A Galerkin solution to a regularized Cauchy singular integro-differential equation
Author:
Jay I. Frankel
Journal:
Quart. Appl. Math. 53 (1995), 245-258
MSC:
Primary 45E05; Secondary 45J05, 65R20
DOI:
https://doi.org/10.1090/qam/1330651
MathSciNet review:
MR1330651
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Abstract: This paper presents a Galerkin approach for solving a regularized version of the Cauchy singular, linear integro-differential equation \[ \frac {{d\Theta }}{{dx}}\left ( x \right ) - f\left ( x \right ) = \lambda \smallint _{y = 0}^1\frac {{\Theta \left ( y \right )}}{{x - y}}dy, \qquad x \in \left ( 0, 1 \right )\], subject to $\Theta \left ( 0 \right ) = \Theta \left ( 1 \right ) = 0$. This equation has appeared in both combined infrared gaseous radiation and molecular conduction, and elastic contact studies. A regularized formulation is produced which suggests the use of an expansion technique where the orthogonal basis functions are chosen as the Chebychev polynomials of the first kind. Accurate results, requiring a minimal computational cost, are formally documented and compared to a purely numerical solution.
T. S. Sankar, S. V. Hoa, and V. I. Fabrikant, Approximate solution of singular integro-differential equations in elastic contact problems, Internat. J. Numer. Methods Engrg. 18, 503–519 (1982)
F. Erdogan, Approximate solutions of systems of singular integral equations, SIAM J. Appl. Math. 17, 1041–1059 (1969)
F. Erdogan and G. D. Gupta, On the numerical solution of singular integral equations, Quart. Appl. Math. 30, 525–534 (1972)
A. Gerasoulis and R. P. Srivastav, A method for the numerical solution of singular integral equations with a principal value integral, Internat. J. Engrg. Sci. 19, 1293–1298 (1981)
S. R. Bland, The two-dimensional oscillating airfoil in a wind tunnel in subsonic flow, SIAM J. Appl. Math. 18, 830–848 (1970)
J. A. Fromme and M. A. Golberg, Aerodynamics interface effects on oscillating airfoils with controls in ventilated wind tunnels, AIAA J. 18, 417–426 (1980)
M. A. Golberg, The convergence of a collocation method for a class of Cauchy singular integral equations, J. Math. Anal. Appl. 100, 500–512 (1984)
M. A. Golberg and J. A. Fromme, On the $L_{2}$ convergence of collocation for the generalized airfoil equation, J. Math. Anal. Appl. 71, 271–286 (1979)
A. Plotkin and S. S. Dodbele, Slender wing in ground effect, AIAA J. 26, 493–494 (1988)
L. Dragos, Numerical solution of the equation for a thin airfoil in ground effect, AIAA J. 28, 2132–2134 (1990)
R. D. Cess and S. N. Tiwari, The interaction of thermal conduction and infrared gaseous radiation, Appl. Sci. Res. 20, 25–39 (1969)
E. M. Sparrow and R. D. Cess, Radiation Heat Transfer, McGraw-Hill, New York, 1978
M. A. Golberg, ed., Solution Methods for Integral Equations, Plenum Press, New York, 1979
M. A. Golberg, ed., Numerical Solution of Integral Equations, Plenum Press, New York, 1990
P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, 1985
C. T. H. Baker, The Numerical Treatment of Integral Equations, Clarendon Press, Oxford, 1978
L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge Univ. Press, Cambridge, 1988
K. E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia, 1976
F. G. Tricomi, Integral Equations, Dover, New York, 1985
H. Hochstadt, Integral Equations, Wiley, New York, 1973
C. D. Green, Integral Equation Methods, Barnes and Noble, New York, 1969
D. Porter and D. S. G. Stirling, Integral Equations, Cambridge Univ. Press, Cambridge, 1990
J. I. Frankel, Regularized and preconditioned boundary integral solution to heat transfer in a participating gas flow between parallel plates, Numer. Heat Transfer 19, 105–126 (1991)
J. I. Frankel, Nonlinear heat transfer: Solution of singular, nonlinear integral equations, Engrg. Anal. 8, 231–238 (1991)
A. S. Peters, A note on the integral equation of the first kind with a Cauchy kernel, Comm. Pure Appl. Math. 16, 57–61 (1963)
A. S. Peters, Abel’s equation and the Cauchy integral equation of the second kind, Comm. Pure Appl. Math. 21, 51–65 (1968)
A. C. Kaya and F. Erdogan, On the solution of integral equations with strongly singular kernels, Quart. Appl. Math. 45, 105–122 (1987)
A. C. Kaya and F. Erdogan, On the solution of integral equations with strong singularities, in Numerical Solution of Singular Integral Equations (A. Gersoulis and V. Vichnevetsky, eds.), IMACS, 1984, pp. 54–57
N. I. Ioakimidis, On the numerical solution of Cauchy type singular integral equations by the collocation method, Appl. Math. Comp. 12, 49–60 (1983)
N. I. Ioakimidis, Further convergence results for the weighted Galerkin method of numerical solution of Cauchy-type singular integral equations, Math. Comp. 41, 79–85 (1983)
Chien-Ke Lu, A class of quadrature formulas of Chebyshev type for singular integrals, J. Math. Anal. Appl. 100, 416–435 (1984)
J. I. Frankel, unpublished analysis and computations, 1992
L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed., McGraw-Hill, New York, 1992
T. J. Rivlin, The Chebyshev Polynomials, Wiley, New York, 1974
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1972
K. E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., Wiley, New York, 1989
T. S. Sankar, S. V. Hoa, and V. I. Fabrikant, Approximate solution of singular integro-differential equations in elastic contact problems, Internat. J. Numer. Methods Engrg. 18, 503–519 (1982)
F. Erdogan, Approximate solutions of systems of singular integral equations, SIAM J. Appl. Math. 17, 1041–1059 (1969)
F. Erdogan and G. D. Gupta, On the numerical solution of singular integral equations, Quart. Appl. Math. 30, 525–534 (1972)
A. Gerasoulis and R. P. Srivastav, A method for the numerical solution of singular integral equations with a principal value integral, Internat. J. Engrg. Sci. 19, 1293–1298 (1981)
S. R. Bland, The two-dimensional oscillating airfoil in a wind tunnel in subsonic flow, SIAM J. Appl. Math. 18, 830–848 (1970)
J. A. Fromme and M. A. Golberg, Aerodynamics interface effects on oscillating airfoils with controls in ventilated wind tunnels, AIAA J. 18, 417–426 (1980)
M. A. Golberg, The convergence of a collocation method for a class of Cauchy singular integral equations, J. Math. Anal. Appl. 100, 500–512 (1984)
M. A. Golberg and J. A. Fromme, On the $L_{2}$ convergence of collocation for the generalized airfoil equation, J. Math. Anal. Appl. 71, 271–286 (1979)
A. Plotkin and S. S. Dodbele, Slender wing in ground effect, AIAA J. 26, 493–494 (1988)
L. Dragos, Numerical solution of the equation for a thin airfoil in ground effect, AIAA J. 28, 2132–2134 (1990)
R. D. Cess and S. N. Tiwari, The interaction of thermal conduction and infrared gaseous radiation, Appl. Sci. Res. 20, 25–39 (1969)
E. M. Sparrow and R. D. Cess, Radiation Heat Transfer, McGraw-Hill, New York, 1978
M. A. Golberg, ed., Solution Methods for Integral Equations, Plenum Press, New York, 1979
M. A. Golberg, ed., Numerical Solution of Integral Equations, Plenum Press, New York, 1990
P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, 1985
C. T. H. Baker, The Numerical Treatment of Integral Equations, Clarendon Press, Oxford, 1978
L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge Univ. Press, Cambridge, 1988
K. E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia, 1976
F. G. Tricomi, Integral Equations, Dover, New York, 1985
H. Hochstadt, Integral Equations, Wiley, New York, 1973
C. D. Green, Integral Equation Methods, Barnes and Noble, New York, 1969
D. Porter and D. S. G. Stirling, Integral Equations, Cambridge Univ. Press, Cambridge, 1990
J. I. Frankel, Regularized and preconditioned boundary integral solution to heat transfer in a participating gas flow between parallel plates, Numer. Heat Transfer 19, 105–126 (1991)
J. I. Frankel, Nonlinear heat transfer: Solution of singular, nonlinear integral equations, Engrg. Anal. 8, 231–238 (1991)
A. S. Peters, A note on the integral equation of the first kind with a Cauchy kernel, Comm. Pure Appl. Math. 16, 57–61 (1963)
A. S. Peters, Abel’s equation and the Cauchy integral equation of the second kind, Comm. Pure Appl. Math. 21, 51–65 (1968)
A. C. Kaya and F. Erdogan, On the solution of integral equations with strongly singular kernels, Quart. Appl. Math. 45, 105–122 (1987)
A. C. Kaya and F. Erdogan, On the solution of integral equations with strong singularities, in Numerical Solution of Singular Integral Equations (A. Gersoulis and V. Vichnevetsky, eds.), IMACS, 1984, pp. 54–57
N. I. Ioakimidis, On the numerical solution of Cauchy type singular integral equations by the collocation method, Appl. Math. Comp. 12, 49–60 (1983)
N. I. Ioakimidis, Further convergence results for the weighted Galerkin method of numerical solution of Cauchy-type singular integral equations, Math. Comp. 41, 79–85 (1983)
Chien-Ke Lu, A class of quadrature formulas of Chebyshev type for singular integrals, J. Math. Anal. Appl. 100, 416–435 (1984)
J. I. Frankel, unpublished analysis and computations, 1992
L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed., McGraw-Hill, New York, 1992
T. J. Rivlin, The Chebyshev Polynomials, Wiley, New York, 1974
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1972
K. E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., Wiley, New York, 1989
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© Copyright 1995
American Mathematical Society
