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Research Papers

A Numerical Scheme for a Class of Parametric Problem of Fractional Variational Calculus

[+] Author and Article Information
Om P. Agrawal

Department of Mechanical Engineering,  Southern Illinois University Carbondale, Carbondale, IL 62901

M. Mehedi Hasan

Department of Mechanical Engineering,  North Dakota State University, Fargo, ND 58108

X. W. Tangpong1

Department of Mechanical Engineering,  North Dakota State University, Fargo, ND 58108Annie.Tangpong@ndsu.edu

1

Corresponding author.

J. Comput. Nonlinear Dynam 7(2), 021005 (Jan 06, 2012) (6 pages) doi:10.1115/1.4005464 History: Received August 31, 2011; Accepted November 17, 2011; Revised November 17, 2011; Published January 06, 2012; Online January 06, 2012

Fractional derivatives (FDs) or derivatives of arbitrary order have been used in many applications, and it is envisioned that in the future they will appear in many functional minimization problems of practical interest. Since fractional derivatives have such properties as being non-local, it can be extremely challenging to find analytical solutions for fractional parametric optimization problems, and in many cases, analytical solutions may not exist. Therefore, it is of great importance to develop numerical methods for such problems. This paper presents a numerical scheme for a linear functional minimization problem that involves FD terms. The FD is defined in terms of the Riemann-Liouville definition; however, the scheme will also apply to Caputo derivatives, as well as other definitions of fractional derivatives. In this scheme, the spatial domain is discretized into several subdomains and 2-node one-dimensional linear elements are adopted to approximate the solution and its fractional derivative at point within the domain. The fractional optimization problem is converted to an eigenvalue problem, the solution of which leads to fractional orthogonal functions. Convergence study of the number of elements and error analysis of the results ensure that the algorithm yields stable results. Various fractional orders of derivative are considered, and as the order approaches the integer value of 1, the solution recovers the analytical result for the corresponding integer order problem.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Function y(t) at different values of N for the integer order problem

Grahic Jump Location
Figure 2

Function y(t) when α=0.8 at different values of N

Grahic Jump Location
Figure 3

Function y(t) when N=32 at different values of α

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