Abstract We consider extension of quasilinearization technique and method of lower and upper solutions for nonlinear weakly singular Volterra integral equations. We obtain linear weakly singular integral equations and discuss on their existence, uniqueness and regularity properties of a solution. On the other hand, we show that the solutions of these linear equations are quadratically convergent to the solution of nonlinear equation. Using smoothing transformation we regularize the linear equations and then to approximate their solutions we apply global product integration method. Because of employing quasilinearization technique and smoothing transformation we yield a sequence of small size linear algebraic systems in the discretization. Error analysis shows the error bound has two parts of quasilinearization error which is quadratically convergent and product integration error where by using smoothing transformation its convergence order is improved. The numerical results obtained from different numerical examples are in agreement with the theoretical results and comparison with the other methods confirms the efficiency of the proposed method.

中文翻译：

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使用拟线性化和乘积积分方法对非线性弱奇异 Volterra 积分方程进行数值解的平滑变换

摘要 我们考虑了非线性弱奇异Volterra积分方程拟线性化技术和上下解方法的推广。我们得到了线性弱奇异积分方程，并讨论了它们的存在性、唯一性和解的规律性。另一方面，我们证明这些线性方程的解二次收敛于非线性方程的解。使用平滑变换，我们正则化线性方程，然后我们应用全局乘积积分方法来逼近它们的解。由于采用了拟线性化技术和平滑变换，我们在离散化中产生了一系列小尺寸线性代数系统。误差分析表明误差界有二次收敛的拟线性化误差和乘积积分误差两部分，通过平滑变换提高了其收敛阶数。不同数值算例的数值结果与理论结果一致，与其他方法的比较证实了所提方法的有效性。