The Puiseux expansion and numerical integration to nonlinear weakly singular Volterra integral equations of the second kind (Q2302413)

The paper is devoted to the following nonlinear weakly singular Volterra integral equation of the second kind: \[ u(x)=f(x)+\int\limits_{0}^{x}\frac{\log^{\mu}(x-t)}{(x-t)^{\alpha}}F(x,t,u(t))dt, \quad 0< x \le T, \tag{1} \] where \(0\le\alpha<1\) and \(\mu\) is a nonnegative integer. An efficient algorithm to solve equation (1) with algebraic and logarithmic singular convolution kernels is suggested. In the first part of this algorithm, the authors derive the asymptotic expansion of the solution near zero via standard Picard iterations \[ u_0(x)=f(x), \; u_n(x)=f(x)+\int\limits_{0}^{x}\frac{\log^{\mu}(x-t)}{(x-t)^{\alpha}}F(x,t,u_{n-1}(t))dt, \quad n=1,2,\ldots. \tag{2} \] The authors 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. When the argument of the solution is near zero, the Puiseux expansion can be used to approximate the solution directly. But when the argument is far away from zero, the algorithm automatically switches to numerical integration method based on trapezoidal rule to discretize the singular integral and derive the Euler-Maclaurin asymptotic expansion using the known Puiseux expansion of the solution. It is proved that the scheme is convergent by extending the Gronwall inequality. 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, this method is applied to solve the Lighthill integral equation and the asymptotic expansions of the solution near zero and infinity are obtained. The computation shows that the trapezoidal rule is only necessary in a small finite range of the semi-infinite interval.