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Exact solutions of fractional oscillator eigenfunction problem with fixed memory length

Klimek Malgorzata, Blaszczyk Tomasz

Journal of Applied Mathematics and Computational Mechanics

2024

Vol. 23, nr 1

45-58

In this paper, the eigenproblem for a fractional oscillator under homogeneous Dirichlet and Neumann boundary conditions is considered. Key properties of fractional operators with fixed memory length are established, such as the connection between left and right operators, the product rule for fractional integrals, and the fractional integration by the parts rule for periodic/antiperiodic functions. Explicit solutions in the form of discrete sets of sine/cosine eigenfunctions are derived. The impact of fractional order and memory length on eigenvalues is presented on graphs. Finally, a comparison of eigenvalues of oscillator with a fixed memory length and infinite memory length is shown.

Towards a non-conformable fractional calculus of n-variables

Martínez Francisco, Nápoles Juan E. Valdés

Journal of Mathematics and Applications

2020

Vol. 43

87--98

In this paper we present an extension of the non-conformable local fractional derivative, to the case of functions of several variables. Results analogous to those known from the classic multivariate calculus are presented. To show the strength of this approach, we show an extension of the Second Lyapunov Method to the non-conformable local fractional case.