(A) numerical solution of abel integral equations of the second kind using continued fraction

Abstract

We consider the special case of $\alpha=\frac{1}{2}$ of Abel integral equations of the second kind. This type has much of physical applications. In many numerical attacks for this problem, we choose the method to approximate the singular kernel $(t - s)^{-\frac{1}{2}}$ with some smooth ones. This observation is quite natural and simple. Our main idea is to approximate the singular kernel $(t - s)^{-\frac{1}{2}}$ with continued fractions. The ν th step continued fraction contains (ν + 1) multiplications, whereas polynomials of degree n contains $\frac{n(n+1)}{2}$ multiplications. So if we use continued fractions instead of polynomials to approximate the singular kernel $(t - s)^{-\frac{1}{2}}$, then we gain more efficiency. We have shown that the degree of convergence is $O(\frac{1}{ν})$ which corresponds to $O(\frac{1}{n^2})$, where ν is the step of continued fractions and n is the degree of polynomials. Since the polynomial approximation yields $O(\frac{1}{n})$, we have an improvement. And many practical examples were treated.