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On analysis and discretization of nonlinear Abel integral equations of first kind. (English) Zbl 0883.65114

Summary: For 0xB, 0<β<1, we consider the integral equation

0 x (x-t) -β K(x,t,y(t))dt=f(x)

under appropriate Lipschitz-like conditions on the function K and some of its derivatives, the most essential condition being

K u (x,t,u)c>0for0txB,u·

After a survey on theorems of existence, uniqueness and stability of the solution, we generalize a numerical method, proposed and investigated by H. W. Branca [Computing 20, 307-324 (1978; Zbl 0394.65047)] for the particular case β=1/2, to all β(0,1) and show it to be O(h 2 ) convergent for all β[0·2118,1) if the solution y is sufficiently smooth. The method is based on piecewise linear interpolation, one-point weighted Gauss quadrature on partition intervals of equal length h, and collocation.

MSC:
65R20Integral equations (numerical methods)
45G05Singular nonlinear integral equations
45M10Stability theory of integral equations