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Research Papers

Simple Recipe for Accurate Solution of Fractional Order Equations

[+] Author and Article Information
Sambit Das

Department of Mechanical Engineering,
IIT Kharagpur 721302, India
e-mail: sambitiitkgp@gmail.com

Anindya Chatterjee

Department of Mechanical Engineering,
IIT Kanpur 208016, India
e-mail: anindya100@gmail.com

Equation (6) can now be used directly to determine x·, which can be used in Eq. (5) to find a·.

1Corresponding author. Former address: Indian Institute of Technology, Kharagpur, 721302.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received May 4, 2012; final manuscript received October 21, 2012; published online December 19, 2012. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 8(3), 031007 (Dec 19, 2012) (7 pages) Paper No: CND-12-1069; doi: 10.1115/1.4023009 History: Received May 04, 2012; Revised October 21, 2012

Fractional order integrodifferential equations cannot be directly solved like ordinary differential equations. Numerical methods for such equations have additional algorithmic complexities. We present a particularly simple recipe for solving such equations using a Galerkin scheme developed in prior work. In particular, matrices needed for that method have here been precisely evaluated in closed form using special functions, and a small Matlab program is provided for the same. For equations where the highest order of the derivative is fractional, differential algebraic equations arise; however, it is demonstrated that there is a simple regularization scheme that works for these systems, such that accurate solutions can be easily obtained using standard solvers for stiff differential equations. Finally, the role of nonzero initial conditions is discussed in the context of the present approximation method.

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Figures

Grahic Jump Location
Fig. 1

Solution of Eq. (4), and Eqs. (5) and (6) with initial conditions a(0) = 0 and. (a) The two solutions are visually close on the graph. (b) The error is small.

Grahic Jump Location
Fig. 2

Solution of Eq. (7), and Eqs. (8) and (9) with initial conditions a(0) = 0 and x(0) = 0. (a) The two solutions are visually close on the graph. (b) The error is small.

Grahic Jump Location
Fig. 3

Galerkin solution of Eq. (10) using Eqs. (13) and (14) with N = 10, and numerical solution of Eqs. (11) and (12) with initial conditions a(0) = 0. (a) The two solutions are visually close on the graph. (b) The error (here the difference between the two solutions) is small.

Grahic Jump Location
Fig. 4

Solution of Eq. (15) (x = t1/3), and Eqs. (16) and (17) with initial conditions a(t0) = 0 and x(t0) = 0, t0 = 10-8. (a) The two solutions are visually close on the graph. (b) The error is small.

Grahic Jump Location
Fig. 5

Solution of Eq. (18) from Eqs. (19), (20), (21) and (22) with initial conditions x(0) = 0, x·(0) = 0 and a1(0) = a2(0)= a3(0) = 0

Grahic Jump Location
Fig. 6

Solution of Eq. (23) (from Maple), and Eqs. (25), (26), and (27) with initial conditions a1(0) = 0, a2(0) = 0, and x(0) = 0. (a) The two solutions are visually close on the graph. (b) The error is small.

Grahic Jump Location
Fig. 7

(a) Two different square pulses used as forcing. (b) Solution of Eq. (18) with initial conditions x(0) = 0 and x·(0) = 0. x(1) is the same in each case, but the subsequent unforced evolutions differ because the system retains a memory of past forcing.

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