Log in :: RSS/Atom Feeds
Journal of Integral Equations and Applications

A Survey of Numerical Methods for Solving Nonlinear Integral Equations

Kendall E. Atkinson
Source: J. Integral Equations Appl. Volume 4, Number 1 (1992), 15-46.
First Page: Show Hide
Primary Subjects: 65R20
Secondary Subjects: 45Gxx, 65J15
Full-text: Open access
PDF File (272 KB)
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.jiea/1181075664
Digital Object Identifier: doi:10.1216/jiea/1181075664
Mathematical Reviews number (MathSciNet): MR1160086
Zentralblatt MATH identifier: 0760.65118

References

E. Allgower, K. Böhmer, F. Potra, and W. Rheinboldt, A mesh independence principle for operator equations and their discretizations, SIAM J. Num. Anal. 23 (1986), 160-169.
Mathematical Reviews (MathSciNet): MR821912
Zentralblatt MATH: 0591.65043
Digital Object Identifier: doi:10.1137/0723011
JSTOR: links.jstor.org
H. Amann, Zum Galerkin-Verfahren für die Hammersteinsche Gleichung, Arch. Rational Mech. Anal. 35 (1969), 114-121.
Mathematical Reviews (MathSciNet): MR273467
Digital Object Identifier: doi:10.1007/BF00247615
--------, Über die Konvergenzgeschwindigkeit des Galerkin-Verfahren für die Hammersteinsche Gleichung, Arch. Rational Mech. Anal. 37 (1970), 143-153.
Mathematical Reviews (MathSciNet): MR254686
Digital Object Identifier: doi:10.1007/BF00249500
P. Anselone, ed., Nonlinear integral equations, University of Wisconsin, Madison, 1964.
Mathematical Reviews (MathSciNet): MR161112
--------, Collectively compact operator approximation theory and applications to integral equations, Prentice-Hall, New Jersey, 1971.
Mathematical Reviews (MathSciNet): MR443383
Zentralblatt MATH: 0228.47001
P. Anselone and R. Ansorge, Compactness principles in nonlinear operator approximation theory, Numer. Funct. Anal. Optim. 6 (1979), 589-618.
Mathematical Reviews (MathSciNet): MR552242
Zentralblatt MATH: 0466.65036
Digital Object Identifier: doi:10.1080/01630567908816036
K. Atkinson, The numerical evaluation of fixed points for completely continuous operators, SIAM J. Num. Anal. 10 (1973), 799-807.
Mathematical Reviews (MathSciNet): MR346602
Zentralblatt MATH: 0237.65040
Digital Object Identifier: doi:10.1137/0710065
JSTOR: links.jstor.org
--------, Iterative variants of the Nyström method for the numerical solution of integral equations, Numer. Math. 22 (1973), 17-31.
Mathematical Reviews (MathSciNet): MR337038
Digital Object Identifier: doi:10.1007/BF01436618
--------, A survey of numerical methods for Fredholm integral equations of the second kind, SIAM, Philadelphia, 1976.
Zentralblatt MATH: 0353.65069
--------, An automatic program for linear Fredholm integral equations of the second kind, ACM Trans. Math. Soft. 2 (1976), 154-171.
Mathematical Reviews (MathSciNet): MR418489
Digital Object Identifier: doi:10.1145/355681.355686
Zentralblatt MATH: 0328.65063
K. Atkinson and A. Bogomolny, The discrete Galerkin method for integral equations, Math. Comp. 48 (1987), 595-616.
Mathematical Reviews (MathSciNet): MR878693
Zentralblatt MATH: 0633.65134
Digital Object Identifier: doi:10.2307/2007830
K. Atkinson and G. Chandler, BIE methods for solving Laplace's equation with nonlinear boundary conditions: The smooth boundary case, Math. Comp. 55 (1990), 451-472.
Mathematical Reviews (MathSciNet): MR1035924
Zentralblatt MATH: 0709.65088
Digital Object Identifier: doi:10.2307/2008428
JSTOR: links.jstor.org
K. Atkinson and J. Flores, The discrete collocation method for nonlinear integral equations, submitted for publication.
K. Atkinson, I. Graham, and I. Sloan, Piecewise continuous collocation for integral equations, SIAM J. Num. Anal. 20, 172-186.
Mathematical Reviews (MathSciNet): MR687375
Zentralblatt MATH: 0514.65094
Digital Object Identifier: doi:10.1137/0720012
JSTOR: links.jstor.org
K. Atkinson and F. Potra, Projection and iterated projection methods for nonlinear integral equations, SIAM J. Num. Anal. 24, 1352-1373.
Mathematical Reviews (MathSciNet): MR917456
Zentralblatt MATH: 0655.65146
Digital Object Identifier: doi:10.1137/0724087
JSTOR: links.jstor.org
--------, The discrete Galerkin method for nonlinear integral equations, J. Integral Equations 1, 17-54.
Mathematical Reviews (MathSciNet): MR955161
Zentralblatt MATH: 0664.65056
Digital Object Identifier: doi:10.1216/JIE-1988-1-1-17
Project Euclid: euclid.jiea/1214948190
C. Baker, The numerical treatment of integral equations, Oxford University Press, Oxford (1977), 685-754.
Mathematical Reviews (MathSciNet): MR467215
H. Brakhage, Über die numerische Behandlung von Integralgleichungen nach der Quadraturformelmethode, Numer. Math. 2 (1960), 183-196.
Mathematical Reviews (MathSciNet): MR129147
Digital Object Identifier: doi:10.1007/BF01386221
F. Browder, Nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type, in Contributions to nonlinear functional analysis (E. Zarantonello, ed.), Academic Press (1971), 425-500.
Mathematical Reviews (MathSciNet): MR394340
H. Brunner and P. van der Houwen, The numerical solution of Volterra equations, North-Holland, Amsterdam, 1986.
Mathematical Reviews (MathSciNet): MR871871
J. Flores, Iteration methods for solving integral equations of the second kind, Ph.D. thesis, University of Iowa, Iowa City, Iowa, 1990.
M. Ganesh and M. Joshi, Discrete numerical solvability of Hammerstein integral equations of mixed type, J. Integral Equations 2 (1989), 107-124.
Mathematical Reviews (MathSciNet): MR1033206
Digital Object Identifier: doi:10.1216/JIE-1989-2-1-107
Project Euclid: euclid.jiea/1214947953
Zentralblatt MATH: 0708.65125
--------, Numerical solvability of Hammerstein integral equations of mixed type, IMA J. Numerical Anal. 11 (1991), 21-31.
Mathematical Reviews (MathSciNet): MR1089546
Digital Object Identifier: doi:10.1093/imanum/11.1.21
Zentralblatt MATH: 0719.65093
M. Golberg, Perturbed projection methods for various classes of operator and integral equations, in Numerical solution of integral equations (M. Golberg, ed.), Plenum Press, New York, 1990.
Mathematical Reviews (MathSciNet): MR1067151
A. Griewank, The local convergence of Broyden-like methods on Lipschitzian problems in Hilbert space, SIAM J. Num. Anal. 24 (1987), 684-705.
Mathematical Reviews (MathSciNet): MR888756
Digital Object Identifier: doi:10.1137/0724045
JSTOR: links.jstor.org
Zentralblatt MATH: 0627.65067
W. Hackbusch, Multi-grid methods and applications, Springer-Verlag, Berlin, 1985.
G. Hsiao and J. Saranen, Boundary element solution of the heat conduction problem with a nonlinear boundary condition, to appear.
O. Hübner, The Newton method for solving the Theodorsen integral equation, J. Comp. Appl. Math. 14 (1986), 19-30.
Mathematical Reviews (MathSciNet): MR829029
Digital Object Identifier: doi:10.1016/0377-0427(86)90129-9
Zentralblatt MATH: 0582.30006
S. Joe, Discrete collocation methods for second kind Fredholm integral equations, SIAM J. Num. Anal. 22 (1985), 1167-1177.
Mathematical Reviews (MathSciNet): MR811190
Digital Object Identifier: doi:10.1137/0722070
JSTOR: links.jstor.org
Zentralblatt MATH: 0597.65095
H. Kaneko, R. Noren, and Y. Xu, Numerical solutions for weakly singular Hammerstein equations and their superconvergence, submitted for publication.
L. Kantorovich and G. Akilov, Functional analysis in normed spaces, Pergamon Press, London, 1964.
Mathematical Reviews (MathSciNet): MR213845
C.T. Kelley, Approximation of solutions of some quadratic integral equations in transport theory, J. Integral Equations 4 (1982), 221-237.
Mathematical Reviews (MathSciNet): MR661170
--------, Operator prolongation methods for nonlinear equations, AMS Lect. Appl. Math., to appear.
Mathematical Reviews (MathSciNet): MR1066292
--------, A fast two-grid method for matrix $H$-equations, Trans. Theory Stat. Physics 18 (1989), 185-204.
Mathematical Reviews (MathSciNet): MR1010023
Digital Object Identifier: doi:10.1080/00411458908204320
C.T. Kelley and J. Northrup, A pointwise quasi-Newton method for integral equations, SIAM J. Num. Anal. 25 (1988), 1138-1155.
Mathematical Reviews (MathSciNet): MR960870
Digital Object Identifier: doi:10.1137/0725065
JSTOR: links.jstor.org
Zentralblatt MATH: 0661.65142
C.T. Kelley and E. Sachs, Broyden's method for approximate solution of nonlinear integral equations, J. Integral Equations 9 (1985), 25-43.
Mathematical Reviews (MathSciNet): MR793102
--------, Fast algorithms for compact fixed point problems with inexact function evaluations, submitted for publication.
--------, Mesh independence of Newton-like methods for infinite dimensional problems, submitted for publication.
H. Keller, Lectures on numerical methods in bifurcation problems, Tata Institute of Fundamental Research, Narosa Publishing House, New Delhi, 1987.
Mathematical Reviews (MathSciNet): MR910499
J. Keller and S. Antman, eds., Bifurcation theory and nonlinear eigenvalue problems, Benjamin Press, New York, 1969.
Mathematical Reviews (MathSciNet): MR241213
M. Krasnoselskii, Topological methods in the theory of nonlinear integral equations, Macmillan, New York, 1964.
Mathematical Reviews (MathSciNet): MR159197
M. Krasnoselskii, G. Vainikko, P. Zabreiko, Y. Rutitskii, and V. Stetsenko, Approximate solution of operator equations, P. Noordhoff, Groningen, 1972.
Mathematical Reviews (MathSciNet): MR385655
M. Krasnoselskii and P. Zabreiko, Geometrical methods of nonlinear analysis, Springer-Verlag, Berlin, 1984.
Mathematical Reviews (MathSciNet): MR736839
S. Kumar, Superconvergence of a collocation-type method for Hammerstein equations, IMA J. Numerical Anal. 7 (1987), 313-326.
Mathematical Reviews (MathSciNet): MR968527
Digital Object Identifier: doi:10.1093/imanum/7.3.313
Zentralblatt MATH: 0637.65140
--------, A discrete collocation-type method for Hammerstein equations, SIAM J. Num. Anal. 25 (1988), 328-341.
Mathematical Reviews (MathSciNet): MR933728
Digital Object Identifier: doi:10.1137/0725023
JSTOR: links.jstor.org
Zentralblatt MATH: 0647.65090
S. Kumar and I. Sloan, A new collocation-type method for Hammerstein integral equations, Math. Comp. 48 (1987), 585-593.
Mathematical Reviews (MathSciNet): MR878692
Digital Object Identifier: doi:10.2307/2007829
Zentralblatt MATH: 0616.65142
H. Lee, Multigrid methods for the numerical solution of integral equations, Ph.D. thesis, University of Iowa, Iowa City, Iowa, 1991.
P. Linz, Analytical and numerical methods for Volterra integral equations, SIAM, Philadelphia, PA, 1985.
Mathematical Reviews (MathSciNet): MR796318
L. Milne-Thomson, Theoretical hydrodynamics, 5th ed., Macmillan, New York, 1968.
R.H. Moore, Newton's method and variations, in Nonlinear integral equations, (P.M. Anselone, ed.) University of Wisconsin Press, Madison, WI, 1964.
Mathematical Reviews (MathSciNet): MR163447
--------, Differentiability and convergence for compact nonlinear operators, J. Math. Anal. Appl. 16 (1966), 65-72.
Mathematical Reviews (MathSciNet): MR196549
Digital Object Identifier: doi:10.1016/0022-247X(66)90186-7
Zentralblatt MATH: 0147.34803
--------, Approximations to nonlinear operator equations and Newton's method, Numer. Math. 12 (1966), 23-34.
Zentralblatt MATH: 0165.17301
I. Moret and P. Omari, A quasi-Newton method for solving fixed point problems in Hilbert spaces, J. Comp. Appl. Math. 20 (1987), 333-340.
Mathematical Reviews (MathSciNet): MR920403
Zentralblatt MATH: 0633.65141
Digital Object Identifier: doi:10.1016/0377-0427(87)90150-6
--------, A projective Newton method for semilinear operator equations in Banach spaces, Tech. Rep. #184, Dept. of Math., University of Trieste, Trieste, Italy, 1989.
Zentralblatt MATH: 0712.65052
W. Petryshyn, On nonlinear $P$-compact operators in Banach space with applications to constructive fixed-point theorems, J. Math. Anal. Appl. 15 (1966), 228-242.
Mathematical Reviews (MathSciNet): MR202014
Zentralblatt MATH: 0149.10602
Digital Object Identifier: doi:10.1016/0022-247X(66)90114-4
--------, Projection methods in nonlinear numerical functional analysis, J. Math. Mech. 17 (1967), 353-372.
Mathematical Reviews (MathSciNet): MR218941
Zentralblatt MATH: 0162.20202
--------, On the approximation solvability of nonlinear equations, Math. Ann. 177 (1968), 156-164.
Mathematical Reviews (MathSciNet): MR226458
Zentralblatt MATH: 0162.20301
Digital Object Identifier: doi:10.1007/BF01350791
L. Rall, Computational solution of nonlinear operator equations, Wiley, New York, 1969.
Mathematical Reviews (MathSciNet): MR240944
K. Ruotsalainen and J. Saranen, On the collocation method for a nonlinear boundary integral equation, J. Comp. App. Math., to appear.
Mathematical Reviews (MathSciNet): MR1038855
Zentralblatt MATH: 0684.65098
Digital Object Identifier: doi:10.1016/0377-0427(89)90345-2
K. Ruotsalainen and W. Wendland, On the boundary element method for some nonlinear boundary value problems, Numer. Math. 53 (1988), 299-314.
Mathematical Reviews (MathSciNet): MR948589
Zentralblatt MATH: 0651.65081
Digital Object Identifier: doi:10.1007/BF01404466
J. Saranen, Projection methods for a class of Hammerstein equations, SIAM J. Num. Anal., to appear.
Mathematical Reviews (MathSciNet): MR1080330
Zentralblatt MATH: 0717.65034
Digital Object Identifier: doi:10.1137/0727083
JSTOR: links.jstor.org
I. Sloan, Improvement by iteration for compact operator equations, Math. Comp. 30 (1976), 758-764.
Mathematical Reviews (MathSciNet): MR474802
Zentralblatt MATH: 0343.45010
Digital Object Identifier: doi:10.2307/2005396
JSTOR: links.jstor.org
D. Stibbs and R. Weir, On the $H$-function for isotropic scattering, Mon. Not. R. Astro. Soc. 119 (1959), 512-525.
Mathematical Reviews (MathSciNet): MR112725
G. Vainikko, Funktionalanalysis der Diskretisierungsmethoden, Teubner, Leipzig, 1976.
Mathematical Reviews (MathSciNet): MR468159
Zentralblatt MATH: 0343.65023
G. Vainikko and O. Karma, The convergence of approximate methods for solving linear and non-linear operator equations, Zh. $v\bar y$ chisl. Mat. mat. Fiz. 14 (1974), 828-837.
Mathematical Reviews (MathSciNet): MR375769
R. Weiss, On the approximation of fixed points of nonlinear compact operators, SIAM J. Num. Anal. 11 (1974), 550-553.
Mathematical Reviews (MathSciNet): MR349049
Digital Object Identifier: doi:10.1137/0711046
JSTOR: links.jstor.org
E. Zeidler, Nonlinear functional analysis and its applications, Vol. I: Fixed-point theorems, Springer-Verlag, Berlin, 1986.
Mathematical Reviews (MathSciNet): MR816732
Zentralblatt MATH: 0583.47050

2013 © Rocky Mountain Mathematics Consortium

Journal of Integral Equations and Applications

Journal of Integral Equations and Applications
Euclid Prime Journal