On variational properties of balanced central fractional derivatives

ABSTRACT

This paper studies variational properties of naturally balanced fractional derivatives based on conventional left-sided and right-sided α-th order formulas, where $\alpha \in (\frac{1}{2},1).$ Approximations of fractional differential equations equipped with such naturally balanced fractional derivatives are investigated via Ritz–Galerkin approaches. It is found that not only the balanced fractional derivatives possess important variational properties, equations equipped with them exhibit desirable dynamic features as a Ritz–Galerkin formula being applied. Simulation experiments are given to illustrate our conclusions and results.

KEYWORDS:

• Fractional derivatives
• left-sided and right-sided formulae
• fractional differential equations
• Ritz–Galerkin method
• weak solutions
• variational principal

2010 AMS SUBJECT CLASSIFICATION:

• 26A33
• 34A08
• 35R11
• 65L60

Acknowledgments

The authors would like also to thank Prof. Guanghui Hu, University of Macau, for his suggestions in the preparation of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. An alternative estimation of $|\phi {|}^2$ is given by $|\phi {|}^2\le \frac{(x-a{)}^{2\alpha -1}}{\left[\mathrm{\Gamma }\right(\alpha \left){]}^2\right(2\alpha -1)}{\int }_a^b{(}_a^c{D}_x^{\alpha }\phi {)}^2\mathrm{d}x.$ Therefore, we have $\begin{array}{r}\gamma =\frac{2\alpha (2\alpha -1)\left[\mathrm{\Gamma }\right(\alpha ){]}^2}{(b-a{)}^{2\alpha }}{\sigma }_{min}.\end{array}$

2. The dual space of ${\mathcal{E}}_0^{\alpha }[a,b]$ is well-defined since ${\mathcal{E}}_0^{\alpha }[a,b]$ is a Hilbert space under usual $L^2$ inner product (see [9, Lemma 2.7] and [1, 1.11–1.12, p.6]). In the sequel, we simply denote it as ${\mathcal{E}}_d^{\alpha }[a,b]$ associated with norm $\parallel \cdot {\parallel }_{{\mathcal{E}}_d^{\alpha }[a,b]}$.

Additional information

Funding

The first author is supported by the National Natural Science Foundation of China (No. 11501581), the “2+6” Program (No. 502042032) of Central South University, and the Project funded by China Postdoctoral Science Foundation (No. 2015M570683). The second and third authors are partially supported by Research Grants MYRG102(Y1-L3)-FST13-SHW from the University of Macau and 105/2012/A3 from the FDCT of Macao.