Abstract
The aim of this paper is to carry out an improved analysis of the convergence of the Nyström and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree ⩽r−1, we obtain convergence order 2r for degenerate kernel and Nyström methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are 3r+1 and 4r, respectively. Moreover, we show that the optimal convergence order 4r is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples.
Highlights
Integro-differential equations emerged at the beginning of the twentieth century thanks to the work of Vito Volterra
Many numerical methods have been developed for solving integro-differential equations
We illustrate the accuracy and effectiveness of theoretical results established in the previous sections for numerically solving Fredholm integro-differential equations
Summary
Integro-differential equations emerged at the beginning of the twentieth century thanks to the work of Vito Volterra. The applications of these equations have proved worthy and effective in the fields of engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory, electrostatics, etc. Many numerical methods have been developed for solving integro-differential equations. Each of these methods has its inherent advantages and disadvantages, and the search for easier and more accurate methods is a continuous and ongoing process. For other methods to solve integro-differential equations, see [11–14]
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