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Mechanics with variable-order differential operators. (English) Zbl 1103.26301
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:
26A33Fractional derivatives and integrals (real functions)
34A30Linear ODE and systems, general
65M06Finite difference methods (IVP of PDE)
74D05Linear constitutive equations (materials with memory)