Abstract
A formal approach is described for obtaining the asymptotic solution to a class of singularly perturbed Fredholm integral equations. The approach is illustrated through application to some example problems which arise in heat transfer, diffraction theory, crack mechanics, Markov processes and low order eigenvalue problems.
Similar content being viewed by others
References
J. S. Angell and W. E. Olmstead,Singular perturbation analysis of an integrodifferential equation modelling filament stretching, Z. angew. Math. Phys.36, 487–490 (1985).
J. S. Angell and W. E. Olmstead,Singularly perturbed Volterra integral equations, SIAM J. Appl. Math.47, 1–14 (1987).
J. S. Angell and W. E. Olmstead,Singularly perturbed Volterra integral equations II, SIAM J. Appl. Math.47, 1150–1162 (1987).
A. Erdelyi,Asymptotic evaluation of integrals involving fractional derivatives, SIAM J. Math. Anal.5, 159–171 (1974).
G. I. Eskin,Boundary value problems for elliptic pseudodifferential equations, Am. Math. Soc. 1981.
A. K. Gautesen,On the asymptotic solution to a class of linear integral equations, SIAM J. Appl. Math. (to appear).
A. K. Gautesen,On a class of Fredholm integral equations—asymptotic expansions of solutions and eigenfunction (recently submitted for publication).
J. Hadamard,Lectures on Cauchy's problem in linear partial differential equations, Dover, New York, 1952.
F. C. Hoppensteadt,An algorithm for approximate solutions to weakly filtered synchronous control systems and nonlinear renewal processes, SIAM J. Appl. Math.43, 834–843 (1983).
A. C. Kaya and F. Erdogan,On the solution of integral equations with strongly singular kernels, Quart. Appl. Math.45, 105–122 (1987).
C. Knessl, B. J. Matkowsky, Z. Schuss and C. Tier,A singular perturbation approach to first passage times for Markov jump processes, J. Stat. Phys.42, 169–184 (1986).
W. T. Koiter,On the diffusion of load from a stiffener into a sheet, Quart. J. Mech. Appl. Math.8, 164–178 (1955).
C. G. Lange and D. R. Smith,Singular perturbation analysis of integral equations, Studies in Appl. Math.79, 1–63 (1988).
A. S. Lodge, J. B. McLeod and J. A. Nohel,A nonlinear singularly perturbed Volterra-integrodifferential equation occurring in polymer rheology, Proc. Roy. Soc. Edinburgh80A, 99–137 (1978).
S. Nemat-Nasser and M. Hori,Toughening by partial or full bridging of cracks in ceramics and fiber reinforced composites, Mech. of Materials6, 245–269 (1987).
S. Nemat-Nasser and M. Hori,Solution of a class of strongly-singular integral equations, SIAM J. Appl. Math. (to appear).
B. Noble.The Wiener Hopf technique, Pergamon Press, Oxford 1958.
W. E. Olmstead and R. A. Handelsman,Singular perturbation analysis of a certain Volterra integral equation, Z. angew. Math. Phys.23, 889–900 (1972).
L. Sirovich and B. A. Knight,Contributions to the eigenvalue problem for slowly varying operators, SIAM J. Appl. Math.42, 356–377 (1982).
H. Widom,Extreme eigenvalues of translation kernels, Trans. Amer. Math. Soc.100, 256–262 (1961).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Olmstead, W.E., Gautesen, A.K. Asymptotic solution of some singularly perturbed Fredholm integral equations. Z. angew. Math. Phys. 40, 230–244 (1989). https://doi.org/10.1007/BF00945000
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00945000