Series equations involving Jacobi polynomials and mixed boundary-value problems of the Laplace equation
Authors:
Masaaki Shimasaki and Takeshi Kiyono
Journal:
Quart. Appl. Math. 31 (1973), 53-73
MSC:
Primary 45F10
DOI:
https://doi.org/10.1090/qam/417713
MathSciNet review:
417713
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Abstract: A formal analysis of series equations involving Jacobi polynomials is given. $\left ( {2N + 1} \right )$ series equations involving Jacobi polynomials are reduced to a set of $N$ simultaneous Fredholm integral equations which can be solved numerically by the use of the Legendre-Gauss quadrature formula. In case of triple series equations the result is in agreement with that of Lowndes. Besides triple series equations, certain quadruple series equations can be also reduced to a single Fredholm integral equation of the second kind. Owing to the introduction of an arbitrary weight factor, the theory is feasible for the analysis of various many-part mixed boundary-value problems of the Laplace equation. As an example, special cases of certain trigonometric series equations are discussed in detail in connection with an electrostatic problem.
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D. W. Kammler, Calculation of characteristic admittances and coupling coefficients for strip transmission lines, IEEE Trans. MTT-16, 925–937 (1968)
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P. C. Chestnut, On determining the capacitances of shielded multiconductor transmission lines, IEEE Trans. MTT-17, 734–745 (1969)
D. W. Kammler, Calculation of characteristic admittances and coupling coefficients for strip transmission lines, IEEE Trans. MTT-16, 925–937 (1968)
T. Kiyono and M. Shimasaki, On the solution of Laplace’s equation by certain dual series equations SIAM J. Appl. Math. 21, 245–257 (1971)
J. S. Lowndes, Some triple series equations involving Jacobi polynomials, Proc. Edin. Math. Soc. 16, 101–108 (1968)
B. Noble, Some dual series equations involving Jacobi polynomials, Proc. Camb. Phil. Soc. 59, 363–371 (1963)
I. N. Sneddon, Mixed boundary value problems in potential theory, North-Holland, 1966
R. P. Srivastav, Dual series relations. IV: dual relations involving series of Jacobi polynomials, Proc. Roy. Soc. Edin. A66, 185–191 (1964)
K. N. Srivastava, On triple series equations involving series of Jacobi polynomials, Proc. Edin. Math. Soc. 15, 221–231 (1967)
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Article copyright:
© Copyright 1973
American Mathematical Society
