An efficient numerical approximation for the linear class of mixed integral equations

Hadizadeh, M.; Asgary, M.

doi:10.1016/j.amc.2004.06.140

Cited by 13 publications

(10 citation statements)

References 15 publications

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“…where m À 1 < a < m (m 2 N þ ; N þ denotes the set of positive integers), l 1 and l 2 are constants, r(t, x), k(s, y), g(s, t), c 1 s; x; f s; x ð Þ À Á and c 2 y; t; f y; t ð Þ À Á are known functions. The mixed type integral equations and fractional integro-differential equations (Maleknejad and JafariBehbahani, 2012;Hadizadeh and Asgari, 2005;Almasieh and Meleh, 2014):…”

Section: Introductionmentioning

confidence: 99%

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Numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types)

Wang

2021

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Purpose This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types). Design/methodology/approach The main idea of the presented algorithm is to combine Bernoulli polynomials approximation with Caputo fractional derivative and numerical integral transformation to reduce the studied two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations to easily solved algebraic equations. Findings Without considering the integral operational matrix, this algorithm will adopt straightforward discrete data integral transformation, which can do good work to less computation and high precision. Besides, combining the convenient fractional differential operator of Bernoulli basis polynomials with the least-squares method, numerical solutions of the studied equations can be obtained quickly. Illustrative examples are given to show that the proposed technique has better precision than other numerical methods. Originality/value The proposed algorithm is efficient for the considered two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations. As its convenience, the computation of numerical solutions is time-saving and more accurate.

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Section: Introductionmentioning

confidence: 99%

“…Example 5.3 Consider the 2D Volterra-Fredholm integral equation of mixed type (Maleknejad and JafariBehbahani, 2012;Hadizadeh and Asgari, 2005;Almasieh and Meleh, 2014):…”

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confidence: 99%

Numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types)

Wang

2021

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“…The analytical solution of the three-dimensional integral equations is usually difficult, and in many cases, it is required to approximate the solutions. Although several numerical methods for approximating the solutions of two-dimensional mixed Volterra-Fredholm integral equations were presented [1][2][3][4][5][6][7][8][9][10][11][12][13], for three-dimensional ones, only a few methods have been discussed in the literature [5]. The analysis of computational methods for several-dimensional integral equations, specially in the nonlinear case, has started more recently and is not so well developed.…”

Section: Introductionmentioning

confidence: 99%

Using operational matrix for solving nonlinear class of mixed Volterra–Fredholm integral equations

Mirzaee

Hadadiyan

2016

Math Methods in App Sciences

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This paper presents a computational technique for the solution of the nonlinear mixed Volterra–Fredholm integral equations of the second kind. Using the properties of three‐dimensional modification of hat functions, these are types of equations to a nonlinear system of algebraic equations. Also, convergence results and error analysis are discussed. The efficiency and accuracy of the proposed method is illustrated by numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

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“…The method was a meshless method, since it did not require any background interpolation or approximation cells and it did not depend on the geometry of domain. Hadizadeh and Asgary [11] using the bivariate Chebyshev collocation method solved the linear Volterra-Fredholm integral equations of the second kind. Alipanah and Esmaeili [2] approximated the solution of the two-dimensional Fredholm integral equation using Gaussian radial basis function based on Legendre-Gauss-Lobatto nodes and weights.…”

Section: Introductionmentioning

confidence: 99%

Utilizing artificial neural network approach for solving two-dimensional integral equations

2014

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This paper surveys the artificial neural networks approach. Researchers believe that these networks have the wide range of applicability, they can treat complicated problems as well. The work described here discusses an efficient computational method that can treat complicated problems. The paper intends to introduce an efficient computational method which can be applied to approximate solution of the linear two-dimensional Fredholm integral equation of the second kind. For this aim, a perceptron model based on artificial neural networks is introduced. At first, the unknown bivariate function is replaced by a multilayer perceptron neural net and also a cost function to be minimized is defined. Then a famous learning technique, namely, the steepest descent method, is employed to adjust the parameters (the weights and biases) to optimize their behavior. The article also examines application of the method which turns to be so accurate and efficient. It concludes with a survey of an example in order to investigate the accuracy of the proposed method.

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An efficient numerical approximation for the linear class of mixed integral equations

Cited by 13 publications

References 15 publications

Numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types)

Numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types)

Using operational matrix for solving nonlinear class of mixed Volterra–Fredholm integral equations

Utilizing artificial neural network approach for solving two-dimensional integral equations

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