On the numerical solutions of Fredholm--Volterra integral equation. (Q1412563)

The authors describe the Toeplitz matrix method and the product Nystrom method for the mixed Fredholm-Volterra singular integral equation of the second kind: \[ \mu\phi(x,t)-\lambda\int_{-1}^1k(x,y)\phi(y,t)\,dy- \lambda\int_0^tF(t, \tau)\phi(x,\tau)\,d\tau= f(x,t),\quad 0\leqslant t\leqslant T,\;| x| \leqslant1,\tag{1} \] where \(k\), \(F\) and \(f\) are given functions and \(\mu\), \(\lambda\) are numerical parameters, while \(\phi(x, t)\) is the unknown function to be determined. The goal is to obtain, numerically, an approximate solution of (1), when the kernel of Fredholm integral term takes the form: \(k(x,y) =| y-x| ^{-\alpha}(\ln| y-x| )^q\), \(0\leqslant \alpha < 1, q\in\mathbb N\). Conditions are discussed to guarantee the existence of a unique solution of (1). A numerical technique is used to reduce (1) to a system of linear Fredholm integral equations of the second kind. The Toeplitz matrix method is applied to this system to get an approximate solution of (1). The product Nyström method is used to solve the same system to obtain another approximate solution of (1). The results are compared with the exact solution of the integral equation. The error of each method is calculated.