Error analysis of a Jacobi modified projection-type method for weakly singular Volterra-Hammerstein integral equations (Q6611974)

The paper addresses a numerical approximation procedure to the nonlinear Volterra-Hammerstein type equation\N\[\N y(t) = \int_{0}^t \frac{1}{(t-s)^\gamma} K(t,s)\Phi(s,y(s)) ds + f(t),\tag{1}\N\]\Nby means of spectral methods based on Jacobi polynomial basis.\N\NThe lack of regularity of solutions to fractional type equations as \(t\to 0^+\) has been repeatedly reported in the literature and is a well-known fact. The authors address this issue for a Volterra equation which is not exactly of fractional type since an additional term \(K(t,s)\) is involved. Therefore, by combining a change of variable and the evaluation of the equation in a convenient power of the time variable they achieve the regularity required for a spectral convergence or super-convergence of the numerical solution. In particular they consider the change of variable \(t=x^r\), for \(r>1\), and then the evaluation \(y(x^r)\) where \(y(t)\) stands for the solution of (1). As a result, they claim that (1) can then be re-written as\N\[\N \bar{y}(x) = y(x^r) = \int_0^{x} \frac{s^{r-1}}{(x^r-s^r)^\gamma} K(x^r,s^r) \Phi(s^r,y(s^r))ds + f(x^r).\N\]\NSeveral computations lead to the final nonlinear equation\N\[\N y(x) = f(x) + \int_{-1}^1 (1-\theta^2) \kappa(x,\mu(x,\theta))\Psi(\mu(x,\theta),y(\mu(x,\theta))) d\theta,\tag{2}\N\]\Nwhich stands for a Volterra-Hammerstein type equation, and therefore far from a pure fractional type equation.\N\NSuch a sequence of transformations enables the authors to increase the regularity of the solution to (1) so that they can implement numerical methods with spectral convergence.\N\NClassical hypotheses for \(\kappa(x,\theta)\), and \(\Psi(t,u)\) in (2) are assumed, and Jacobi polynomial basis adapted to the shape of (2) are proposed. The spectral convergence of a Galerkin-type method (and a modification of it) and a collocation-type method (and a modification of it) derived from such a basis are presented. Moreover orders of convergence \(O(n^{-2r})\) and \(O(n^{-r})\) are respectively shown throughout the paper.\N\NThe theoretical results are conveniently illustrated in the last section of the paper.