High accuracy combination algorithm and a posteriori error estimation for solving the first kind Abel integral equations

This paper presents high accuracy combination algorithm for solving the first kind Abel integral equations. To avoid solving ill-posed problems, we transform the first kind Abel integral equation to the second kind Volterra integral equation with a continuous kernel expressed by a weakly singular integral. By using integration rules the approximation of this kernel can be easily computed. Then two quadrature algorithms for solving Abel integral equations are proposed, which possess accuracy order O(h1+α)(0 < α < 1) and asymptotic expansion of the errors. By means of combination algorithm, we may obtain an approximate solution with a higher accuracy order O(h2). Moreover a posteriori error estimate for the algorithms is derived. Both theory and numerical examples show that it is effective and saves storage capacity and computational work.