On a Novel Fractional Calculus and its Applications to Well-known Problems

Abstract

The objective of this study is to introduce three novel and comprehensive models utilizing the concept of conformable fractional calculus. These models encompass the fractional population growth model (FPGM), the fractional body cooling model (FBCM), and the fractional heat differential equation (FHDE). Firstly, using new results, a type of new conformable fractional derivative introduced in \cite{5}:
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$\left(\mathcal{D}^\alpha H\right)(y)=\lim _{z \longrightarrow 0} \frac{H\left(y+z e^{(\alpha-1) y}\right)-H(y)}{z},$
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where $\alpha \in(0,1]$ and $H$ is a function. We obtain exact solutions of the considered models. The results feature excellent agreement of exact solutions. Finally, it is shown that the proposed method provides a more powerful mathematical tool for solving models in mathematical physics.