Delay differential equations (DDEs) are infinite-dimensional systems, therefore analyzing their stability is a difficult task. The delays can be discrete or distributed in nature. DDEs with distributed delays are referred to as delay integro-differential equations (DIDEs) in the literature. In this work, we propose a method to convert the DIDEs into a system of ordinary differential equations (ODEs). The stability of the DIDEs can then be easily studied from the obtained system of ODEs. By using a space-time transformation, we convert the DIDEs into a partial differential equation (PDE) with a time-dependent boundary condition. Then, by using the Galerkin method, we obtain a finite-dimensional approximation to the PDE. The boundary condition is incorporated into the Galerkin approximation using the Tau method. The resulting system of ODEs will have time-periodic coefficients, provided the coefficients of the DIDEs are time periodic. Thus, we use Floquet theory to analyze the stability of the resulting ODE systems. We study several numerical examples of DIDEs with different kernel functions. We show that the results obtained using our method are in close agreement with those existing in the literature. The theory developed in this work can also be used for the integration of DIDEs. The computational complexity of our numerical integration method is , whereas the direct brute-force integration of DIDE has a computational complexity of .
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November 2015
Research-Article
Galerkin Approximations for Stability of Delay Differential Equations With Distributed Delays
Anwar Sadath,
Anwar Sadath
Department of Mechanical
and Aerospace Engineering,
Telangana 502205,
and Aerospace Engineering,
Indian Institute of Technology Hyderabad
,Ordnance Factory Estate
,Telangana 502205,
India
Search for other works by this author on:
C. P. Vyasarayani
C. P. Vyasarayani
1
Department of Mechanical
and Aerospace Engineering,
Telangana 502205,
e-mail: vcprakash@iith.ac.in
and Aerospace Engineering,
Indian Institute of Technology Hyderabad
,Ordnance Factory Estate
,Telangana 502205,
India
e-mail: vcprakash@iith.ac.in
1Corresponding author.
Search for other works by this author on:
Anwar Sadath
Department of Mechanical
and Aerospace Engineering,
Telangana 502205,
and Aerospace Engineering,
Indian Institute of Technology Hyderabad
,Ordnance Factory Estate
,Telangana 502205,
India
C. P. Vyasarayani
Department of Mechanical
and Aerospace Engineering,
Telangana 502205,
e-mail: vcprakash@iith.ac.in
and Aerospace Engineering,
Indian Institute of Technology Hyderabad
,Ordnance Factory Estate
,Telangana 502205,
India
e-mail: vcprakash@iith.ac.in
1Corresponding author.
Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 30, 2014; final manuscript received March 19, 2015; published online June 9, 2015. Assoc. Editor: Hiroshi Yabuno.
J. Comput. Nonlinear Dynam. Nov 2015, 10(6): 061024 (8 pages)
Published Online: November 1, 2015
Article history
Received:
October 30, 2014
Revision Received:
March 19, 2015
Online:
June 9, 2015
Citation
Sadath, A., and Vyasarayani, C. P. (November 1, 2015). "Galerkin Approximations for Stability of Delay Differential Equations With Distributed Delays." ASME. J. Comput. Nonlinear Dynam. November 2015; 10(6): 061024. https://doi.org/10.1115/1.4030153
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