Some dynamical models involving fractional-order derivatives with the Mittag-Leffler type kernels and their applications based upon the Legendre spectral collocation method
Date
2021
Authors
Srivastava, H.M.
Alomari, Abedel-Karrem N.
Saad, Khaled M.
Hamanah, Waleed M.
Journal Title
Journal ISSN
Volume Title
Publisher
Fractal and Fractional
Abstract
Fractional derivative models involving generalized Mittag-Leffler kernels and opposing
models are investigated. We first replace the classical derivative with the GMLK in order to obtain
the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a
spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions
of the pr. We then construct a scheme for the fractional-order models by using the spectral method
involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic
equations, which can be approximated by the Newton-Raphson method. For the second model, we
also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by
comparing it with our analytical solution. In the second and third models, the residual error functions
are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid
technique of numerical and analytical approach that is applicable for partial differential equations
with multi-order of fractional derivatives involving GMLK with three parameters.
Description
Keywords
fractional derivative, generalized Mittag-Leffler kernel (GMLK), Legendre polynomials, Legendre spectral collocation method
Citation
Srivastava, H. M., Alomari, A. N., Saad, K. M., & Hamanah, W. M. (2021). “Some dynamical models involving fractional-order derivatives with the Mittag-Leffler type kernels and their applications based upon the Legendre spectral collocation method.” Fractal and Fractional, 5(3), 131. https://doi.org/10.3390/fractalfract5030131