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A spectral element approach for the stability of delay systems. (English) Zbl 1242.70007
Summary: We describe a spectral element approach to study the stability and equilibria solutions of Delay differential equations (DDEs). In contrast to the prototypical temporal finite element analysis (TFEA), the described spectral element approach admits spectral rates of convergence and allows exploiting hp-convergence schemes. The described approach also avoids the limitations of analytical integrations in TFEA by using highly accurate numerical quadratures — enabling the study of more complicated DDEs. The effectiveness of this new approach is compared with well-established methods in the literature using various case studies. Specifically, the stability results are compared with the conventional TFEA and Legendre collocation methods whereas the equilibria solutions are compared with the numerical simulations and the homotopy perturbation method (HPM) solutions. Our results reveal that the presented approach can have higher rates of convergence than both collocation methods and the HPM.

MSC:
70-08 Computational methods for problems pertaining to mechanics of particles and systems
70J25 Stability for problems in linear vibration theory
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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