Solution of a class of Volterra integral equations with singular and weakly singular kernels
The purpose of the paper is to derive the solution to a Volterra integral equation \[ y(t)=g(t)+\int^t_0\frac{s^{\mu-\nu}}{t^\mu}y(s)\,ds\quad (0<t<T), \] where \(\mu\) and \(\nu\) are constants, and \(g(t)\) a given function. Set \[ \varphi(t)=\int^t_0{s^{\mu-\nu}}y(s)\,ds \quad (0<t<T). \] Then the Volterra integral equation is transformed into a first-order ordinary differential equation \[ t^\nu\frac{d\varphi}{dt}=t^\mu g(t)+\varphi(t)\quad (0<t<T). \] Based on such a transformation, the authors derive analytically a general solution to the Volterra integral equation.
scientific article; zbMATH DE number 5286409