1. Galerkin approximations with embedded boundary conditions for retarded delay differential equations.
- Author
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Ahsan, Zaid, Uchida, Thomas, and Vyasarayani, C. P.
- Subjects
- *
DELAY differential equations , *DIFFERENTIAL equations , *BOUNDARY value problems , *LEGENDRE'S functions - Abstract
Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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