Geodesic
Operational Müntz-Galerkin approximation for Abel-Hammerstein integral equations of the second kind
Electronic transactions on numerical analysis, Tome 45 (2016), pp. 183-200
Since solutions of Abel integral equations exhibit singularities, existing spectral methods for these equations suffer from instability and low accuracy. Moreover, for nonlinear problems, solving the resulting complex nonlinear algebraic systems numerically requires high computational costs. To overcome these drawbacks, in this paper we propose an operational Galerkin strategy for solving Abel-Hammerstein integral equations of the second kind which applies Müntz-Legendre polynomials as natural basis functions to discretize the problem and to obtain a sparse nonlinear system with upper-triangular structure that can be solved directly. It is shown that our approach yields a well-posed and easy-to-implement approximation technique with a high order of accuracy regardless of the singularities of the exact solution. The numerical results confirm the superiority and effectiveness of the proposed scheme.
Classification : 45E10, 41A25
Keywords: Abel-Hammerstein integral equations, Galerkin method, müntz-Legendre polynomials, well-posedness 
@article{ETNA_2016__45__a16,
     author = {Mokhtary,  P.},
     title = {Operational {M\"untz-Galerkin} approximation for {Abel-Hammerstein} integral equations of the second kind},
     journal = {Electronic transactions on numerical analysis},
     pages = {183--200},
     year = {2016},
     volume = {45},
     zbl = {1346.45003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2016__45__a16/}
}
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Mokhtary,  P. Operational Müntz-Galerkin approximation for Abel-Hammerstein integral equations of the second kind. Electronic transactions on numerical analysis, Tome 45 (2016), pp. 183-200. http://geodesic.mathdoc.fr/item/ETNA_2016__45__a16/