On the numerical solution of singular integro-differential equations

Ioakimidis, N. I.; Theocaris, P. S.

doi:10.1090/qam/548991

Authors: N. I. Ioakimidis and P. S. Theocaris
Journal: Quart. Appl. Math. 37 (1979), 325-331
MSC: Primary 65R20
DOI: https://doi.org/10.1090/qam/548991
MathSciNet review: 548991
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Abstract: A method of numerical solution of a sufficiently wide class of Cauchy-type singular integrodifferential equations along a straight finite interval is presented. This method consists of approximating the integrals in such an equation by using appropriate numerical integration rules and appropriately-selected collocation points and reducing such an equation to a system of linear algebraic equations. This technique constitutes a direct generalization of the corresponding methods of numerical solution of Cauchy-type singular integral equations and presents some advantages over the classical Multhopp method of numerical solution of Cauchy-type singular integrodifferential equations, to which it reduces in some special cases. An application to a specific equation is also made.

A. I. Kalandiya, Approximate solution of a class of singular integral equations, Dokl. Akad. Nauk SSSR 125, 715–718 (1959) (English translation available through the British Library-Lending Division: RTS 8730, 1974)

A. I. Kalandiya, Mathematical methods of two-dimensional elasticity, Mir Publishers, Moscow, 1975. Translated from the Russian by M. Konyaeva [M. Konjaeva]. MR 0400846
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
P. S. Theocaris and N. I. Ioakimidis, Numerical integration methods for the solution of singular integral equations, Quart. Appl. Math. 35 (1977/78), no. 1, 173–187. MR 445873, DOI https://doi.org/10.1090/S0033-569X-1977-0445873-X
N. I. Ioakimidis and P. S. Theocaris, On the numerical evaluation of Cauchy principal value integrals, Rev. Roumaine Sci. Tech. Sér. Méc. Appl. 22 (1977), no. 6, 803–818. MR 483321
N. I. Ioakimidis and P. S. Theocaris, On the numerical solution of a class of singular integral equations, J. Mathematical and Physical Sci. 11 (1977), no. 3, 219–235. MR 483590
P. S. Theocaris, On the numerical solution of Cauchy-type singular integral equations, Serdica 2 (1976), no. 3, 252–275. MR 448968
P. S. Theocaris and N. I. Ioakimidis, On the numerical solution of Cauchy type singular integral equations and the determination of stress intensity factors in case of complex singularities, Z. Angew. Math. Phys. 28 (1977), no. 6, 1085–1098 (English, with French summary). MR 464623, DOI https://doi.org/10.1007/BF01601675
P. S. Theocaris and N. I. Ioakimidis, Numerical solution of Cauchy type singular integral equations, Trans. Acad. Athens 40 (1977), no. 1, 7–39 (English, with Greek summary). MR 0659483
N. I. Ioakimidis and P. S. Theocaris, The numerical evaluation of a class of generalized stress intensity factors by use of the Lobatto-Jacobi numerical integration rule, Internat. J. Fracture 14 (1978), no. 5, 469–484 (English, with French summary). MR 599793, DOI https://doi.org/10.1007/BF01390469
P. S. Theocaris and G. Tsamasphyros, Numerical solution of systems of singular integral equations with variable coefficients, Applicable Anal. 9 (1979), no. 1, 37–52. MR 536690, DOI https://doi.org/10.1080/00036817908839250

H. Multhopp, Die Berechnung der Auftriebsverteilung von Tragflügeln, Luftfahrtforschung 15, 153–166 (1938)

S. M. Sharfuddin, A two-dimensional discontinuous boundary-value problem for circular regions and Prandtl’s integral equation, Acta Mech. 4, 374–381 (1967)

M. Stippes, H. B. Wilson Jr., and F. N. Krull, A contact stress problem for a smooth disk in an infinite plate, Proc. 4th U.S. Nat. Congr. Appl. Mech. (Univ. California, Berkeley, Calif., 1962) Amer. Soc. Mech. Engrs., New York, 1962, pp. 799–806. MR 0152183
Steen Krenk, On the use of the interpolation polynomial for solutions of singular integral equations, Quart. Appl. Math. 32 (1974/75), 479–484. MR 474919, DOI https://doi.org/10.1090/S0033-569X-1975-0474919-7