Numeric solution of Volterra integral equations of the first kind with discontinuous kernels

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• 作者：Ildar Muftahov^a ; Aleksandr Tynda^b ; Denis Sidorov^a ; ^c ; ^{contact.dns@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Volterra integral equations ; Discontinuous kernels ; Direct quadrature method ; Regularization ; Evolving dynamical systems ; Midpoint quadrature
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：313
• 期：Complete
• 页码：119-128
• 全文大小：424 K

文摘

Numeric methods for solution of the weakly regular linear and nonlinear evolutionary (Volterra) integral equations of the first kind are proposed. The kernels of such equations have jump discontinuities along the continuous curves (endogenous delays) which start at the origin. In order to linearize these equations the modified Newton–Kantorovich iterative process is employed. Two direct quadrature methods based on the piecewise constant and piecewise linear approximation of the exact solution are proposed for linear solutions. The accuracy of proposed numerical methods is O(1/N)$\mathcal{O}(1/N)$ and O(1/N²)$\mathcal{O}(1/N^2)$ respectively. A certain iterative numerical scheme enjoying the regularization properties is suggested. Furthermore, generalized numerical method for nonlinear equations is adduced. The midpoint quadrature rule in all the cases is employed. In conclusion several numerical examples are applied in order to demonstrate the efficiency of proposed numerical methods.