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Research Papers

Exponentially Accurate Rayleigh–Ritz Method for Fractional Variational Problems

[+] Author and Article Information
Zhi Mao

Hunan Key Laboratory for Computation
and Simulation in Science and Engineering,
Xiangtan University,
Xiangtan, Hunan 411105, China
School of Mathematical Sciences,
Tongren University,
Tongren, Guizhou 554300, China
e-mail: mzxtu1982@163.com

Aiguo Xiao

Hunan Key Laboratory for Computation
and Simulation in Science and Engineering,
Xiangtan University,
Xiangtan, Hunan 411105, China
e-mail: xag@xtu.edu.cn

Dongling Wang

School of Mathematics,
Northwest University,
Xi’an, Shaanxi 710127, China

Zuguo Yu, Long Shi

Hunan Key Laboratory for Computation
and Simulation in Science and Engineering,
Xiangtan University,
Xiangtan, Hunan 411105, China

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 6, 2014; final manuscript received September 11, 2014; published online April 2, 2015. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 10(5), 051009 (Sep 01, 2015) (6 pages) Paper No: CND-14-1119; doi: 10.1115/1.4028581 History: Received May 06, 2014; Revised September 11, 2014; Online April 02, 2015

A high accurate Rayleigh–Ritz method is developed for solving fractional variational problems (FVPs). The Jacobi poly-fractonomials proposed by Zayernouri and Karniadakis (2013, “Fractional Sturm–Liouville Eigen-Problems: Theory and Numerical Approximation,” J. Comput. Phys., 252(1), pp. 495–517.) are chosen as basis functions to approximate the true solutions, and the Rayleigh–Ritz technique is used to reduce FVPs to a system of algebraic equations. This method leads to exponential decay of the errors, which is superior to the existing methods in the literature. The fractional variational errors are discussed. Numerical examples are given to illustrate the exponential convergence of the method.

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Figures

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Fig. 1

The numerical solution yN*(t) and exact solution y*(t) for α = 0.4 and β = 5.7

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Fig. 2

The absolute error with N = 18 for α = 0.6 and β = 5.7

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Fig. 3

L-norm error versus N for α = 0.4 and β = 10.8

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Fig. 4

The numerical solution y*N (t) and exact solution y*(t) for α = 0.6

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Fig. 5

L-norm error versus N for α = 0.6

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Fig. 6

The numerical solution y*N (t) and exact solution y*(t) for α = 0.2 and β = 3.5

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Fig. 7

The absolute error with N = 12 for α = 0.2 and β = 10.5

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Fig. 8

L-norm error versus N for α = 0.2 and β = 10.5

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