| Abstract: |
| The solutions of singular Volterra integral equations of the second kind usually behave like singular features about derivative at the left endpoint of the interval, which leads to obvious decreasing of computational accuracy when standard algorithms are used to solve these equations. This talk discusses the high accuracy computation to singular Volterra integral equations of the second kind. First, we derive the general Puiseux expansion of the solution at the singularity by Picard iteration and series decomposition, which is the accurate measurement of the singular type and singular degree of the solution. This asymptotic expansion can be used to approximate the solution when the variable is small, but the error increases rapidly as the variable becomes large. Second, we use some numerical integration methods 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 formulas, we can obtain high accuracy evaluations to the nonlinear Volterra integral equation. Finally, some examples are provided to demonstrate that the combination of the Puiseux expansion and the numerical integration can effectively solve nonlinear singular Volterra integral equations of the second kind. |
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