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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Vilnius Gediminas Technical University
2025-01-01
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| Series: | Mathematical Modelling and Analysis |
| Subjects: | |
| Online Access: | https://gc.vgtu.lt/index.php/MMA/article/view/21036 |
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| Summary: | 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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| ISSN: | 1392-6292 1648-3510 |