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.


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
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