A collocation method for solving singular integro-differential equations (Q1960204)

The authors propose a collocation method for numerical solution of the equation \[ a(x)\varphi(x)+\frac{1}{\pi}{VP}\int_{-1}^{1}\frac{\varphi'(t)}{t-x}dt + \int_{-1}^{1}k(x,t)\varphi'(t)dt + \int_{-1}^{1}l(x,t)\varphi(t)dt = f(x), \] with \(|x|<1\) and boundary conditions \(\varphi(-1)=\varphi(1)=0\). They use the change of the variable \(u(x)=\varphi'(x)\). Concerning the new variable the origin integro-differential equation becomes the integral equation, also Cauchy-type. In accordance with the theory of singular integral equations [\textit{N. I. Muskhelishvili}, Singular integral equations. (1953; Zbl 0051.33203)] the required solution has the form \[ u(x)=\frac{g(x)}{\sqrt{1-x^2}}. \] The proposed numerical method is based on the approximation of the unknown function \(g(x)\) by Chebyshev polynomials of the first type. Uniform convergence of this method is proved. Numerical examples illustrate the theoretical results.