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Numerical solving of several types of two‐dimensional integral equations and estimation of error bound

Berenguer

Gámez

2018

Math Methods in App Sciences

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A numerical method based on hybrid orthonormal Bernstein and improved block‐pulse functions for solving Volterra–Fredholm integral equations

Ramadan

Osheba

Hadhoud

2022

Numerical Methods Partial

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This paper deals with the numerical solution of the integral equations of linear second kind Volterra–Fredholm. These integral equations are commonly used in engineering and mathematical physics to solve many of the problems. A hybrid of Bernstein and improved block‐pulse functions method is introduced and used where the key point is to transform linear second‐type Volterra–Fredholm integral equations into an algebraic equation structure that can be solved using classical methods. Numeric examples are given which demonstrate the related features of the process.

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Numerical solution of the linear two-dimensional Fredholm integral equations of the second kind via two-dimensional triangular orthogonal functions

Cited by 21 publications

References 10 publications

Combined probabilistic algorithm for solving high dimensional problems

Combined probabilistic algorithm for solving high dimensional problems

Numerical solving of several types of two‐dimensional integral equations and estimation of error bound

A numerical method based on hybrid orthonormal Bernstein and improved block‐pulse functions for solving Volterra–Fredholm integral equations

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