Abstract
This paper proposes to model fractional behaviors using Volterra equations. As fractional differentiation-based models that are commonly used to model such behaviors exhibit several drawbacks and are particular cases of Volterra equations (in the kernel definition), it appears legitimate in a modeling approach to work directly with Volterra equations. In this paper, a numerical method is thus developed to identify the kernel associated to a Volterra equation that describes the input–output behavior of a system. This method is used to model a lithium-ion cell using real data. The resulting model is compared to a fractional differentiation-based model with the same number of tunable parameters.
Highlights
Fractional behaviors are induced by numerous physical phenomena, often of a stochastic nature, such as diffusion, collision, adsorption, freezing, aggregation, and fragmentation [1,2]Many examples of this type of behavior have been revealed in various areas, including electrochemistry [3,4], thermal science [5,6], biology [7,8], mechanics [9], acoustics [10], and electrical engineering [11]
The application of the previous section shows that both fractional and Volterra models accurately capture the dynamics of a system known for its fractional behaviour, with the same number of parameters and with a slightly better quadratic error for the Volterra model; one advantage of Volterra-based models is their ability to overcome several drawbacks of the fractional models [12,13,15]
Starting from the idea that a fractional differentiation-based model is a particular type of Volterra equation [22], this paper proposes a method to directly derive an explicit form for the kernel involved in a Volterra equation from real input–output data
Summary
Fractional behaviors are induced by numerous physical phenomena, often of a stochastic nature, such as diffusion, collision, adsorption, freezing, aggregation, and fragmentation [1,2]. Physical systems with fractional behaviors are often modeled using fractional differentiation-based models, or “fractional models” for short. Volterra equations have been used in various applications: energy [18,19], sorption kinetics [20], and mechanical systems [21] These equations are more general than the fractional differentiation-based models, since fractional pseudostate space descriptions are special cases of Volterra equations [22] involving a particular kernel. Of Lorenzi [28] in which some results on identification of unknown terms in integrodifferential equations are gathered None of these works allow the direct computation of an analytic function for the kernel in the Volterra model to be obtained. An identification method that permits the direct determination of the kernel involved in a first-kind Volterra equation is proposed. A comparison of the obtained model with a fractional differentiation based model is carried out
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