  This paper is concerned with the numerical solution for a class of nonlinear
weakly singular Volterra integral equation of the first kind. The existence and
uniqueness issue of the nonlinear Abel integral equations is studied
completely. An $hp$-version collocation method in conjunction with Jacobi
polynomials is introduced so as an appropriate numerical solution to be found.
We analyze it properly and find an error estimation in $L^2$-norm. The
efficiency of the method is illustrated by some numerical experiments.
