Fractional Order Numerical Methods
In this project we consider various quadrature methods that can be used in approximating the Riemann-Liouville fractional derivative operator. For more details regarding fractional order methods and their definitions, see the paper by Mashayekhi et. al.. The following methods have been implemented for approximating the fractional derivative
- Grunwald-Letnikov (GL)
- Riemann-Sum (RS)
- Gaussian Quadrature (GQ)
- Gauss-Laguerre Quadrature (GLQ)
Preliminary analysis of different combinations of these approaches led to the conference proceeding by Miles et. al., where the following observations were made with respect to a nonlinear viscoelastic model. It is observed in the table that the optimal computational performance is achieved when combining Gaussian Quadrature (GQ) with the Riemann-Sum (RS) approximation.
| Method | Rel. Err. | CPU Time (sec) |
|---|---|---|
| GL | - | 1.19 |
| RS | 1.88e-1 | 1.13 |
| GQRS | 2.05e-1 | 0.27 |
| GQGL | 1.09e-1 | 1.63 |
1. Mashayekhi, Somayeh, Paul Miles, M. Yousuff Hussaini, and William S. Oates. "Fractional viscoelasticity in fractal and non-fractal media: Theory, experimental validation, and uncertainty analysis." Journal of the Mechanics and Physics of Solids 111 (2018): 134-156.
2. Paul Miles, Graham Pash, William Oates, Ralph C. Smith. "Numerical Techniques to Model Fractional-Order Nonlinear Viscoelasticity in Soft Elastomers." ASME Smart Materials, Adaptive Structures, and Intelligent Systems, 2018. Pending