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Research Papers

Numerical Scheme for a Quadratic Type Generalized Isoperimetric Constraint Variational Problems With A-Operator

[+] Author and Article Information
Rajesh K. Pandey

Department of Mathematical Sciences,
Indian Institute of Technology (BHU),
Varanasi, Uttar Pradesh 221005, India

Om P. Agrawal

Mechanical Engineering and Energy Process,
Southern Illinois University Carbondale,
Carbondale, IL 62901
e-mail: om@engr.siu.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 17, 2013; final manuscript received September 21, 2014; published online January 12, 2015. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 10(2), 021003 (Mar 01, 2015) (6 pages) Paper No: CND-13-1142; doi: 10.1115/1.4028630 History: Received June 17, 2013; Revised September 21, 2014; Online January 12, 2015

This paper presents a numerical scheme for a class of isoperimetric constraint variational problems (ICVPs) defined in terms of an A-operator introduced recently. In this scheme, Bernstein's polynomials are used to approximate the desired function and to reduce the problem from a functional space to an eigenvalue problem in a finite dimensional space. Properties of the eigenvalues and eigenvectors of this problem are used to obtain approximate solutions to the problem. Results for two examples are presented to demonstrate the effectiveness of the proposed scheme. In special cases, the A-operator reduces to Riemann–Liouville, Caputo, Riesz–Riemann–Liouville, and Riesz–Caputo, and several other fractional derivatives defined in the literature. Thus, the approach presented here provides a general scheme for ICVPs defined using different types of fractional derivatives. Although, only Bernstein's polynomials are used here to approximate the solutions, many other approximation schemes are possible. Effectiveness of these approximation schemes will be presented in the future. While the presented numerical scheme is applied to a quadratic type generalized ICVPs, it can also be applied to other types of problems.

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Figures

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Fig. 6

Approximate solution for order α = 0.8 problem at different no. of polynomials for a = 0.001

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Fig. 5

Integer order analytical solution Y(t) for α = 1 and approximate solutions for α = 0.8 at different values of a

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Fig. 4

Analytical solution Y(t) for α = 1 and other approximate solutions for a = 1/2, at different values of α

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Fig. 3

Approximate solution for α = 0.8 at different no. of polynomials for a = 0.001

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Fig. 2

Approximate solution for α = 0.8 at different values of a

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Fig. 1

Exact solution Y(t) for integer order problem and other solutions for different α for a = 0

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Fig. 7

Integer order analytical solution Y(t) for α = 1 and approximate solutions at different α

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Fig. 8

Integer order analytical solution Y(t) for α = 1 and approximate solutions at different α

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