Using Chebyshev’s polynomials for solving Fredholm integral equations of the second kind
The main problem with the Newton method is the computation of the inverse of the first derivative of the operator involved at each iteration step. Thus, when we want to apply the Newton method directly to solve an integral equation, the existence of the inverse of the first derivative is guaranteed...
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Vilnius Gediminas Technical University
2025-01-01
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| Series: | Mathematical Modelling and Analysis |
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| Online Access: | https://gc.vgtu.lt/index.php/MMA/article/view/21036 |
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| author | José Antonio Ezquerro Miguel Ángel Hernández-Verón |
| author_facet | José Antonio Ezquerro Miguel Ángel Hernández-Verón |
| author_sort | José Antonio Ezquerro |
| collection | DOAJ |
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The main problem with the Newton method is the computation of the inverse of the first derivative of the operator involved at each iteration step. Thus, when we want to apply the Newton method directly to solve an integral equation, the existence of the inverse of the first derivative is guaranteed, when the kernel is sufficiently differentiable into any of its two components, through its approximation by Taylor’s polynomial. In this paper, we study the case in which the kernel is not differentiable in any of its two components. So, we present a strategy that consists of approximating the kernel of the nonlinear integral equation by a Chebyshev interpolation polynomial, which is separable. This allows us to explicitly calculate the inverse of the first derivative operator in each step of the Newton method and then approximate a solution of the approximate integral equation.
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| format | Article |
| id | doaj-art-cea10b7314f143b894f0bd34694e8921 |
| institution | Kabale University |
| issn | 1392-6292 1648-3510 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Vilnius Gediminas Technical University |
| record_format | Article |
| series | Mathematical Modelling and Analysis |
| spelling | doaj-art-cea10b7314f143b894f0bd34694e89212025-01-27T16:30:17ZengVilnius Gediminas Technical UniversityMathematical Modelling and Analysis1392-62921648-35102025-01-0130110.3846/mma.2025.21036Using Chebyshev’s polynomials for solving Fredholm integral equations of the second kindJosé Antonio Ezquerro0Miguel Ángel Hernández-Verón1Department of Mathematics and Computation, University of La Rioja, La Rioja, SpainDepartment of Mathematics and Computation, University of La Rioja, La Rioja, Spain The main problem with the Newton method is the computation of the inverse of the first derivative of the operator involved at each iteration step. Thus, when we want to apply the Newton method directly to solve an integral equation, the existence of the inverse of the first derivative is guaranteed, when the kernel is sufficiently differentiable into any of its two components, through its approximation by Taylor’s polynomial. In this paper, we study the case in which the kernel is not differentiable in any of its two components. So, we present a strategy that consists of approximating the kernel of the nonlinear integral equation by a Chebyshev interpolation polynomial, which is separable. This allows us to explicitly calculate the inverse of the first derivative operator in each step of the Newton method and then approximate a solution of the approximate integral equation. https://gc.vgtu.lt/index.php/MMA/article/view/21036Fredholm integral equationthe Newton methodexistence domainuniqueness domain |
| spellingShingle | José Antonio Ezquerro Miguel Ángel Hernández-Verón Using Chebyshev’s polynomials for solving Fredholm integral equations of the second kind Mathematical Modelling and Analysis Fredholm integral equation the Newton method existence domain uniqueness domain |
| title | Using Chebyshev’s polynomials for solving Fredholm integral equations of the second kind |
| title_full | Using Chebyshev’s polynomials for solving Fredholm integral equations of the second kind |
| title_fullStr | Using Chebyshev’s polynomials for solving Fredholm integral equations of the second kind |
| title_full_unstemmed | Using Chebyshev’s polynomials for solving Fredholm integral equations of the second kind |
| title_short | Using Chebyshev’s polynomials for solving Fredholm integral equations of the second kind |
| title_sort | using chebyshev s polynomials for solving fredholm integral equations of the second kind |
| topic | Fredholm integral equation the Newton method existence domain uniqueness domain |
| url | https://gc.vgtu.lt/index.php/MMA/article/view/21036 |
| work_keys_str_mv | AT joseantonioezquerro usingchebyshevspolynomialsforsolvingfredholmintegralequationsofthesecondkind AT miguelangelhernandezveron usingchebyshevspolynomialsforsolvingfredholmintegralequationsofthesecondkind |