Asymptotic expansions of the error for hyper-singular integrals with an interval variable - Chong Chen; Jin Huang; Yanying Ma (Journal of Inequalities and Applications, 2016)
Abstract In this paper, we present high accuracy quadrature formulas for hyper-singular integrals ∫ a b g ( x ) q α ( x , t ) d x $\int_{a}^{b}g(x)q^{α}(x,t)\, dx$ , where q ( x , t ) = | x − t | $q(x,t)=|x-t|$ (or x − t $x-t$ ), t ∈ ( a , b ) $t\in(a,b)$ , and α ≤ − 1 $α≤-1$ (or α < − 1 $α<-1$ ). If g ( x ) $g(x)$ is 2 m + 1 $2m+1$ times differentiable on [ a , b ] $[a,b]$ , the asymptotic expansions of the error show that the convergence order is O ( h 2 μ + 1 + α ) $O(h^{2μ+1+α})$ with q ( x , t ) = | x − t | $q(x,t)=|x-t|$ (or x − t $x-t$ ) for α ≤ − 1 $α≤-1$ (or α < − 1 $α<-1$ and α being non-integer), and the error power is O ( h η ) $O(h^{η})$ with q ( x , t ) = x − t $q(x,t)=x-t$ for α being integers less than −1, where η = min ( 2 μ , 2 μ + 2 + α ) $η =\min(2μ,2μ+2+α)$ and μ = 1 , … , m $μ=1,…,m$ . Since the derivatives of the density function g ( x ) $g(x)$ in the quadrature formulas can be eliminated by means of the extrapolation method, the formulas can easily be applied to solving corresponding hyper-singular boundary integral equations. The reliability and efficiency of the proposed formulas in this paper are demonstrated by some numerical examples.