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Research Papers

# Galerkin Approximations for Stability of Delay Differential Equations With Distributed Delays

[+] Author and Article Information

Department of Mechanical
and Aerospace Engineering,
Ordnance Factory Estate,
Telangana 502205, India

C. P. Vyasarayani

Department of Mechanical
and Aerospace Engineering,
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 10(6), 061024 (Nov 01, 2015) (8 pages) Paper No: CND-14-1269; doi: 10.1115/1.4030153 History: Received October 30, 2014; Revised March 19, 2015; Online June 09, 2015

## Abstract

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 $O(t)$, whereas the direct brute-force integration of DIDE has a computational complexity of $O(t2)$.

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## Figures

Fig. 1

Stability plot for the first-order time periodic DIDE (Eq. (25)) for ϵ = 5, ω = 2π, and ν = 2π/ω. The number of terms in the series solutions are (a) N = 2, (b) N = 3, and (c) N = 5.

Fig. 2

(a) Eigenvalue of state transition matrix showing a transcritical bifurcation when the parameters δ is varied from −8 to −2 with β = 3.5. (b) Eigenvalue of state transition matrix showing a flip bifurcation when the parameter δ is varied from −8 to 4 with β = −3.5. The blue unit circle shows the stability boundary where the absolute value of an eigenvalue is unity.

Fig. 3

Stability diagram of first-order scalar DIDE (Eq. (31)). The red dots belong to the stable region obtained using the Galerkin method, while the blue boundary is obtained from the literature [23].

Fig. 4

Convergence of e(λ∧) with increasing N for Eq. (31). The parameters are (a) α = −6 and β = −5, (b) α = −2 and β = −10, and (c) α = 1, and β = −15.

Fig. 5

Stability diagram of second-order time-periodic DIDE (Eq. (34)) for ω = 2π, ν = 1, ϵ = 5, and T = 1

Fig. 6

Stability diagram of the system given by Eq. (35) for the kernel given by Eq. (36). Red dots indicate the stable region obtained using the Galerkin approximation, which shows a good agreement with the results reported in Ref. [23]. The parameters used are (a) ϵ = 20 and N = 15 and (b) ϵ = 60 and N = 20.

Fig. 7

Stability diagram of the system given by Eq. (35) for the kernel given by Eq. (37). Red dots denote the stable region obtained by Galerkin approximation, which shows a good agreement with those reported in Ref. [23]. The parameters used are (a) ϵ = 20 and N = 20 and (b) ϵ = 60 and N = 15.

Fig. 8

Time response of the system given by Eq. (35) with kernel function Eq. (36). The parameters used for the simulation are (a) γ = 10, δ = 12.5, and ϵ = 20 (stable region) and (b) γ = −20, δ = 10, and ϵ = 20 (unstable region).

Fig. 9

Time response of the system Eq. (35) with kernel function Eq. (37). The parameters used for the simulation are (a) γ = −50, δ = 20, and ϵ = 20 (stable region) and (b) γ = −100, δ = 10, and ϵ = 20 (unstable region).

Fig. 10

(a) Time response of the system given by Eq. (35) with kernel function Eq. (37). (b) Error plot between the Galerkin method and the direct numerical method. (c) Comparison of computational time between the Galerkin method and the numerical method with simulation time t.

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