A modified approach to numerical solution of Fredholm integral equations of the second kind

Panda, Srikumar; Martha, S. C.; Chakrabarti, A.

doi:10.1016/j.amc.2015.08.111

Cited by 9 publications

(8 citation statements)

References 16 publications

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“…This method is also being used by both Panda and Martha et el. [2] and Bhattacharya and Mondal [3] and they approximated the solution in terms of linear combination of Lagrange's and Bernstein polynomials respectively. All of them have found excellent agreement in approximate solution with exact solution.…”

Section: Introductionmentioning

confidence: 99%

Numerical Approximation of Fredholm Integral Equation (FIE) of 2nd Kind using Galerkin and Collocation Methods

Molla

Saha

2019

GANIT: J. Bangladesh Math. Soc.

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In this research work, Galerkin and collocation methods have been introduced for approximating the solution of FIE of 2nd kind using LH (product of Laguerre and Hermite) polynomials which are considered as basis functions. Also, a comparison has been done between the solutions of Galerkin and collocation method with the exact solution. Both of these methods show the outcome in terms of the approximate polynomial which is a linear combination of basis functions. Results reveal that performance of collocation method is better than Galerkin method. Moreover, five different polynomials such as Legendre, Laguerre, Hermite, Chebyshev 1st kind and Bernstein are also considered as a basis functions. And it is found that all these approximate solutions converge to same polynomial solution and then a comparison has been made with the exact solution. In addition, five different set of collocation points are also being considered and then the approximate results are compared with the exact analytical solution. It is observed that collocation method performed well compared to Galerkin method. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 11-25

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Section: Introductionmentioning

confidence: 99%

Numerical Approximation of Fredholm Integral Equation (FIE) of 2nd Kind using Galerkin and Collocation Methods

Molla

Saha

2019

GANIT: J. Bangladesh Math. Soc.

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“…( 2015 ), Okayama et al. ( 2011 ), Panda ( 2013 ), Javidi and Golbabai ( 2009 ), Ghorbani and Saberi-Nadjafi ( 2006 ), Mohamad Nor et al. ( 2013 ).…”

Section: Introductionmentioning

confidence: 99%

“…To improve the efficiency of the HPM, a few modifications have been made by many researches. For instance, Javidi and Golbabai ( 2009 ) added the accelerating parameter to the perturbation equation for obtaining the approximate solution for nonlinear Fredholm integral equation. Ghorbani and Saberi-Nadjafi ( 2006 ) added a series of parameter and selective functions to HPM to find the semi-analytical solutions of nonlinear Fredholm and Volterra integral equations.…”

Section: Introductionmentioning

confidence: 99%

Modified homotopy perturbation method for solving hypersingular integral equations of the first kind

et al. 2016

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Modified homotopy perturbation method (HPM) was used to solve the hypersingular integral equations (HSIEs) of the first kind on the interval [−1,1] with the assumption that the kernel of the hypersingular integral is constant on the diagonal of the domain. Existence of inverse of hypersingular integral operator leads to the convergence of HPM in certain cases. Modified HPM and its norm convergence are obtained in Hilbert space. Comparisons between modified HPM, standard HPM, Bernstein polynomials approach Mandal and Bhattacharya (Appl Math Comput 190:1707−1716, 2007), Chebyshev expansion method Mahiub et al. (Int J Pure Appl Math 69(3):265–274, 2011) and reproducing kernel Chen and Zhou (Appl Math Lett 24:636–641, 2011) are made by solving five examples. Theoretical and practical examples revealed that the modified HPM dominates the standard HPM and others. Finally, it is found that the modified HPM is exact, if the solution of the problem is a product of weights and polynomial functions. For rational solution the absolute error decreases very fast by increasing the number of collocation points.

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“…u x y x y = We use the present method to approximate the exact solution of two-dimensional integral equation. We compare the absolution errors obtained by the present method and the method in [6] and the method in [7] for different n . Form the Tab.1, it is easy to find that when 6 n = the method in [6] is better than the present method, but with the increase of n the present method can reach a higher accuracy than the method in [6] and the method in [7].…”

Section: Examplesmentioning

confidence: 99%

“…A modified interpolation method and Nystrom method have been introduced in [6,7]. In [6], the author has presented an approach which is different and more general as well as more efficient than the Nystöm method, in the sense that Nystöm method becomes a particular case of the method developed here. Imran et al in [8] developed the haar wavelet method for the numerical solutions of two-dimensional nonlinear integral equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of two-dimensional Fredholm integral equations via modification of barycentric rational interpolation

Pan¹,

Huang²

2017

Proceedings of the 2017 2nd International Conference on Automation, Mechanical Control and Computational Engineering (AMCCE 201

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Abstract. We have presented a modified barycentric rational interpolation method for solving two-dimensional integral equations. The present method can accurately approximate the exact solution. We also compare with the Lagrange interpolation method and Nystöm method. The proposed method can achieve higher numerical accuracy than other two methods. At last, we give some examples to illustrate the validity of the presented method.

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A modified approach to numerical solution of Fredholm integral equations of the second kind

Cited by 9 publications

References 16 publications

Numerical Approximation of Fredholm Integral Equation (FIE) of 2nd Kind using Galerkin and Collocation Methods

Numerical Approximation of Fredholm Integral Equation (FIE) of 2nd Kind using Galerkin and Collocation Methods

Modified homotopy perturbation method for solving hypersingular integral equations of the first kind

Numerical solution of two-dimensional Fredholm integral equations via modification of barycentric rational interpolation

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