THE $G$-TRANSFORM APPROACH TO FRACTIONAL HEAT AND MASS TRANSFER EQUATIONS, WITH A NOTE ON FRACTIONAL-ORDER LEARNING DYNAMICS

In a previous work [5], we introduced the $G$-transform. This is a generalized Laplace-type integral transform defined by $G(f) = u^{\alpha} \int_{0}^{\infty} e^{-t/u} f(t) \, dt$. This integrates the Laplace, Sumudu, and Elzaki transforms within a single framework parameterized by an integer $\alpha$. The present paper extends this framework to fractional differential equations (FDEs), which are directly applicable to anomalous heat and mass transfer. Many transport phenomena in porous media, nanofluids, and biological tissues deviate from classical Fourier-Fick behavior and are more accurately modeled by time fractional equations involving the Caputo derivative. First, we derive the formula for the $G$-transform for the Caputo fractional derivatives. This formula for the order $\beta \in (0, 1]$ maintains a net algebraic structure similar to classical differential formulas (an extension to $1 < \beta \le 2$, covering the second initial condition $f’(0)$, is given as Corollary 2.4). By applying these results, we solve the time-fractional differential heat conduction equation and the time-fractional mass diffusion equation, demonstrating that the $G$-transform provides an integrated and computationally efficient tool for these problems regardless of the choice of $\alpha$. The parameter $\alpha$ can be adjusted to minimize computational complexity for specific problem types, and we show that choosing $\alpha = -1$ is particularly convenient for fractional relaxation transport equations. Finally, we note that the same fractional relaxation equation underlies fractional-order gradient descent in machine learning, giving a direct bridge between the heat-transfer framework developed here and AI optimization dynamics.