Dimensional consistency of derivatives with non-singular kernels
DOI:
https://doi.org/10.31349/RevMexFis.72.060702Keywords:
Caputo fractional derivative, Caputo-Fabrizio derivative, Atangana-Baleanu derivative, RC circuitAbstract
The Caputo-Fabrizio (CF) and Atangana-Baleanu (AB) derivatives were proposed as alternatives to the usual Caputo and Riemann-Liouville fractional derivatives. An advantage of CF and AB derivatives is that they are defined in terms of non-singular kernels, such as the exponential function and the Mittag-Leffler function, respectively. However, the use and application of fractional derivatives with non-singular kernels has resulted in many models in terms of fractional differential equations that do not comply with the principle of dimensional homogeneity and therefore their solutions are not entirely satisfactory. In this work, we propose New Derivatives of the Caputo-Fabrizio (NDCF) and Atangana-Baleanu (NDAB) types, derive their Laplace transforms and apply them to the solution of an RC circuit as an illustrative example. Furthermore, we confirm that the price to pay for attempting to avoid the singularity of the Caputo derivative is that the initial condition is not satisfied for kernels without singularities. The initial condition depends on the order of the fractional derivative γ and only reduces to the initial condition when γ = 1. To provide a more realistic response, a shift in the dynamic variable was introduced, yielding physical solutions consistent with the dimensionality, the initial condition, and the steady state.
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