Discrete collocation for a first kind Cauchy singular integral equation with weakly singular solution
A discrete collocation method for the numerical solution of the first kind Cauchy singular integral equation \[ {1\over 2\pi} \int^{2\pi}_0 u(s)\text{cot}{t- s\over 2} ds= f(t),\quad t\in[0, 2\pi] \] is investigated. Represented in the form \[ {1\over 2\pi} \int^{2\pi}_0 \{u(s)- u(t)\}\text{cot}{t- s\over 2} ds= f(t), \] and using the sequences of grids by the knots \[ x^n_0= 0< x^n_1<\cdots< x^n_n= 2\pi,\quad n\in\mathbb{N} \] and \[ t^n_i:= (1- \lambda^n_i) x^n_{i- 1}+ \lambda^n_i x^n_i,\quad 0<\lambda^n_i< 1,\quad i= 1,2,\dots, n \] the discrete collocation method \[ {1\over 2\pi} \sum^n_{j= 1} \omega^n_j\text{cot}{t^n_i- x^n_j\over 2} (u^n_j- (1- \lambda^n_i) u^n_{i- 1}- \lambda^n_i u^n_i)= f(t^n_i),\quad i= 1,2,\dots, n,\tag{\(*\)} \] where \(u^n_i= u_n(x^n_i)\), \(i= 0,1,\dots, n\), \(u^n_0= u^n_n\) is received. The choice of the collocation points \(t^n_i\), \(i= 1,\dots, n\), stability, convergence and exactness of the discrete collocation method \((*)\) are investigated.
scientific article; zbMATH DE number 1185928