In this paper, a new modified definition of the fractional derivative is presented. The Laplace transform of the modified fractional derivative involves the initial values of the integer-order derivatives, but does not involve the initial values of the fractional derivatives as the Caputo fractional derivative. Using this new definition, Nutting’s law of viscoelastic materials can be derived from the Scott-Blair stress-strain law as the Riemann-Liouville fractional derivative. Moreover, as the order a approaches n− and (n−1)+, the new modified fractional derivative †Da t f (t) approaches the corresponding integer-order derivatives f (n)(t) and f (n−1)(t), respectively. Therefore, the proposed modified fractional derivative preserves the merits of the Riemann-Liouville fractional derivative and the Caputo fractional derivative, while avoiding their demerits. By solving a fractional vibration equation, we confirm the advantages of the proposed fractional derivative.
Digital Object Identifier (DOI)
"A Modified Fractional Derivative and its Application to Fractional Vibration Equation,"
Applied Mathematics & Information Sciences: Vol. 10
, Article 27.
Available at: https://dc.naturalspublishing.com/amis/vol10/iss5/27