Fractional Calculus with Applications in Mechanics

Fractional Burgers models in creep and stress relaxation tests

“…on model parameters appearing in the classical Burgers model (1) are derived in [16] by requiring non-negativity of the storage and loss modulus that is obtained as a consequence of the dissipativity inequality in the steady state regime, see [5]. The fractional generalization of Burgers model is derived in [16] by considering the Scott-Blair (fractional) element instead of the dash-pot element in the rheological representation from Figure 1, with the orders of fractional differentiation corresponding to the fractional elements and their sums being replaced by the arbitrary orders of fractional derivatives α, β, µ, γ, and ν.…”

A numerical‐analytical solution of multi‐term fractional‐order differential equations

“…The roots z l of the denominator to the function (32), the multiplicity q l of the l th root, and the coefficients C r k,l of the partial fraction decomposition is known as the generalized Bagley-Torvik equation. 1 We consider this equation for α = 7 5 , a 0 = 1.0, a 1 = 0.5, and f(t) = sinωt with ω = 8. Assuming ν = 1 5 , we find m 1 = 7 and m 2 = 10.…”

A nonlocal problem with integral gluing condition for a third‐order loaded equation with parabolic‐hyperbolic operator involving fractional derivatives

“…In fact, using these more general concepts, we can describe better certain real-world phenomena that are not possible using integer-order derivatives. For example, we can find applications of fractional calculus in mechanics, 1 engineering, 2 viscoelasticity, 3 dynamical systems, 4 etc. Although the Riemann-Liouville fractional derivative is of great importance, from historical reasons, it may not be suitable to model some real-world phenomena.…”