Smoothing transformation and spline collocation for weakly singular Volterra integro-differential equations

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• 作者：T. Diogo^a ; ^{tdiogo@math.ist.utl.pt} ; P.M. Lima^a ; ^{plima@math.ist.utl.pt} ; A. Pedas^b ; ^{arvet.pedas@ut.ee} ; G. Vainikko^b ; ^gen@ut.ee
• 关键词：Weakly singular Volterra integro-differential equation ; Smoothing ; Collocation method
• 刊名：Applied Numerical Mathematics
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：114
• 期：Complete
• 页码：63-76
• 全文大小：411 K
• 卷排序：114

文摘

This work is concerned with the construction and analysis of high order numerical methods for solving initial value problems for linear Volterra integro-differential equations with different types of singularities. Using an integral reformulation of the initial value problem, a smoothing transformation is applied so that the exact solution of the resulting equation does not contain any singularities in its derivatives up to a certain order. After that, the regularized equation is solved by a piecewise polynomial collocation method on a uniform or mildly graded grid. Finally, the obtained spline approximations can be used to define (typically non-polynomial) approximations for the initial value problem. The theoretical results are tested by some numerical examples.