&=& B_{\omega}e^{\frac{\gamma_{1\omega}}{\mu}(d-x_1)} \\
&-& \int_x^d\frac{C_{1\omega}\Delta \widetilde{T}(x'_1) + Q^0_{\omega}e^{-x/\delta}\tau_{1\omega}}{4\pi\Lambda_{1\omega}\mu}e^{\frac{\gamma_{1\omega}}{\mu}(x'_1-x_1)}dx'_1\ (\mu \in [-1, 0]),\end{aligned}$$ where $\gamma_{1\omega} = (1+i\eta\tau_{1\omega})/\Lambda_{1\omega}$, $d$ is the film thickness, and $B_{\omega}$ are the unknown coefficients determined by the interface condition at $x_1 = d$. Here, $\widetilde{g}_{1\omega}^+(x_1,\mu)$ indicates the forward-going phonons and $\widetilde{g}_{1\omega}^-(x_1,\mu)$ the backward-going phonons.

To close the problem, we plug Eqs. (\[eq:ODEsolution\_gPlus\]) & (\[eq:ODEsolution\_gMinus\]) into Eq. (\[eq:EnergyConservation\]) and obtain an integral equation for temperature as: $$\label{eq:FilmTemperature}
\Delta \widetilde{T}(\widehat{x}_1) - \int_0^1\Delta \widetilde{T}(\widehat{x}'_1)K(\widehat{x}'_1,\widehat{x}_1)d\widehat{x}'_1 = \int_0^{\omega_m}B_{\omega}F^1_{\omega}(\widehat{x}_1)d\omega + F^2(\widehat{x}_1),$$ where $\widehat{x}_1 = x_1/d$. The kernel function $K(\widehat{x}'_1,\widehat{x})$ is given by $$\label{eq:Kernel}
K(\widehat{x}'_1,\widehat{x}_1) = \frac{1}{2\int_0^{\omega_m}\frac{C_{1\omega}}{\tau_{1\omega}}d\omega}\int_0^{\omega_m}\frac{C_{1\omega}}{\tau_{1\omega}\text{Kn}_{1\omega}}\{E_1[\widehat{\gamma}_{1\omega}(\widehat{x}'_1+\widehat{x}_1)]+E_1[\widehat{\gamma}_{1\omega}|\widehat{x}'_1-\widehat{x}_1|]\}d\omega$$ and the two inhomogeneous functions are given by $$\begin{aligned}
\label{eq:InhomoF1}
F^1_{\omega}(\widehat{x}_1) &=& \frac{2\pi}{\int_0^{\omega_m}\frac{C_{1\omega}}{\tau_{1\omega}}d\omega}\frac{1}{\tau_{1\omega}}\{E_2[\widehat{\gamma}_{1\omega}(1+\widehat{x}_1)]+E_2[\widehat{\gamma}_{1\omega}(1-\widehat{x}_1)]\}\\ \label{eq:InhomoF2}\nonumber
F^2(\widehat{x}_1) &=&  \frac{2\pi}{\int_0^{\omega_m}\frac{C_{1\omega}}{\tau_{1\omega}}d\omega}\int_0^1\int_0^{\omega_m}\frac{Q^0_{\omega}e^{-\rho\widehat{x}'_1}}{\text{Kn}_{1\omega}}\{E_1[\widehat{\gamma}_{1\omega}(\widehat{x}'_1+\widehat{x}_1)]+E_1[\widehat{\gamma}_{1\omega}|\widehat{x}'_1-\widehat{x}_1|]\}d\omega d\widehat{x}'\\\end{aligned}$$ where Kn$_{1\omega} = \Lambda_{1\omega}/d$ is the Knudsen number, $\widehat{\gamma}_{1\omega} =\frac{1+i\eta\tau_{1\omega}}{\text{Kn}_{1\omega}}$, and $E_n(x)$ is the exponential integral given by[@Gangbook]: $$\label{eq:ExpInt}
E_n(x) = \int_0^1 \mu^{n-2}e^{-\frac{x}{\mu}}d\mu.$$

Recently, we have developed a spectral method to efficiently solve Eq. (\[eq:FilmTemperature\]) in Ref. 60. Briefly, the functions in Eq. (\[eq:FilmTemperature\]) can be expanded as a finite cosine series, such as $$\label{eq:TempExp}
\Delta \widetilde{T}_{1(N)}(\widehat{x}_1) \approx \sum\limits_{n = 0}^{N}t_n \text{cos}(n\pi\widehat{x}_1)$$ and $$\begin{aligned}
\label{eq:KernelExp}\nonumber
K_{(N)}(\widehat{x},\widehat{x}')=\frac{1}{4}k_{00}&+&\frac{1}{2}\sum\limits_{m=1}^{N}k_{m0}cos(m\pi \widehat{x})+\frac{1}{2}\sum\limits_{n=1}^{N}k_{0n}cos(n\pi \widehat{x}'), \\
&+&\sum\limits_{m=1}^N\sum\limits_{n=1}^N k_{mn}cos(m\pi \widehat{x})cos(n\pi \widehat{x}')\end{aligned}$$ where $N$ is the truncated basis number, and $t_n$’s and $k_{nm}$’s are the Fourier coefficient. Similarly, 