This paper presents a numerical scheme for the solutions of Fractional Differential Equations (FDEs) of order , 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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April 2006
Research Papers
Numerical Scheme for the Solution of Fractional Differential Equations of Order Greater Than One
Pankaj Kumar,
Pankaj Kumar
Mechanical Engineering and Energy Processes,
Southern Illinois University
, Carbondale, Illinois 62901
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Om P. Agrawal
Om P. Agrawal
Mechanical Engineering and Energy Processes,
e-mail: om@engr.siu.edu
Southern Illinois University
, Carbondale, Illinois 62901
Search for other works by this author on:
Pankaj Kumar
Mechanical Engineering and Energy Processes,
Southern Illinois University
, Carbondale, Illinois 62901
Om P. Agrawal
Mechanical Engineering and Energy Processes,
Southern Illinois University
, Carbondale, Illinois 62901e-mail: om@engr.siu.edu
J. Comput. Nonlinear Dynam. Apr 2006, 1(2): 178-185 (8 pages)
Published Online: December 16, 2005
Article history
Received:
June 14, 2005
Revised:
December 16, 2005
Citation
Kumar, P., and Agrawal, O. P. (December 16, 2005). "Numerical Scheme for the Solution of Fractional Differential Equations of Order Greater Than One." ASME. J. Comput. Nonlinear Dynam. April 2006; 1(2): 178–185. https://doi.org/10.1115/1.2166147
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