**4. Problem Solution**

For the case considered here, a cylindrical coordinate system corresponding to the geometry of the physical model was used, and thus Equation (3) can be written as:

$$\frac{\partial^{\mu}\theta}{\partial\tau^{a}} = \frac{1}{r}\frac{\partial\theta}{\partial r} + \frac{\partial^{2}\theta}{\partial r^{2}} + \frac{\partial^{2}\theta}{\partial z^{2}}, \text{ where } \tau = \kappa \, t \tag{18}$$

Equation (18) was solved by adopting the Marchi–Zgrablich transform, the finite Fourier cosine transform, and the Laplace transform. The general form of the finite integral transform by Marchi-Zgrablich [34] is:

$$\mathcal{H}(f(\mathbf{x})) = \overline{f}(n) = \int\_{r\_i}^{r\_0} r \, f(\mathbf{x}) \, \mathbb{S}\_p(\lambda, \mathbf{a}, k\_\mathbf{n} \, r) dr \tag{19}$$

The inverse integral transform proposed by Marchi-Zgrablich can be expressed as:

$$\mathcal{H}^{-1}\left(\overline{f}\left(\mathbf{x}\right)\right) = f(\mathbf{x}) = \sum\_{n=1}^{\infty} a\_n \mathcal{S}\_p(\lambda, \mathfrak{a}, k\_n \cdot r)\_{\star} \tag{20}$$

where

$$a\_n = \frac{\overline{f}\_p(n)}{\mathbb{C}\_n},\tag{21}$$

$$\mathbb{C}\_{n} = \int\_{r\_{i}}^{r\_{0}} r \left[ \mathbb{S}\_{p}(\lambda, \alpha, k\_{n}r) \right]^{2} dr. \tag{22}$$

Thus, Equation (18) is written as:

$$\frac{\partial^{\alpha}\overline{\theta}}{\partial\tau^{\alpha}} = \left(\frac{\partial^{2}\overline{\theta}}{\partial z^{2}} - k^{2}{}\_{n}\overline{\theta}\right). \tag{23}$$

The finite Fourier cosine transform is described in the general form as:

$$\mathcal{F}\_{\mathfrak{C}}[\mathfrak{g}(\mathfrak{x})] = \mathfrak{g}^\*(m) = \int\_0^L \mathfrak{g}(\mathfrak{x}) \cos \left(\frac{m\pi\mathfrak{x}}{L}\right) d\mathfrak{x},\tag{24}$$

where *m* = 1, 2, 3, . . .

$$\mathcal{F}\_{\varepsilon}\left[\frac{d^2\mathbf{g}(\mathbf{x})}{d\mathbf{x}^2}\right] = (-1)^m \left.\frac{d\mathbf{g}(\mathbf{x})}{d\mathbf{x}}\right|\_{\mathbf{x}=L} - \left.\frac{d\mathbf{g}(\mathbf{x})}{d\mathbf{x}}\right|\_{\mathbf{x}=0} - \frac{m^2\pi^2}{L^2} \underbrace{\mathbf{g}^\*(m)}\_{\mathbf{g}^\*(n) = \mathcal{F}\_{\varepsilon}[\mathbf{g}(\mathbf{x})]}.\tag{25}$$

The inverse transform is given in the general form as:

$$\mathcal{F}\_c^{-1}[g^\*(m)] = g(\mathbf{x}) = \frac{g^\*(m=0)}{L} + \frac{2}{L} \sum\_{m=1}^{\infty} g^\*(m) \cos\left(\frac{m\pi\mathbf{x}}{L}\right). \tag{26}$$

Applying the finite Fourier cosine transform to write Equation (23) and assuming the boundary conditions given in Equation (14), we obtain:

$$\frac{\partial^{\alpha}\overline{\theta}^{\*}}{\partial\tau^{\alpha}} = \left(\frac{\overline{\varphi}\_{q}}{\lambda} - \frac{m^{2}\pi^{2}}{L^{2}}\overline{\theta}^{\*}(m) - k^{2}{}\_{n}\overline{\theta}^{\*}(m)\right). \tag{27}$$

With the Laplace transform [25] being:

$$\mathcal{L}\{D\_C^a f(t)\} = s^a f(s) - \sum\_{k=0}^{n-1} f^{(k)}\left(0^+\right) s^{a-1-k}, n-1 < a < n,\tag{28}$$

we can rewrite Equation (27) as:

$$s^{\alpha}\hat{\overline{\theta}}^{\*}(s) = \frac{\overline{\varphi}\_{q}}{\lambda} - \frac{m^{2}\pi^{2}}{\left(L^{r}\right)^{2}}\hat{\overline{\theta}}^{\*}(s) - k^{2}\,\_{n}\hat{\overline{\theta}}^{\*}(s). \tag{29}$$

Once the transformations are completed, Equation (18) has the following form:

$$\stackrel{\scriptstyle \mathfrak{S}^{\ast}}{\partial}(s) = \frac{\overline{\Psi}\_{\emptyset}}{\lambda} \frac{1}{\omega^{2}} \left( \frac{1}{s} - \frac{s^{a-1}}{(s^{a} + \omega^{2})} \right) \text{ where} \left( \frac{m^{2}\pi^{2} + L^{2}k^{2}\_{\text{n}}}{L^{2}} \right) = \omega^{2} \text{ and } \omega\_{0}{}^{2} = k^{2}\_{\text{n}} \tag{30}$$

After the three inverse integral transforms are employed, the equation describing the temperature distribution for the rotor takes the final form:

$$T' = T\_0 + \left(\begin{array}{c} \frac{1}{L} \sum\_{n=1}^{\infty} \frac{\int\_{r\_l}^{r\_l} \overline{\eta}\_q S\_p(\lambda^n \mu^l, k\_n r) dr}{\int\_{r\_l}^{r\_l} \overline{\eta}\_p (\lambda \mu, k\_n r)^2 dr} S\_p(\lambda, \mu, k\_n r) \left(\frac{1}{\omega\_0^2} - \frac{1}{\omega\_0 r} E\_a(-\kappa \omega\_0^2 \mu^l)\right) + \\\ + \frac{\tau\_q}{L} \sum\_{n=1}^{\infty} \frac{\tau\_l}{\int\_{r\_l}^{r\_l} \overline{\eta}\_p S\_p(\lambda^n \mu^l, k\_n r) dr} S\_p(\lambda, \mu, k\_n r) \sum\_{m=1}^{\infty} \cos\left(\frac{m \pi x}{L}\right) \left(\frac{1}{\omega^2} - \frac{1}{\omega^2} E\_a(-\kappa \omega^2 \mu^l)\right) \end{array} \right) \tag{31}$$

where *Eα*,*β*(*z*) is the function of the Mittag–Leffler type [35,36]:

$$E\_{\alpha,\beta}(z) = \sum\_{v=1}^{\infty} \frac{z^v}{\Gamma(\alpha v + \beta)} \,\,\alpha > 0,\,\,\beta > 0. \tag{32}$$

For *α*, *β* = 1, *Eα*,*β*(*z*) = e*z*, and thus Equation (31) is solved using the classical heat equation with predetermined initial and boundary conditions.

$$\begin{array}{c} \mathcal{S}\_{\mathbb{P}}(\lambda, \mathfrak{a}, k\_{\mathfrak{n}} \cdot \boldsymbol{r}) = \left(-k\_{\mathfrak{n}} \, \mathrm{Y}\_{1}(k\_{\mathfrak{n}} \, \boldsymbol{r}\_{\mathfrak{i}}) + \mathfrak{a} \, \mathrm{Y}\_{0}(k\_{\mathfrak{n}} \, \boldsymbol{r}\_{\mathfrak{o}}) - \lambda \, k\_{\mathfrak{n}} \, \mathrm{Y}\_{1}(k\_{\mathfrak{n}} \, \boldsymbol{r}\_{\mathfrak{o}})\right) I\_{0}(k\_{\mathfrak{n}} \, \boldsymbol{r}) \\ \qquad + (k\_{\mathfrak{n}} \, \mathrm{J}\_{1}(k\_{\mathfrak{n}} \, \boldsymbol{r}\_{\mathfrak{i}}) - \mathsf{a} \, \mathrm{J}\_{0}(k\_{\mathfrak{n}} \, \boldsymbol{r}\_{\mathfrak{o}}) + \lambda \, k\_{\mathfrak{n}} \, \mathrm{J}\_{1}(k\_{\mathfrak{n}} \, \boldsymbol{r}\_{\mathfrak{o}})) \mathrm{Y}\_{0}(k\_{\mathfrak{n}} \, \boldsymbol{r}) \end{array} \tag{33}$$

Equation (33) is dependent on the boundary conditions assumed for the rotor or stator model. The distribution of temperature in the stator is defined as:

$$T^{\sf S} = T\_0 + \sum\_{n=1}^{\infty} \frac{\cosh(k\_n z) \int\_{r\_i}^{r\_o} r \,\theta^f \, \mathcal{S}\_p \left(\lambda^s, \alpha^f, k\_n r\right) dr}{\cosh(k\_n L) \int\_{r\_i}^{r\_o} r \left[\mathcal{S}\_p \left(\lambda^s, \alpha^f, k\_n \cdot r\right)\right]^2 dr} \mathcal{S}\_p \left(\lambda^s, \alpha^f, k\_n r\right). \tag{34}$$

where *<sup>θ</sup> <sup>f</sup>* is the excess temperature of the medium in the gap, calculated as *<sup>θ</sup> <sup>f</sup>* = *<sup>T</sup><sup>f</sup>* − *To*.
