Research Papers

On a Numerical Approach to Solve Multi-Order Fractional Differential Equations With Initial/Boundary Conditions

[+] Author and Article Information
M. A. Firoozjaee

Department of Mathematics,
Shahid Beheshti University,
G.C., Tehran 19839-63113, Iran
e-mail: m_firoozjaee@sbu.ac.ir

S. A. Yousefi

Department of Mathematics,
Shahid Beheshti University,
G.C., Tehran 19839-63113, Iran
e-mail: s-yousefi@sbu.ac.ir

H. Jafari

Department of Mathematics,
University of Mazandaran,
Babolsar 47416-95447, Iran
Department of Mathematical Sciences,
University of South Africa,
PO Box 392,
UNISA 0003, Pretoria, South Africa
e-mail: jafari@umz.ac.ir

D. Baleanu

Department of Mathematics and
Computer Science,
Çankaya University,
Balgat 0630, Ankara, Turkey
Institute of Space Sciences,
P.O. BOX MG-23,
R 76900 Magurele-Bucharest, Romania
e-mail: dumitru@cankaya.edu.tr

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 19, 2014; final manuscript received February 2, 2015; published online June 25, 2015. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 10(6), 061025 (Nov 01, 2015) (6 pages) Paper No: CND-14-1294; doi: 10.1115/1.4029785 History: Received November 19, 2014; Revised February 02, 2015; Online June 25, 2015

In this manuscript, a new method is introduced for solving multi-order fractional differential equations. By transforming the fractional differential equations into an optimization problem and using polynomial basis functions, we obtain the system of algebraic equation. Then, we solve the system of nonlinear algebraic equation and obtain the coefficients of polynomial expansion. Also, we show the convergence of the method. Some numerical examples are presented which illustrate the theoretical results and the performance of the method.

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Grahic Jump Location
Fig. 3

Exact solution for α = 1 (solid line) and numerical solution for m = 7 (dotted)

Grahic Jump Location
Fig. 4

Numerical solution y(x) for m = 7 and α = 0.5, 0.7, 0.9, and 1

Grahic Jump Location
Fig. 7

Exact solution for α = 2, β = 1 (solid line) and numerical solution for m = 7 (dotted)

Grahic Jump Location
Fig. 8

Numerical solution y(x) for β = 1, and m = 7, α = 1.35, 1.5, 1.75, and 2

Grahic Jump Location
Fig. 1

Exact solution for α = 1 (solid line) and numerical solution for m = 7 (dotted)

Grahic Jump Location
Fig. 2

Numerical solution y(x) for m = 7 and α = 0.5, 0.7, 0.9, and 1

Grahic Jump Location
Fig. 5

Exact solution for α = 1.2 (solid line) and numerical solution for m = 7 (dotted)

Grahic Jump Location
Fig. 6

Exact solution for α = 1.6 (solid line) and numerical solution for m = 7 (dotted)



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