A new numerical method for a class of Volterra and Fredholm integral equations (Q2184019)

The present paper is devoted to the following nonlinear quadratic Volterra integral equation: \[ \phi(x)=g(x)+f(x,\phi(x))\int\limits_{0}^{x}v(x,t,\phi(t))dt, \quad x\in I=[0,1], \] and to the Fredholm-Hammerstein integral equation of the second kind: \[ \phi(x)=f(x)+\int\limits_{0}^{1}k(x,t)\psi(t,\phi(t))dt, \quad x\in I, \] with the components satisfying some conditions guaranteeing the existence of theoretical monotonic non-negative solutions. In order to solve these equations numerically a new method based on a strong version of the mean-value theorem for integrals is introduced. By means of an equality theorem, the integral that appears in the aforementioned equations is transformed into one that enables a more accurate numerical solution with fewer calculations than other previously described methods. Convergence analysis is given and several numerical examples are also presented.