On the numerical solution of singular integro-differential equations

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.

**[1]**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)**[2]**A. I. Kalandiya,*Mathematical methods of two-dimensional elasticity*, Mir Publishers, Moscow, 1975. Translated from the Russian by M. Konyaeva [M. Konjaeva]. MR**0400846****[3]**F. Erdogan and G. D. Gupta,*On the numerical solution of singular integral equations*, Quart. Appl. Math.**29**(1971/72), 525–534. MR**0408277**, https://doi.org/10.1090/S0033-569X-1972-0408277-4**[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**0445873**, https://doi.org/10.1090/S0033-569X-1977-0445873-X**[5]**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**0483321****[6]**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**0483590****[7]**P. S. Theocaris,*On the numerical solution of Cauchy-type singular integral equations*, Serdica**2**(1976), no. 3, 252–275. MR**0448968****[8]**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**0464623**, https://doi.org/10.1007/BF01601675**[9]**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****[10]**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**, https://doi.org/10.1007/BF01390469**[11]**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**, https://doi.org/10.1080/00036817908839250**[12]**H. Multhopp,*Die Berechnung der Auftriebsverteilung von Tragflügeln*, Luftfahrtforschung**15**, 153-166 (1938)**[13]**S. M. Sharfuddin,*A two-dimensional discontinuous boundary-value problem for circular regions and Prandtl's integral equation*, Acta Mech.**4**, 374-381 (1967)**[14]**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****[15]**Steen Krenk,*On the use of the interpolation polynomial for solutions of singular integral equations*, Quart. Appl. Math.**32**(1974/75), 479–484. MR**0474919**, https://doi.org/10.1090/S0033-569X-1975-0474919-7

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DOI:
https://doi.org/10.1090/qam/548991

Article copyright:
© Copyright 1979
American Mathematical Society