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Research Papers

Numerical Simulation and Convergence Analysis of Fractional Optimization Problems With Right-Sided Caputo Fractional Derivative

[+] Author and Article Information
Samer S. Ezz-Eldien

Faculty of Science,
Department of Mathematics,
Assiut University,
New Valley Branch,
El-Kharja 72511, Egypt
e-mail: s_sezeldien@yahoo.com

Ahmed A. El-Kalaawy

Faculty of Science,
Department of Mathematics,
Beni-Suef University,
Beni-Suef 62511, Egypt
e-mail: ah.elkelaawy@yahoo.co.uk

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 2, 2017; final manuscript received July 10, 2017; published online October 9, 2017. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(1), 011010 (Oct 09, 2017) (8 pages) Paper No: CND-17-1146; doi: 10.1115/1.4037597 History: Received April 02, 2017; Revised July 10, 2017

This paper presents an efficient approximation schemes for the numerical solution of a fractional variational problem (FVP) and fractional optimal control problem (FOCP). As basis function for the trial solution, we employ the shifted Jacobi orthonormal polynomial. We state and derive a new operational matrix of right-sided Caputo fractional derivative of such polynomial. The new methodology of the present schemes is based on the derived operational matrix with the help of the Gauss–Lobatto quadrature formula and the Lagrange multiplier technique. Accordingly, the aforementioned problems are reduced into systems of algebraic equations. The error bound for the operational matrix of right-sided Caputo derivative is analyzed. In addition, the convergence of the proposed approaches is also included. The results obtained through numerical procedures and comparing our method with other methods demonstrate the high accuracy and powerful of the present approach.

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Figures

Grahic Jump Location
Fig. 1

Exact and approximate solutions of x(t) with γ = 4.50, ν = 0.70, ρ = σ = 1/2, and N = 5, 10, 15, 20 for Example 1

Grahic Jump Location
Fig. 2

Exact and approximate solutions of u(t) with γ = 4.50, ν = 0.70, ρ = σ = 1/2, and N = 5, 10, 15, 20 for Example 1

Grahic Jump Location
Fig. 3

Log10MAE for x(t) at γ = 5.50 and ρ = σ = 0 with different values of N and ν for Example 1

Grahic Jump Location
Fig. 4

Log10MAE for u(t) at γ = 5.50 and ρ = σ = 0 with different values of N and ν for Example 1

Grahic Jump Location
Fig. 5

Exact and approximate solutions of x(t) with γ = 3, ν = 0.80, ρ = σ = 1, and N = 4, 8, 12, 16 for Example 2

Grahic Jump Location
Fig. 6

Log10MAE at γ = 3, ρ = σ = 0.70 with different values of N and ν for Example 2

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