AtanganaDerivative

To define the Atangana derivative with memory in a more precise, measure-theoretic set notation, we need to understand it in the context of fractional calculus. The Atangana derivative is a modification of the classical fractional derivative, which itself extends the traditional notion of derivatives to non-integer orders.

In the classical sense, a fractional derivative of order $\alpha$ of a function $f$ can be represented using either the Riemann-Liouville or the Caputo definition. Let's first consider the Riemann-Liouville definition:

$$D^\alpha f(t) = \frac{1}{\Gamma(n - \alpha)} \frac{d^n}{dt^n} \int_{a}^{t} \frac{f(\tau)}{(t - \tau)^{\alpha - n + 1}} d\tau$$

where $\alpha > 0$, $n = \lceil \alpha \rceil$ is the smallest integer greater than or equal to $\alpha$, and $\Gamma$ is the Gamma function.

Alternatively, the Caputo derivative, which is often used in practical applications due to its ability to incorporate initial conditions in a more straightforward manner, is defined as:

$$^C D^\alpha f(t) = \frac{1}{\Gamma(n - \alpha)} \int_{a}^{t} \frac{f^{(n)}(\tau)}{(t - \tau)^{\alpha - n + 1}} d\tau$$

where $f^{(n)}(\tau)$ denotes the $n$-th derivative of $f$ with respect to $\tau$.

The Atangana derivative with memory modifies these formulations by incorporating a memory function, say $m(t, \tau)$, which represents the effect of the past states of the function on its current state. A simplified version of this derivative, which can be applied to either the Riemann-Liouville or Caputo frameworks, can be written as:

$$\mathcal{A}^\alpha f(t) = \int_{a}^{t} m(t, \tau) \omega_\alpha(t - \tau) f(\tau) d\tau$$

In this formulation:

• $m(t, \tau)$ is the memory function that depends on the current time $t$ and a past time $\tau$.
• $\omega_\alpha(t - \tau)$ is a kernel function that depends on the fractional order $\alpha$ and the difference between the current and past times. This kernel is what modifies the classical fractional derivative to incorporate memory.

The exact form of the memory function $m(t, \tau)$ and the kernel $\omega_\alpha$ can vary based on the specific application and the properties of the system being modeled. This formulation allows the Atangana derivative to model physical and other processes where historical states significantly influence current behavior, providing a more accurate and nuanced approach than traditional derivatives.

It's important to note that this representation is a simplified version to convey the concept in a measure-theoretic framework. In actual applications, both the memory function and the kernel might have more complex forms, and additional more specific structures are required for mathematical rigor and physical relevance; in this way an analogy can be made to Java programming where this could be said to be an interface, and an actual implementation class would be needed to realize it.