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Zbl 1103.26301
Coimbra, Carlos F. M.
Mechanics with variable-order differential operators. (English)
[J] Ann. Phys. (8) 12, No. 11-12, 692-703 (2003). ISSN 0003-3804; ISSN 1521-3889/e

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 2000:: *26A33 Fractional derivatives and integrals (real functions)
34A30 Linear ODE and systems
65M06 Finite difference methods (IVP of PDE)
74D05 Linear constitutive equations

Keywords: Variable-order calculus; fractional calculus; memory effects.

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