Abstract
In this paper, we consider the Galerkin and iterated Galerkin methods for solving Fredholm-Hammestein integral equations with a Green’s kernel, whose first derivative has singularity. We obtain error bounds and convergence rates for both the Galerkin and iterated Galerkin methods using graded mesh. In fact, by choosing the grading exponent appropriately, we obtain superconvergence results in iterated Galerkin method.
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References
Ahues, M., Largillier, A., Limaye, B.V.: Spectral Computations for Bounded Operators. Chapman and Hall/CRC, New York (2001)
Anselone, P.M.: Collectively Compact Operator Approximation Theory and Application to Integral Equations. Prentice Hall, Englewood Cliffs (1971)
Atkinson, K.E., Potra, F.A.: Projection and iterated projection methods for nonlinear integral equations. SIAM J. Numer. Anal. 24(6), 1352–1373 (1987)
Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)
Brunner, H.: The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes. Math. Comput. 45(172), 417–437 (1985)
Cen, Z.: Numerical study for a class of singular two-point boundary value problems using Green’s functions. Appl. Math. Comput. 183(1), 10–16 (2006)
Chatelin, F.: Spectral Approximation of Linear Operators CI. SIAM, Philadelphia (1983)
Ben-Yu, G.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)
Gray, B.: The distribution of heat sources in the human head-theoretical considerations. J. Theor. Biol. 82(3), 473–476 (1980)
Kaneko, H., Xu, Y.: Superconvergence of the iterated Galerkin methods for Hammerstein equations. SIAM J. Numer. Anal. 33(3), 1048–1064 (1996)
Lin, S.: Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. J. Theor. Biol. 60(2), 449–457 (1976)
Adomian, G.: Solution of the Thomas-Fermi equation. Appl. Math. Lett. 11(3), 131–133 (1998)
Singh, R., Kumar, J., Nelakanti, G.: Numerical solution of singular boundary value problems using Green’s function and improved decomposition method. J. Appl. Math. Comput. 43(1–2), 409–425 (2013)
Shen, J., Tang, T.: Spectral and High-Order Methods with Applications. Science Press, Beijing (2006)
Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics. Springer, New York (2011)
Vainikko, G.M.: A perturbed Galerkin method and the general theory of approximate methods for nonlinear equations. USSR Comput. Math. Math. Phys. 7(4), 1–41 (1967)
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Das, P., Nahid, N., Nelakanti, G. (2020). Superconvergence of Iterated Galerkin Method for a Class of Nonlinear Fredholm Integral Equations. In: Castillo, O., Jana, D., Giri, D., Ahmed, A. (eds) Recent Advances in Intelligent Information Systems and Applied Mathematics. ICITAM 2019. Studies in Computational Intelligence, vol 863. Springer, Cham. https://doi.org/10.1007/978-3-030-34152-7_5
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DOI: https://doi.org/10.1007/978-3-030-34152-7_5
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