On the solution of integral equations with a generalized Cauchy kernel
Authors:
A. C. Kaya and F. Erdogan
Journal:
Quart. Appl. Math. 45 (1987), 455-469
MSC:
Primary 73C40; Secondary 45E05
DOI:
https://doi.org/10.1090/qam/910453
MathSciNet review:
910453
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: In this paper a certain class of singular integral equations that may arise from the mixed boundary value problems in nonhomogeneous materials is considered. The distinguishing feature of these equations is that in addition to the Cauchy singularity, the kernels contain terms that are singular only at the end points. In the form of the singular integral equations adopted, the density function is a potential or a displacement and consequently the kernel has strong singularities of the form ${\left ( {t - x} \right )^{ - 2}},{x^{n - 2}}{\left ( {t + x} \right )^n}\left ( {n \ge 2,0 < x,t < b} \right )$. The complex function theory is used to determine the fundamental function of the problem for the general case and a simple numerical technique is described to solve the integral equation. Two examples from the theory of elasticity are then considered to show the application of the technique.
- Ian N. Sneddon, Mixed boundary value problems in potential theory, North-Holland Publishing Co., Amsterdam; Interscience Publishers John Wiley & Sons, Inc., New York, 1966. MR 0216018
- N. I. Muskhelishvili, Singular integral equations, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR 0355494
F. Erdogan, Mixed boundary value problems, Mechanics Today, S. Nemat-Nasser, ed. Vol. 4, 1–86 (1978)
- F. Erdogan and G. D. Gupta, On the numerical solution of singular integral equations, Quart. Appl. Math. 29 (1971/72), 525–534. MR 408277, DOI https://doi.org/10.1090/S0033-569X-1972-0408277-4
- A. C. Kaya and F. Erdogan, On the solution of integral equations with strongly singular kernels, Quart. Appl. Math. 45 (1987), no. 1, 105–122. MR 885173, DOI https://doi.org/10.1090/S0033-569X-1987-0885173-4
F. Erdogan and T. S. Cook, Antiplane shear crack terminating at and going through a bimaterial interface, Int. Journal of Fracture 10, 227–240 (1974)
- H. F. Bueckner, On a class of singular integral equations, J. Math. Anal. Appl. 14 (1966), 392–426. MR 193460, DOI https://doi.org/10.1016/0022-247X%2866%2990002-3
T. S. Cook and F. Erdogan, Stresses in bonded materials with a crack perpendicular to the interface, Int. J. Engng. Sci. 10, 667–697 (1972)
T. W. Chou, Dislocation pileups and elastic cracks at a bimaterial interface, Metallurgical Transactions 1, 1245–1248 (1970)
P. S. Theocaris and N. I. Ioakimidis, Stress intensity factors at the tips of an antiplane shear crack terminating at a bimaterial interface, Int. Journal of Fracture 13, 549–552 (1977)
J. L. Bassani and F. Erdogan, Stress intensity factors in bonded half planes containing inclined cracks and subjected to antiplane shear loading, Int. Journal of Fracture 15, 145–158 (1979)
I. N. Sneddon, Mixed boundary value problems in potential theory, North Holland, Amsterdam, 1966
N. I. Muskhelishvili, Singular integral equations, Noordhoff, Groningen, The Netherlands, 1953
F. Erdogan, Mixed boundary value problems, Mechanics Today, S. Nemat-Nasser, ed. Vol. 4, 1–86 (1978)
F. Erdogan and G. D. Gupta, On the numerical solution of singular integral equations, Quart. Appl. Math. 29, 525–534 (1972)
A. C. Kaya and F. Erdogan, On the solution of integral equations with strongly singular kernels, Quart. Appl. Math. 45, 105–122 (1987)
F. Erdogan and T. S. Cook, Antiplane shear crack terminating at and going through a bimaterial interface, Int. Journal of Fracture 10, 227–240 (1974)
H. F. Bueckner, On a class of singular integral equations, J. Math. Analysis and Applications 14, 392–426 (1966)
T. S. Cook and F. Erdogan, Stresses in bonded materials with a crack perpendicular to the interface, Int. J. Engng. Sci. 10, 667–697 (1972)
T. W. Chou, Dislocation pileups and elastic cracks at a bimaterial interface, Metallurgical Transactions 1, 1245–1248 (1970)
P. S. Theocaris and N. I. Ioakimidis, Stress intensity factors at the tips of an antiplane shear crack terminating at a bimaterial interface, Int. Journal of Fracture 13, 549–552 (1977)
J. L. Bassani and F. Erdogan, Stress intensity factors in bonded half planes containing inclined cracks and subjected to antiplane shear loading, Int. Journal of Fracture 15, 145–158 (1979)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
73C40,
45E05
Retrieve articles in all journals
with MSC:
73C40,
45E05
Additional Information
Article copyright:
© Copyright 1987
American Mathematical Society