Journal of Integral Equations and Applications

A collocation method for solving nonlinear Volterra integro-differential equations of neutral type by sigmoidal functions

Danilo Costarelli and Renato Spigler

Full-text: Open access

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.

Article information

Source
J. Integral Equations Applications, Volume 26, Number 1 (2014), 15-52.

Dates
First available in Project Euclid: 17 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1397764953

Digital Object Identifier
doi:10.1216/JIE-2014-26-1-15

Mathematical Reviews number (MathSciNet)
MR3195114

Zentralblatt MATH identifier
1288.65185

Subjects
Primary: 65R20: Integral equations 41A30: Approximation by other special function classes 45D05: Volterra integral equations [See also 34A12] 45G10: Other nonlinear integral equations

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

Citation

Costarelli, Danilo; Spigler, Renato. A collocation method for solving nonlinear Volterra integro-differential equations of neutral type by sigmoidal functions. J. Integral Equations Applications 26 (2014), no. 1, 15--52. doi:10.1216/JIE-2014-26-1-15. https://projecteuclid.org/euclid.jiea/1397764953


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