Fractional Calculus on Time Scales

Abstract

We introduce a discrete-time fractional calculus of variations on the time scales $\mathbb{Z}$ and $(h\mathbb{Z}{)}_a$. First and second order necessary optimality conditions are established. Some numerical examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. We also give new definitions of fractional derivatives and integrals on time scales via the inverse generalized Laplace transform.

PhD thesis, Doctoral Programme in Mathematics and Applications (PDMA), University of Aveiro and University of Minho, 2012. Supervisor: Delfim F. M. Torres. Defended and approved 13/Feb/2012

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Cited by in corpus (5)

• A Fractional Calculus on Arbitrary Time Scales: Fractional Differentiation and Fractional Integration
• Exponentials and Laplace transforms on nonuniform time scales
• Generalized Fractional Operators on Time Scales with Application to Dynamic Equations
• Nonsymmetric and Symmetric Fractional Calculi on Arbitrary Nonempty Closed Sets
• Time-Fractional Optimal Control of Initial Value Problems on Time Scales