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RESEARCH PAPERS

Numerical Scheme for the Solution of Fractional Differential Equations of Order Greater Than One

[+] Author and Article Information
Pankaj Kumar

Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, Illinois 62901

Om P. Agrawal1

Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, Illinois 62901om@engr.siu.edu

1

Corresponding author.

J. Comput. Nonlinear Dynam 1(2), 178-185 (Dec 16, 2005) (8 pages) doi:10.1115/1.2166147 History: Received June 14, 2005; Revised December 16, 2005

This paper presents a numerical scheme for the solutions of Fractional Differential Equations (FDEs) of order α, 1<α<2 which have been expressed in terms of Caputo Fractional Derivative (FD). In this scheme, the properties of the Caputo derivative are used to reduce an FDE into a Volterra-type integral equation. The entire domain is divided into several small domains, and the distribution of the unknown function over the domain is expressed in terms of the function values and its slopes at the node points. These approximations are then substituted into the Volterra-type integral equation to reduce it to algebraic equations. Since the method enforces the continuity of variables at the node points, it provides a solution that is continuous and with a slope that is also continuous over the entire domain. The method is used to solve two problems, linear and nonlinear, using two different types of polynomials, cubic order and fractional order. Results obtained using both types of polynomials agree well with the analytical results for problem 1 and the numerical results obtained using another scheme for problem 2. However, the fractional order polynomials give more accurate results than the cubic order polynomials do. This suggests that for the numerical solutions of FDEs fractional order polynomials may be more suitable than the integer order polynomials. A series of numerical studies suggests that the algorithm is stable.

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Figures

Grahic Jump Location
Figure 1

Comparison of y(t) for h=0.1s and α=1.5 obtained using COPs and an analytical method (O, analytical; ×, numerical)

Grahic Jump Location
Figure 2

Comparison of y′(t) for h=0.1s and α=1.5 obtained using COPs and an analytical method (O, analytical; ×, numerical)

Grahic Jump Location
Figure 3

Comparison of y(t) for h=0.1s and different values of α obtained using COPs and analytically [COPs-Δ: α=1.25, +: α=1.75, ×: α=1.95, O: α=2; ∇: analytical (α=2)]

Grahic Jump Location
Figure 4

Comparison of y′(t) for h=0.1s and different values of α obtained using COPs and analytically [COPs-Δ: α=1.25, +: α=1.75, ×: α=1.95, O: α=2; ∇: analytical (α=2)]

Grahic Jump Location
Figure 5

Comparison of errors en′ for h=0.1s and different values of α obtained using COPs (∇: 1.25, Δ: α=1.5, +: α=1.75, ×: α=1.95, O: α=2)

Grahic Jump Location
Figure 6

Comparison of y(t) and y′(t) for h=0.1s and α=1.5 obtained using FOPs and ananytically [O: y(t) analytical, ×: y(t) FOPs, Δ: y′(t) analytical, +: y′(t) FOPs]

Grahic Jump Location
Figure 7

Comparison of y(t) for h=0.1s and different α obtained using FOPs and analytically (FOPs-O: α=2, ×: α=1.95, +: α=1.75, Δ: α=1.25; analytical ∇: α=2)

Grahic Jump Location
Figure 8

Comparison of y′(t) for h=0.1s and different α=obtained using FOPs and analytically (FOPs-O: α=2, ×: α=1.95, +: α=1.75, Δ: α=1.25; analytical ∇: α=2)

Grahic Jump Location
Figure 9

Comparison of errors in y′(t) for h=0.1s and different values of α obtained using FOPs (∇: α=1.25, Δ: α=1.5, +: α=1.75, ×: α=1.95, O: α=2)

Grahic Jump Location
Figure 10

Comparison of y(t) for h=0.1s and different values of α obtained using FOPs (∇: α=1.25, Δ: α=1.5, +: α=1.75, ×: α=1.95, O: α=2)

Grahic Jump Location
Figure 11

Comparison of y′(t) for h=0.1s and different values of α obtained using FOPs (∇: α=1.25, Δ: α=1.5, +: α=1.75, ×: α=1.95, O: α=2)

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