Electron. J. Differential Equations, Vol. 2017 (2017), No. 68, pp. 1-25.

Nonlinear differential equations with deviating arguments and approximations via a Parker-Sochacki approach

Vincenzo M. Isaia

The Parker-Sochacki method has been successful in generating approximations for a wide variety of ODEs, and even PDEs of evolution type, by achieving an autonomous polynomial vector field and implementing the Picard iteration. The intent of this article is to extend PSM to a large family of differential equations with deviating arguments. Results will be given for problems with delays which are linear in time and state independent, and also have constant initial data and nonlinear differential equations which are retarded, neutral or advanced. The goal of the proofs is to motivate a numerically efficient DDE solver. In addition, an explicit a priori error estimate that does not require derivatives of the vector field is presented. The non-constant initial data cases and the state dependent delay cases are discussed formally.

Submitted April 1, 2016. Published March 8, 2017.
Math Subject Classifications: 34K07, 34K40, 65L03.
Key Words: Delay differential equations; lag; PSM method; method of steps; method of successive approximation; deviating argument.

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Vincenzo M. Isaia
Department of Mathematics
Rose-Hulman Institute of Technology
Terre Haute, IN 47803, USA
email: isaia@rose-hulman.edu

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