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SPRING 2014 A collocation method for solving nonlinear Volterra integro-differential equations of neutral type by sigmoidal functions
Danilo Costarelli, Renato Spigler
J. Integral Equations Applications 26(1): 15-52 (SPRING 2014). DOI: 10.1216/JIE-2014-26-1-15

Abstract

A numerical collocation method is developed for solving nonlinear Volterra integro-differential equations (VIDEs) of the neutral type, as well as other non-standard and classical VIDEs. A sigmoidal functions approximation is used to suitably represent the solutions. Special computational advantages are obtained using unit step functions, and important applications can also be obtained by using other sigmoidal functions, such as logistic and Gompertz functions. The method allows one to obtain a simultaneous approximation of the solution to a given VIDE and its first derivative, by means of an explicit formula. A priori as well as a posteriori estimates are derived for the numerical errors, and numerical examples are given for the purpose of illustration. A comparison is made with the classical piecewise polynomial collocation method as for accuracy and CPU time.

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Danilo Costarelli. Renato Spigler. "A collocation method for solving nonlinear Volterra integro-differential equations of neutral type by sigmoidal functions." J. Integral Equations Applications 26 (1) 15 - 52, SPRING 2014. https://doi.org/10.1216/JIE-2014-26-1-15

Information

Published: SPRING 2014
First available in Project Euclid: 17 April 2014

zbMATH: 1288.65185
MathSciNet: MR3195114
Digital Object Identifier: 10.1216/JIE-2014-26-1-15

Subjects:
Primary: 41A30 , 45D05 , 45G10 , 65R20

Keywords: collocation methods , Gompertz functions , logistic functions , neutral type integro-differential equations , non standard integro-differential equations , nonlinear Volterra integro-differential equations , sigmoidal function approximation , unit step functions

Rights: Copyright © 2014 Rocky Mountain Mathematics Consortium

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Vol.26 • No. 1 • SPRING 2014
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