Λεπτομέρειες βιβλιογραφικής εγγραφής
| Τίτλος: |
Orthogonal systems based schemes for the stable inversion of Abel's integral operator. |
| Συγγραφείς: |
Karoui, Abderrazek1 (AUTHOR) abderrazek.karoui@fsb.ucar.tn, Khazri, Ibtissem1,2 (AUTHOR) khazri.ibtissem89@gmail.com |
| Πηγή: |
Mathematical Foundations of Computing. Aug2026, Vol. 12, p1-29. 29p. |
| Θεματικοί όροι: |
*Jacobi polynomials, *Inverse problems, *Orthogonal polynomials, *Numerical analysis, *Random matrices, *Singular integrals, *Least squares |
| Περίληψη: |
In this work, we study two methods for solving the ill-posed inverse problem given by an Abel integral equation with a general kernel defined on the lower unit triangle and given by $ { k(x, y) = \frac{1}{(x-y)^{\frac{1}{2}-\alpha}}} $ with $ { -\frac{1}{2} <\alpha<\frac{1}{2}.} $ Here, the emphases will be on the crucial issues of the stability of the methods in the presence of perturbed observed output functions, as well as the handling of eventually non-smooth, including discontinuous solutions of the integral equation. The first method is based on a series expansion of the solution $ f(\cdot) $ with respect to a Jacobi orthogonal polynomials system. The main feature of this method is that the expansion coefficients of the solution are given in terms of the Fourier coefficients of a $ 2\pi- $periodic function that involves the output function $ g(\cdot). $ Another feature of this method is the use of a least-squares based scheme for the stable and accurate numerical approximations of these expansion coefficients. The second proposed method is based on the expansion of the solution in the weighted Jacobi polynomials system. A combined least-squares and random matrix scheme is used for the stable and accurate numerical computation of the expansion coefficients of the solution with respect to the considered orthogonal system. Finally, we give some numerical simulations that illustrate the theoretical results of this work. [ABSTRACT FROM AUTHOR] |
| Βάση Δεδομένων: |
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