Dimensional consistency in fractional differential equations with non singular kernels

Abstract: The purpose of this article is to address the issues of dimensional consistency that arise in the process of replacing the ordinary time derivative operator by a fractional derivative operator in order to write a fractional differential equation. We show that by performing a simple change of variables fulfilling certain conditions ensures the consistency in physical dimensions for fractional differential equations with non singular kernels. An example of the proposed method is given.

• The paper introduces an order-dependent time-scale function and non-local time transformation that make Caputo–Fabrizio fractional equations dimensionally consistent while recovering the classical derivative at α=1.
• The method converts a first-order linear fractional equation into an ordinary differential equation with variable coefficients and produces a closed-form solution for a fractional RC charging circuit.
• The RC model approaches the textbook charge law as α→1, but the non-unique choice of the time-scale function means dimensional consistency alone does not determine a physically unique model.

Overview

The paper addresses a persistent technical obstacle in fractional modeling: the dimensional inconsistency that arises when an ordinary time derivative $d/dt$ is replaced by a fractional operator in a differential equation governing a physical system. While the standard remedy for singular-kernel operators (Riemann–Liouville, Caputo) is the introduction of a scaling parameter $\sigma$ with units of time, the author observes that this route is not directly applicable to derivatives with non-singular kernels, such as the Caputo–Fabrizio operator, which is itself dimensionless. The central contribution is a systematic substitution scheme based on a change of the independent variable, $\tau(t,\alpha)=\int_0^t dt'/\varphi(t',\alpha)$, where $\varphi$ is a time-dimensional function depending on the fractional order $\alpha$. The method is illustrated on a first-order linear fractional differential equation and applied to a charging RC circuit (2604.02432).

Background and motivation

Fractional derivatives with power-law kernels capture algebraic memory decay but are singular at the origin, complicating the treatment of initial conditions. The Caputo–Fabrizio derivative replaces the kernel $(t-s)^{-\alpha}$ with an exponential $\exp[-\alpha(t-s)/(1-\alpha)]$ and the prefactor $1/\Gamma(1-\alpha)$ with $(2-\alpha)M(\alpha)/2(1-\alpha)$, where $M(\alpha)=2/(2-\alpha)$. The resulting operator is linear, annihilates constants, interpolates correctly between $\sigma$0 (yielding $\sigma$1) and $\sigma$2 (recovering $\sigma$3), and has the Laplace transform

$\sigma$4

which facilitates analytical solution of FDEs. Crucially for dimensional analysis, the exponential kernel renders the operator dimensionless, so the naive replacement $\sigma$5 cannot preserve units, unlike the singular-kernel case where the operator carries units of $\sigma$6 and a single parameter $\sigma$7 (with $\sigma$8) suffices to restore homogeneity (2604.02432).

The change-of-variables scheme

The proposed construction introduces an auxiliary function $\sigma$9 with dimensions of time, defines the non-local time $\tau(t,\alpha)=\int_0^t dt'/\varphi(t',\alpha)$0, and replaces

$\tau(t,\alpha)=\int_0^t dt'/\varphi(t',\alpha)$1

Dimensional consistency requires the classical limit to be recovered at $\tau(t,\alpha)=\int_0^t dt'/\varphi(t',\alpha)$2, imposing the constraint

$\tau(t,\alpha)=\int_0^t dt'/\varphi(t',\alpha)$3

Applying the scheme to the linear first-order equation $\tau(t,\alpha)=\int_0^t dt'/\varphi(t',\alpha)$4 with constant coefficients yields a fractional equation in $\tau(t,\alpha)=\int_0^t dt'/\varphi(t',\alpha)$5 with time-dependent coefficients $\tau(t,\alpha)=\int_0^t dt'/\varphi(t',\alpha)$6 and $\tau(t,\alpha)=\int_0^t dt'/\varphi(t',\alpha)$7. Differentiating the integral form of the Caputo–Fabrizio equation converts it into an ordinary first-order ODE in $\tau(t,\alpha)=\int_0^t dt'/\varphi(t',\alpha)$8,

$\tau(t,\alpha)=\int_0^t dt'/\varphi(t',\alpha)$9

which is solved via the integrating factor $\varphi$0. The general solution is $\varphi$1 with $\varphi$2 (2604.02432).

Application to an RC circuit

For a series RC circuit driven by a DC source $\varphi$3, Kirchhoff's voltage law gives the fractional equation $\varphi$4, with $\varphi$5 and $\varphi$6. The author chooses

$\varphi$7

which satisfies the classical-limit constraint. The resulting fractional equation,

$\varphi$8

is solved by the general formula above with the initial condition $\varphi$9. In terms of ordinary time, the capacitor charge is

$\alpha$0

and the corresponding capacitor voltage is $\alpha$1. The solution passes the essential consistency check: taking $\alpha$2 recovers the textbook result $\alpha$3 (2604.02432). The plotted voltage curves show classical behavior as $\alpha$4 and increasingly dissipative, non-local dynamics for $\alpha$5, interpreted as internal friction associated with the fractional order.

Limitations and open questions

The author is explicit that the choice of $\alpha$6 is not unique: different admissible functions lead to different fractional differential equations and hence different classes of solutions, so the method guarantees dimensional consistency but does not by itself select a physically canonical model. The physical interpretation of $\alpha$7 and of the resulting non-local time $\alpha$8 remains open, paralleling the interpretational difficulties of the parameter $\alpha$9 in singular-kernel treatments. The analysis is restricted to first-order linear FDEs; extension to higher-order non-singular derivatives is stated as a direction but not developed. The claim that the fractional order $(t-s)^{-\alpha}$0 represents energy dissipation is asserted rather than derived from a thermodynamic argument, and no experimental data are used to validate the RC-circuit solution beyond its classical limit.

Conclusion

The paper provides a systematic, dimensionally consistent procedure for formulating fractional differential equations with non-singular (exponential) kernels, based on a change of the independent variable governed by an order-dependent time-scale function $(t-s)^{-\alpha}$1 subject to a classical-limit constraint. The method reproduces standard singular-kernel results at $(t-s)^{-\alpha}$2, yields a closed-form solution for the fractional RC circuit, and generalizes the auxiliary-parameter approach of Gómez-Aguilar and others to the non-singular case. The principal open issue is the non-uniqueness of $(t-s)^{-\alpha}$3 and the associated ambiguity in the physical meaning of the resulting fractional dynamics.