Uniform error estimates of Galerkin methods for monotone Abel-Volterra integral equations on the half-line

Author:
P. P. B. Eggermont

Journal:
Math. Comp. **53** (1989), 157-189

MSC:
Primary 65R20; Secondary 45D05, 47H17

MathSciNet review:
969485

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Abstract: We consider Galerkin methods for monotone Abel-Volterra integral equations of the second kind on the half-line. The theory follows from Kolodner's theory of monotone Hammerstein, equations. We derive the theory from the theory by relating the - and -spectra of operators of the form to one another. Here denotes convolution, and and . As an extra condition we need , with . We also prove the discrete analogue. In particular, we verify that the Galerkin matrix satisfies the "discrete" conditions.

**[1]**Christopher T. H. Baker and Geoffrey F. Miller (eds.),*Treatment of integral equations by numerical methods*, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1982. MR**755336****[2]**V. L. Bakke and Z. Jackiewicz,*Stability analysis of product 𝜃-methods for Abel integral equations of the second kind*, Numer. Math.**48**(1986), no. 2, 127–136. MR**824164**, 10.1007/BF01389867**[3]**Melvin S. Berger,*Nonlinearity and functional analysis*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Lectures on nonlinear problems in mathematical analysis; Pure and Applied Mathematics. MR**0488101****[4]**Hermann Brunner,*The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes*, Math. Comp.**45**(1985), no. 172, 417–437. MR**804933**, 10.1090/S0025-5718-1985-0804933-3**[5]**H. Brunner and P. J. van der Houwen,*The numerical solution of Volterra equations*, CWI Monographs, vol. 3, North-Holland Publishing Co., Amsterdam, 1986. MR**871871****[6]**N. G. de Bruijn and P. Erdös,*On a recursion formula and on some Tauberian theorems*, J. Research Nat. Bur. Standards**50**(1953), 161–164. MR**0054745****[7]**John Rozier Cannon,*The one-dimensional heat equation*, Encyclopedia of Mathematics and its Applications, vol. 23, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. With a foreword by Felix E. Browder. MR**747979****[8]**P. P. B. Eggermont, "A generalization of W. A. J. Luxemburg's asymptotic problem concerning the Laplace transform,"*Indag. Math.*, v. 42, 1981, pp. 257-265.**[9]**P. P. B. Eggermont,*On monotone Abel-Volterra integral equations on the half line*, Numer. Math.**52**(1988), no. 1, 65–79. MR**918317**, 10.1007/BF01401022**[10]**Paul P. B. Eggermont,*On Galerkin methods for Abel-type integral equations*, SIAM J. Numer. Anal.**25**(1988), no. 5, 1093–1117. MR**960868**, 10.1137/0725063**[11]**P. P. B. Eggermont, "Approximation of non-smooth solutions of Abel-Volterra integral equations," manuscript, 1986.**[12]**A. Erdélyi et al.,*Higher Transcendental Functions*, Vol. 3, McGraw-Hill, New York, 1955.**[13]**George A. Hufford,*An integral equation approach to the problem of wave propagation over an irregular surface*, Quart. Appl. Math.**9**(1952), 391–404. MR**0044350****[14]**D. Kershaw,*Some results for Abel-Volterra integral equations of the second kind*, Treatment of integral equations by numerical methods (Durham, 1982), Academic Press, London, 1982, pp. 273–282. MR**755362****[15]**Ignace I. Kolodner,*Equations of Hammerstein type in Hilbert spaces*, J. Math. Mech.**13**(1964), 701–750. MR**0171184****[16]**Stig-Olof Londen,*On some nonintegrable Volterra kernels with integrable resolvents including some applications to Riesz potentials*, J. Integral Equations**10**(1985), no. 1-3, suppl., 241–289. Integro-differential evolution equations and applications (Trento, 1984). MR**831246****[17]**Ch. Lubich,*Fractional linear multistep methods for Abel-Volterra integral equations of the second kind*, Math. Comp.**45**(1985), no. 172, 463–469. MR**804935**, 10.1090/S0025-5718-1985-0804935-7**[18]**Ch. Lubich,*On the numerical solution of Volterra equations with unbounded nonlinearity*, J. Integral Equations**10**(1985), no. 1-3, suppl., 175–183. Integro-differential evolution equations and applications (Trento, 1984). MR**831243****[19]**W. A. J. Luxemburg,*On an asymptotic problem concerning the Laplace transform*, Applicable Anal.**8**(1978/79), no. 1, 61–70. MR**516522**, 10.1080/00036817808839212**[20]**Chiang C. Mei and E. O. Tuck,*Forward scattering by long thin bodies*, SIAM J. Appl. Math.**39**(1980), no. 1, 178–191. MR**585838**, 10.1137/0139016**[21]**Olavi Nevanlinna,*On the stability of discrete Volterra equations*, Treatment of integral equations by numerical methods (Durham, 1982), Academic Press, London, 1982, pp. 139–147. MR**755349****[22]**S. L. Paveri-Fontana and Rossella Rigacci,*A singularly perturbed weakly-singular integro-differential problem from analytical chemistry*, Numerical analysis of singular perturbation problems (Proc. Conf., Math. Inst., Catholic Univ., Nijmegen, 1978) Academic Press, London-New York, 1979, pp. 475–484. MR**556538****[23]**Elias M. Stein and Guido Weiss,*Introduction to Fourier analysis on Euclidean spaces*, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR**0304972****[24]**Herman J. J. te Riele,*Collocation methods for weakly singular second-kind Volterra integral equations with nonsmooth solution*, IMA J. Numer. Anal.**2**(1982), no. 4, 437–449. MR**692290**, 10.1093/imanum/2.4.437

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1989-0969485-X

Article copyright:
© Copyright 1989
American Mathematical Society