Mechanics with variable-order differential operators. (English) Zbl 1103.26301

Coimbra, Carlos F. M.

Ann. Phys. (8) 12, No. 11-12, 692-703 (2003).

Summary: This work presents the novel concept of Variable-Order (VO) Calculus through the description of a simple problem in Mechanics. A mathematical definition for the VO-differential operator that is suitable to mechanical modelling is proposed, and an example concerning the effect of nonuniform viscoelastic frictional forces is described. A numerical method for the solution of Variable Order Differential Equations (VODEs) is proposed. The physical model under study requires mathematical tools that lie beyond the traditional methods of Constant-Order (CO) differential equations. The VO-Calculus formulation is compared to a CO-Calculus model in order to show the limitations of the latter in resolving the transition between the relevant dynamic regimes.

MSC:

26A33	Fractional derivatives and integrals
34A30	Linear ordinary differential equations and systems
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
74D05	Linear constitutive equations for materials with memory

Keywords:

Variable-order calculus; fractional calculus; memory effects.