Numerical solutions for weakly singular Hammerstein equations and their superconvergence. (English) Zbl 0764.65085
G. Vainikko and P. Uba [J. Aust. Math. Soc., Ser. B 22, 431- 438 (1981; Zbl 0475.65084)] used piecewise continuous polynomial collocation for solving linear Fredholm integral equations of the second kind whose kernels contain weak singularities of the types \(| t- s|^{-\alpha}\) (\(0<\alpha<1\)) or \(\log | t-s|\), and they proved that local superconvergence can occur if the underlying mesh is suitably graded.
The present paper extends these results to weakly singular nonlinear Fredholm integral equations of Hammerstein type. In addition, the authors analyze the order of convergence of the modified collocation method of S. Kumar and I. H. Sloan [Math. Comp. 48, 585-593 (1987; Zbl 0616.65142)] when the method is applied to these nonlinear integral equations. Two numerical examples illustrate the theory.
The present paper extends these results to weakly singular nonlinear Fredholm integral equations of Hammerstein type. In addition, the authors analyze the order of convergence of the modified collocation method of S. Kumar and I. H. Sloan [Math. Comp. 48, 585-593 (1987; Zbl 0616.65142)] when the method is applied to these nonlinear integral equations. Two numerical examples illustrate the theory.
Reviewer: H.Brunner (St.John’s)
Keywords:
weakly singular kernels; local superconvergence; weakly singular nonlinear Fredholm integral equations of Hammerstein type; collocation method; numerical examplesReferences:
| [1] | K. Atkinson, The numerical solution of Fredholm integral equations of the second kind , SIAM J. Num. Anal. 4 (1967), 337-349. JSTOR: · Zbl 0155.47404 · doi:10.1137/0704029 |
| [2] | ——–, The numerical solution of Fredholm integral equations of the second kind with singular kernels , Num. Math. 19 (1972), 248-259. · Zbl 0258.65117 · doi:10.1007/BF01404695 |
| [3] | ——–, The numerical evaluation of fixed points of completely continuous operators , SIAM J. Num. Anal. 10 (1973), 799-807. JSTOR: · Zbl 0237.65040 · doi:10.1137/0710065 |
| [4] | G.A. Chandler, Galerkin method for boundary integral equations on polygonal domains , J. Austral. Math. Soc. Ser. B 26 (1984), 1-13. · Zbl 0559.65083 · doi:10.1017/S033427000000429X |
| [5] | I. Graham, Singular expansions for the solution of second kind Fredholm integral equations with weakly singular convolution kernels , J. Integral Equations Appl. 4 (1982), 1-30. · Zbl 0482.45003 |
| [6] | I. Graham and C. Schneider, Product integration for weakly singular integral equations , in Constructive methods for the practical treatment of integral equations (G. Hammerlin and K. Hoffmann, eds.), ISNM 73, Birkhauser, 1985, 156-164. · Zbl 0596.65095 |
| [7] | H. Kaneko, R. Noren and Y. Xu, Regularity of the solution of Hammerstein equations with weakly singular kernel , Integral Equations Operator Theory, 13 (1990), 660-670. · Zbl 0706.45005 · doi:10.1007/BF01732317 |
| [8] | M. Krasnoselskii, Topological methods for nonlinear integral equations , Pergamon Press, New York, 1964. · Zbl 0111.30303 |
| [9] | S. Kumar, Superconvergence of a collocation-type method for Hammerstein equations , IMA J. Num. Anal. 7 (1987), 313-325. · Zbl 0637.65140 · doi:10.1093/imanum/7.3.313 |
| [10] | S. Kumar and I. Sloan, A new collocation-type method for Hammerstein integral equations , Math. Comp. 48 (1987), 585-593. · Zbl 0616.65142 · doi:10.2307/2007829 |
| [11] | R. Piessens and M. Branders, Numerical solution of integral equations of mathematical physics using Chebyshev polynomials , J. Comp. Phys. 21 (1976), 178-196. · Zbl 0331.65096 · doi:10.1016/0021-9991(76)90010-3 |
| [12] | J. Rice, On the degree of convergence of nonlinear spline approximations in Approximation with special emphasis on spline functions (I.J. Schoenberg, ed.), Academic Press, New York, 1969, 349-365. · Zbl 0266.41012 |
| [13] | G. Richter, On weakly singular Fredholm integral equations with displacement kernels, J. Math. Anal. Appl. 55 (1976), 32-42. · Zbl 0355.45005 · doi:10.1016/0022-247X(76)90275-4 |
| [14] | C. Schneider, Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Integral Equations Operator Theory 2 (1979), 62-68. · Zbl 0403.45002 · doi:10.1007/BF01729361 |
| [15] | ——–, Product integration for weakly singular integral equations , Math. Comp. 36 (1981), 207-213. JSTOR: · Zbl 0474.65095 · doi:10.2307/2007737 |
| [16] | I. Sloan, On the numerical evaluation of singular integrals , BIT 18 (1978), 91-102. · Zbl 0386.65002 · doi:10.1007/BF01947747 |
| [17] | G. Vainikko, Perturbed Galerkin method and general theory of approximate methods for nonlinear equations , Zh. Vychisl. Mat. i Mat. Fiz. 7 (1967), 723-751. Engl. Translation, U.S.S.R. Comp. Math. and Math. Phys. 7 , No. 4 (1967), 1-41. |
| [18] | G. Vainikko and P. Uba, A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel , J. Austral. Math. Soc. Series B, 22 (1981), 431-438. · Zbl 0475.65084 · doi:10.1017/S0334270000002770 |
| [19] | R. Weiss, On the approximation of fixed points of nonlinear compact operators , SIAM J. Num. Anal. 11 (1974), 550-553. JSTOR: · Zbl 0285.41019 · doi:10.1137/0711046 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
