Homogenizing atomic dynamics by fractional differential equations

Spline Collocation for Multi-Term Fractional Integro-Differential Equations with Weakly Singular Kernels

“…, n − 1) and z = 0. Therefore, the homogeneous system corresponding to (13) has only the trivial solution c 0 = c 1 = • • • = c n−1 = 0 (and thus M is regular) if and only if from all polynomials y of degree n − 1 only y = 0 satisfies (14).…”

“…. , n − 1) has in C[0, b] only the trivial solution y = 0, and from all polynomials y of degree n − 1 only y = 0 satisfies the conditions (14).…”

“…For details, including basic theory of fractional calculus and references to some applications, see the monographs [1][2][3][4] and review papers [5][6][7]. Some concrete examples can be found in the works [8] (modelling the behaviour of viscoelastic materials), [9,10] (kinetics of polymers), [11,12] (modelling the behaviour of humans), [13] (fractional processes in financial economics), [14] (atomic wall dynamics), [15] (viscoelastic laws for arterial wall mechanics), [16] (models of supercapacitor energy storage), [17] (transition of flow in fluid dynamics) and [18][19][20] (anomalous diffusion).…”

An Operator-Based Approach for the Construction of Closed-Form Solutions to Fractional Differential Equations

“…Viscoelastic constitutive laws for arterial wall mechanics are investigated via fractional-order models in [36]. Atomic chain dynamics are approximated using fractional differential equations in [32]. Dynamics of the transition from laminar to turbulent fluid flow is described using fractional models in [15].…”

“…Analytical solutions to the fractional modified Telegraph and Rayleigh equations are constructed in terms of Mittag-Leffler, Hypergeometric, Hermite and Fox's H functions in [33]. The solution to a Hilfer-generalized Riemann-Liouville fractional diffusion equation is obtained using variable separation, Laplace transform and Sturm-Liouville analysis in [31].…”