An effective and simple scheme for solving nonlinear Fredholm integral equations
In this paper, a simple scheme is constructed for finding approximate solution of the nonlinear Fredholm integral equation of the second kind. To this end, the Lagrange interpolation polynomials together with the Gauss-Legendre quadrature rule are used to transform the source problem to a system of...
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| Format: | Article |
| Language: | English |
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Vilnius Gediminas Technical University
2022-04-01
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| Series: | Mathematical Modelling and Analysis |
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| Online Access: | https://tede.vgtu.lt/index.php/MMA/article/view/14194 |
| _version_ | 1818013356917784576 |
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| author | Ahmad Shahsavaran Forough Fotros |
| author_facet | Ahmad Shahsavaran Forough Fotros |
| author_sort | Ahmad Shahsavaran |
| collection | DOAJ |
| description |
In this paper, a simple scheme is constructed for finding approximate solution of the nonlinear Fredholm integral equation of the second kind. To this end, the Lagrange interpolation polynomials together with the Gauss-Legendre quadrature rule are used to transform the source problem to a system of nonlinear algebraic equations. Afterwards, the resulting system can be solved by the Newton method. The basic idea is to choose the Lagrange interpolation points to be the same as the points for the Gauss-Legendre integration. This facilitates the evaluation of the integral part of the equation. We prove that the approximate solution converges uniformly to the exact solution. Also, stability of the approximate solution is investigated. The advantages of the method are simplicity, fastness and accuracy which enhance its applicability in practical situations. Finally, we provide some test examples.
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| first_indexed | 2024-04-14T06:32:10Z |
| format | Article |
| id | doaj.art-ee0df738de334f34965590dd197825b1 |
| institution | Directory Open Access Journal |
| issn | 1392-6292 1648-3510 |
| language | English |
| last_indexed | 2024-04-14T06:32:10Z |
| publishDate | 2022-04-01 |
| publisher | Vilnius Gediminas Technical University |
| record_format | Article |
| series | Mathematical Modelling and Analysis |
| spelling | doaj.art-ee0df738de334f34965590dd197825b12022-12-22T02:07:36ZengVilnius Gediminas Technical UniversityMathematical Modelling and Analysis1392-62921648-35102022-04-0127210.3846/mma.2022.14194An effective and simple scheme for solving nonlinear Fredholm integral equationsAhmad Shahsavaran0Forough Fotros1Young Researchers and Elite Club, Borujerd Branch, Islamic Azad University, Takhti Street, Borujerd, IranYoung Researchers and Elite Club, Borujerd Branch, Islamic Azad University, Takhti Street, Borujerd, Iran In this paper, a simple scheme is constructed for finding approximate solution of the nonlinear Fredholm integral equation of the second kind. To this end, the Lagrange interpolation polynomials together with the Gauss-Legendre quadrature rule are used to transform the source problem to a system of nonlinear algebraic equations. Afterwards, the resulting system can be solved by the Newton method. The basic idea is to choose the Lagrange interpolation points to be the same as the points for the Gauss-Legendre integration. This facilitates the evaluation of the integral part of the equation. We prove that the approximate solution converges uniformly to the exact solution. Also, stability of the approximate solution is investigated. The advantages of the method are simplicity, fastness and accuracy which enhance its applicability in practical situations. Finally, we provide some test examples. https://tede.vgtu.lt/index.php/MMA/article/view/14194Fredholm integral equationLagrange polynomialsGauss-Legendre integrationinterpolationconvergence and stability |
| spellingShingle | Ahmad Shahsavaran Forough Fotros An effective and simple scheme for solving nonlinear Fredholm integral equations Mathematical Modelling and Analysis Fredholm integral equation Lagrange polynomials Gauss-Legendre integration interpolation convergence and stability |
| title | An effective and simple scheme for solving nonlinear Fredholm integral equations |
| title_full | An effective and simple scheme for solving nonlinear Fredholm integral equations |
| title_fullStr | An effective and simple scheme for solving nonlinear Fredholm integral equations |
| title_full_unstemmed | An effective and simple scheme for solving nonlinear Fredholm integral equations |
| title_short | An effective and simple scheme for solving nonlinear Fredholm integral equations |
| title_sort | effective and simple scheme for solving nonlinear fredholm integral equations |
| topic | Fredholm integral equation Lagrange polynomials Gauss-Legendre integration interpolation convergence and stability |
| url | https://tede.vgtu.lt/index.php/MMA/article/view/14194 |
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