The Puiseux Expansion and Numerical Integration to Nonlinear Weakly Singular Volterra Integral Equations of the Second Kind

This paper develops an efficient algorithm to solve nonlinear Volterra integral equation of the second kind with weakly singular convolution kernel. First, we show that the general Puiseux series for the solution about zero exists under smooth assumptions for the nonlinear function, and then design an algorithm to recover the finite-term truncation of the asymptotic expansion by Picard iteration. This asymptotic expansion can easily yield a more accurate Pade approximation. Second, we use trapezoidal rule to discretize the singular integral and derive the Euler-Maclaurin asymptotic expansion using the known Puiseux expansion of the solution. By accumulating some lower order error terms to the quadrature formula, we obtain high precision evaluations to the nonlinear Volterra integral equation. Third, we prove that the scheme is convergent by extending the Gronwall inequality to be held for the scheme. Fourth, an example is provided to illustrate that the combination of the Puiseux expansion and the numerical integration can effectively increase the computational accuracy of the equation. Finally, we apply the method to solve the Lighthill integral equation, and obtain the asymptotic expansions of the solution near zero and infinity, respectively. The computation shows that the trapezoidal rule is only necessary in a small finite range of the semi-infinite interval.