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On analysis and discretization of nonlinear Abel integral equations of first kind. (English) Zbl 0883.65114
Summary: For $0\le x\le B$, $0<\beta<1$, we consider the integral equation $$\int^x_0(x- t)^{-\beta} K(x,t,y(t))dt= f(x)$$ under appropriate Lipschitz-like conditions on the function $K$ and some of its derivatives, the most essential condition being $$K_u(x, t,u)\ge c>0\quad\text{for} \quad 0\le t\le x\le B,\ u\in\bbfR.$$ After a survey on theorems of existence, uniqueness and stability of the solution, we generalize a numerical method, proposed and investigated by {\it H. W. Branca} [Computing 20, 307-324 (1978; Zbl 0394.65047)] for the particular case $\beta= 1/2$, to all $\beta\in(0, 1)$ and show it to be $O(h^2)$ convergent for all $\beta\in[0.2118,1)$ 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.

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