×

zbMATH — the first resource for mathematics

A survey of numerical methods for solving nonlinear integral equations. (English) Zbl 0760.65118
The author gives a survey of numerical methods for solving nonlinear integral equations of the second kind such as the following: \[ x(t)=y(t)+\int_ D K(t,s)f(s,x(s))ds,\quad t\in D. \] Projection methods (such as Nyström technique and iterated projection method) are described and convergence results are given. Also Galerkin’s method is detailed and the concrete example \[ x(t)=y(t)+\int^ 1_ 0(t+s+x(s))^{-1}ds,\quad 0\leq t\leq 1 \] is solved numerically.
We also find a discrete collocation method, a discrete Galerkin method and grid methods, but I did not find a description of a new and powerful method for solving nonlinear functional equations: Adomian’s method. This technique is a decomposition method giving the exact solution as a series of functions. Practically one uses a truncated series and thus an approximated solution [see Ph.D. thesis of Lionel Gabet, Ecole Centrale de Paris (1992)].

MSC:
65R20 Numerical methods for integral equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
45G10 Other nonlinear integral equations
PDF BibTeX Cite
Full Text: DOI
References:
[1] 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. JSTOR: · Zbl 0591.65043
[2] H. Amann, Zum Galerkin-Verfahren für die Hammersteinsche Gleichung , Arch. Rational Mech. Anal. 35 (1969), 114-121. · Zbl 0186.20902
[3] ——–, Über die Konvergenzgeschwindigkeit des Galerkin-Verfahren für die Hammersteinsche Gleichung , Arch. Rational Mech. Anal. 37 (1970), 143-153. · Zbl 0242.49029
[4] P. Anselone, ed., Nonlinear integral equations , University of Wisconsin, Madison, 1964. · Zbl 0149.11502
[5] ——–, Collectively compact operator approximation theory and applications to integral equations , Prentice-Hall, New Jersey, 1971. · Zbl 0228.47001
[6] P. Anselone and R. Ansorge, Compactness principles in nonlinear operator approximation theory , Numer. Funct. Anal. Optim. 6 (1979), 589-618. · Zbl 0466.65036
[7] K. Atkinson, The numerical evaluation of fixed points for completely continuous operators , SIAM J. Num. Anal. 10 (1973), 799-807. JSTOR: · Zbl 0237.65040
[8] ——–, Iterative variants of the Nyström method for the numerical solution of integral equations , Numer. Math. 22 (1973), 17-31. · Zbl 0267.65089
[9] ——–, A survey of numerical methods for Fredholm integral equations of the second kind , SIAM, Philadelphia, 1976. · Zbl 0353.65069
[10] ——–, An automatic program for linear Fredholm integral equations of the second kind , ACM Trans. Math. Soft. 2 (1976), 154-171. · Zbl 0328.65063
[11] K. Atkinson and A. Bogomolny, The discrete Galerkin method for integral equations , Math. Comp. 48 (1987), 595-616. · Zbl 0633.65134
[12] 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. JSTOR: · Zbl 0709.65088
[13] K. Atkinson and J. Flores, The discrete collocation method for nonlinear integral equations , submitted for publication. · Zbl 0771.65090
[14] K. Atkinson, I. Graham, and I. Sloan, Piecewise continuous collocation for integral equations , SIAM J. Num. Anal. 20 , 172-186. JSTOR: · Zbl 0514.65094
[15] K. Atkinson and F. Potra, Projection and iterated projection methods for nonlinear integral equations , SIAM J. Num. Anal. 24 , 1352-1373. JSTOR: · Zbl 0655.65146
[16] ——–, The discrete Galerkin method for nonlinear integral equations , J. Integral Equations 1 , 17-54. · Zbl 0664.65056
[17] C. Baker, The numerical treatment of integral equations , Oxford University Press, Oxford (1977), 685-754.
[18] H. Brakhage, Über die numerische Behandlung von Integralgleichungen nach der Quadraturformelmethode , Numer. Math. 2 (1960), 183-196. · Zbl 0142.11903
[19] 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. · Zbl 0267.47038
[20] H. Brunner and P. van der Houwen, The numerical solution of Volterra equations , North-Holland, Amsterdam, 1986. · Zbl 0611.65092
[21] J. Flores, Iteration methods for solving integral equations of the second kind , Ph.D. thesis, University of Iowa, Iowa City, Iowa, 1990.
[22] M. Ganesh and M. Joshi, Discrete numerical solvability of Hammerstein integral equations of mixed type , J. Integral Equations 2 (1989), 107-124. · Zbl 0708.65125
[23] ——–, Numerical solvability of Hammerstein integral equations of mixed type , IMA J. Numerical Anal. 11 (1991), 21-31. · Zbl 0719.65093
[24] 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. · Zbl 0732.65046
[25] A. Griewank, The local convergence of Broyden-like methods on Lipschitzian problems in Hilbert space , SIAM J. Num. Anal. 24 (1987), 684-705. JSTOR: · Zbl 0627.65067
[26] W. Hackbusch, Multi-grid methods and applications , Springer-Verlag, Berlin, 1985. · Zbl 0595.65106
[27] G. Hsiao and J. Saranen, Boundary element solution of the heat conduction problem with a nonlinear boundary condition , · Zbl 0848.35063
[28] O. Hübner, The Newton method for solving the Theodorsen integral equation , J. Comp. Appl. Math. 14 (1986), 19-30. · Zbl 0582.30006
[29] S. Joe, Discrete collocation methods for second kind Fredholm integral equations , SIAM J. Num. Anal. 22 (1985), 1167-1177. JSTOR: · Zbl 0597.65095
[30] H. Kaneko, R. Noren, and Y. Xu, Numerical solutions for weakly singular Hammerstein equations and their superconvergence , submitted for publication. · Zbl 0764.65085
[31] L. Kantorovich and G. Akilov, Functional analysis in normed spaces , Pergamon Press, London, 1964. · Zbl 0127.06104
[32] C.T. Kelley, Approximation of solutions of some quadratic integral equations in transport theory , J. Integral Equations 4 (1982), 221-237. · Zbl 0495.45010
[33] ——–, Operator prolongation methods for nonlinear equations , AMS Lect. Appl. Math., · Zbl 0697.65050
[34] ——–, A fast two-grid method for matrix \(H\)-equations , Trans. Theory Stat. Physics 18 (1989), 185-204. · Zbl 0684.65114
[35] C.T. Kelley and J. Northrup, A pointwise quasi-Newton method for integral equations , SIAM J. Num. Anal. 25 (1988), 1138-1155. JSTOR: · Zbl 0661.65142
[36] C.T. Kelley and E. Sachs, Broyden’s method for approximate solution of nonlinear integral equations , J. Integral Equations 9 (1985), 25-43. · Zbl 0576.65055
[37] ——–, Fast algorithms for compact fixed point problems with inexact function evaluations , submitted for publication. · Zbl 0754.65054
[38] ——–, Mesh independence of Newton-like methods for infinite dimensional problems , submitted for publication. · Zbl 0756.65085
[39] H. Keller, Lectures on numerical methods in bifurcation problems , Tata Institute of Fundamental Research, Narosa Publishing House, New Delhi, 1987.
[40] J. Keller and S. Antman, eds., Bifurcation theory and nonlinear eigenvalue problems , Benjam · Zbl 0181.00105
[41] M. Krasnoselskii, Topological methods in the theory of nonlinear integral equations , Macmillan, New York, 1964. · Zbl 0111.30303
[42] M. Krasnoselskii, G. Vainikko, P. Zabreiko, Y. Rutitskii, and V. Stetsenko, Approximate solution of operator equations , P. Noordhoff, Groningen, 1972.
[43] M. Krasnoselskii and P. Zabreiko, Geometrical methods of nonlinear analysis , Springer-Verlag, Berlin, 1984.
[44] S. Kumar, Superconvergence of a collocation-type method for Hammerstein equations , IMA J. Numerical Anal. 7 (1987), 313-326. · Zbl 0637.65140
[45] ——–, A discrete collocation-type method for Hammerstein equations , SIAM J. Num. Anal. 25 (1988), 328-341. JSTOR: · Zbl 0647.65090
[46] S. Kumar and I. Sloan, A new collocation-type method for Hammerstein integral equations , Math. Comp. 48 (1987), 585-593. · Zbl 0616.65142
[47] H. Lee, Multigrid methods for the numerical solution of integral equations , Ph.D. thesis, University of Iowa, Iowa City, Iowa, 1991.
[48] P. Linz, Analytical and numerical methods for Volterra integral equations , SIAM, Philadelphia, PA, 1985. · Zbl 0566.65094
[49] L. Milne-Thomson, Theoretical hydrodynamics , 5th ed., Macmillan, New York, 1968. · Zbl 0164.55802
[50] R.H. Moore, Newton’s method and variations , in Nonlinear integral equations , (P.M. Anselone, ed.) University of Wiscons · Zbl 0123.31803
[51] ——–, Differentiability and convergence for compact nonlinear operators , J. Math. Anal. Appl. 16 (1966), 65-72. · Zbl 0147.34803
[52] ——–, Approximations to nonlinear operator equations and Newton’s method , Numer. Math. 12 (1966), 23-34. · Zbl 0165.17301
[53] 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. · Zbl 0633.65141
[54] ——–, A projective Newton method for semilinear operator equations in Banach spaces , Tech. Rep. #184, Dept. of Math., University of Trieste, Trieste, Italy, 1989. · Zbl 0712.65052
[55] 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. · Zbl 0149.10602
[56] ——–, Projection methods in nonlinear numerical functional analysis , J. Math. Mech. 17 (1967), 353-372. · Zbl 0162.20202
[57] ——–, On the approximation solvability of nonlinear equations , Math. Ann. 177 (1968), 156-164. · Zbl 0162.20301
[58] L. Rall, Computational solution of nonlinear operator equations , Wiley, New York, 1969. · Zbl 0175.15804
[59] K. Ruotsalainen and J. Saranen, On the collocation method for a nonlinear boundary integral equation , J. Comp. App. Math., · Zbl 0684.65098
[60] K. Ruotsalainen and W. Wendland, On the boundary element method for some nonlinear boundary value problems , Numer. Math. 53 (1988), 299-314. · Zbl 0651.65081
[61] J. Saranen, Projection methods for a class of Hammerstein equations , SIAM J. Num. Anal., to appear. JSTOR: · Zbl 0717.65034
[62] I. Sloan, Improvement by iteration for compact operator equations , Math. Comp. 30 (1976), 758-764. JSTOR: · Zbl 0343.45010
[63] D. Stibbs and R. Weir, On the \(H\)-function for isotropic scattering , Mon. Not. R. Astro. Soc. 119 (1959), 512-525. · Zbl 0092.22403
[64] G. Vainikko, Funktionalanalysis der Diskretisierungsmethoden , Teubner, Leipzig, 1976. · Zbl 0343.65023
[65] 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.
[66] R. Weiss, On the approximation of fixed points of nonlinear compact operators , SIAM J. Num. Anal. 11 (1974), 550-553. JSTOR: · Zbl 0285.41019
[67] E. Zeidler, Nonlinear functional analysis and its applications , Vol. I: Fixed-point theorems , Springer-Verlag, Berlin, 1986. · Zbl 0583.47050
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.