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Research Papers

# Unified Galerkin- and DAE-Based Approximation of Fractional Order Systems

[+] Author and Article Information
Satwinder Jit Singh1

Mechanical Engineering, Indian Institute of Science, Bangalore 560012, Indiasjitsingh@rediffmail.com

Anindya Chatterjee2

Mechanical Engineering, Indian Institute of Science, Bangalore 560012, Indiaanindya100@gmail.com

A Maple-8 worksheet to compute matrices $Aα$, $Bα$, and $cα$ is available on request (email).

1

Corresponding author. Present address: SMMEE, IIT Ropar, Punjab 140001, India.

2

Present address: Mechanical Engineering, IIT Kharagpur, Kharagpur 721302, India.

J. Comput. Nonlinear Dynam 6(2), 021010 (Oct 28, 2010) (7 pages) doi:10.1115/1.4002516 History: Received September 15, 2009; Revised August 18, 2010; Published October 28, 2010; Online October 28, 2010

## Abstract

We consider numerical solutions of nonlinear multiterm fractional integrodifferential equations, where the order of the highest derivative is fractional and positive but is otherwise arbitrary. Here, we extend and unify our previous work, where a Galerkin method was developed for efficiently approximating fractional order operators and where elements of the present differential algebraic equation (DAE) formulation were introduced. The DAE system developed here for arbitrary orders of the fractional derivative includes an added block of equations for each fractional order operator, as well as forcing terms arising from nonzero initial conditions. We motivate and explain the structure of the DAE in detail. We explain how nonzero initial conditions should be incorporated within the approximation. We point out that our approach approximates the system and not a specific solution. Consequently, some questions not easily accessible to solvers of initial value problems, such as stability analyses, can be tackled using our approach. Numerical examples show excellent accuracy.

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## Figures

Figure 1

DAE-based numerical solution and series solution of Eq. 29

Figure 2

DAE-based numerical solution and exact solution of Eq. 32 with Eq. 33

Figure 3

(a) Stability boundary of Eq. 34 and (b) absolute errors for 15 and 30 finite elements

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