Numerical solution of the abel integral equation
Tóm tắt
A numerical method for the solution of the Abel integral equation is presented. The known function is approximated by a sum of Chebyshev polynomials. The solution can then be expressed as a sum of generalized hypergeometric functions, which can easily be evaluated, using a simple recurrence relation.
Tài liệu tham khảo
F. G. Tricomi,Integral Equations, Interscience Publ., New York, 1957.
M. E. Fettis,On the numerical solution of equations of the Abel type, Math. Comp. 18 (1964), 491–496.
G. N. Minerbo and M. E. Levy,Inversion of Abel's integral equation by means of orthogonal polynomials, SIAM J. Numer. Anal. 6 (1969), 598–616.
O. H. Nestor and H. N. Olsen,Numerical methods for reducing line and surface probe data, SIAM Rev. 2 (1960), 200–207.
K. Bockasten,Transformation of observed radiances into radial distribution of the emission of a plasma, J. Opt. Soc. Amer. 51 (1961), 943–947.
C. D. Maldonado, A. P. Caron and H. N. Olsen,New method for obtaining emission coefficients from emitted spectral intensities, J. Opt. Soc. Amer. 55 (1965), 1247–1254.
G. E. Roberts and H. Kaufman,Table of Laplace transforms, Saunders Co., Philadelphia, 1966.
E. D. Rainville,Special functions, Macmillan Co., New York, 1960.
L. M. Milne-Thomson,The calculus of finite differences, Macmillan and Co., London, 1965.
J. Denef and R. Piessens,On the asymptotic behaviour of solutions of difference equations of Poincaré type, to be published.
L. Fox and I. B. Parker,Chebyshev polynomials in numerical analysis, Oxford Univ. Press, London, 1968.
C. W. Clenshaw,Curve fitting with a digital computer, Computer J. 2 (1960), 170–173.