What is the difference between fractional-order derivative and integer-order derivative?

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Ma'Mon Abu Hammad

Fractional-order derivatives have both geometric and physical interpretations in mathematical models. Let's explore each interpretation:

Geometric Interpretation:

Fractional derivative as a generalization of integer-order derivatives: Geometrically, the fractional derivative can be seen as a generalization of integer-order derivatives. Just as the first derivative represents the slope of a tangent line and the second derivative represents the curvature of a curve, fractional derivatives capture more intricate properties of curves. They provide a way to measure the "rate of change" of a function in a fractional order.

Fractional derivative as a measure of fractal dimension: In some cases, fractional derivatives are linked to fractal geometry. Fractal objects exhibit self-similarity at different scales, and fractional derivatives can capture the complexity and irregularity of such objects. The fractional derivative order can be related to the fractal dimension of a curve or a set, providing insight into its intricacy and self-similarity.

Physical Interpretation:

Memory and hereditary effects: Fractional derivatives account for memory and hereditary effects in physical systems. In many physical processes, the current state is influenced by past states, and fractional derivatives capture this memory property. They represent the "history" of a system and its influence on the current behavior.

Anomalous diffusion and subdiffusion: Fractional derivatives are used to model anomalous diffusion and subdiffusion phenomena observed in various physical systems. Anomalous diffusion refers to non-classical behaviors where the spreading of particles or quantities does not follow standard diffusion laws. Fractional derivatives provide a mathematical tool to describe these non-Gaussian and non-Fickian diffusion processes.

Viscoelasticity and material behavior: Fractional derivatives are employed in the modeling of viscoelastic materials, which exhibit both elastic and viscous properties. Fractional-order constitutive equations capture the fractional relaxation and creep behaviors of viscoelastic materials, allowing for more accurate modeling of their mechanical response.

These interpretations highlight how fractional derivatives extend our understanding of geometric properties and physical phenomena beyond integer-order derivatives. They provide valuable tools to describe complex behaviors and memory effects in mathematical models and physical systems.

Seyed Hossein Ahmadi

سلام

The fractional derivative is simply a conclusion of the gamma function in the Laplace transform, and when the derivative is derived from this function, it can be said that a derivative of the fractional order is obtained.