In this manuscript, we propose a simplified mathematical model based on the heat transfer laws to predict the temperature profiles of a liquid controlled by a simple thermostat. The model result in a set of linear ordinary differential equations ODEs with forcing which turn on and off at a priori unknown times \({\mathcal {T}}_M=\left\{ \zeta _0,\zeta _1,\ldots ,\zeta _M\right\} \) . The pth switch-time \(\zeta _p\in {\mathcal {T}}_p\) is calculated from the zeros of a function \({\mathcal {Q}}(\chi )={\mathcal {Q}}(\chi ;\zeta _1,\ldots ,\zeta _{p-1})\) coming from analytical solutions of the system depending on the previous times \(\zeta _1,\ldots ,\zeta _{p-1}\) . The mathematical problem can be solved by using standard techniques for solving ODEs once \({\mathcal {T}}_M\) is calculated by M-successive iterations of the conditional expression \({\mathcal {Q}}(\chi =\zeta _p)=0\) and the Newton–Raphson method. We provide analytical expressions for the temperature as a function of time and \({\mathcal {T}}_M\) considering direct (DC) and alternate (AC) feeding voltages. We solve the system using this numerical-analytical approach and compare it with the results of the 4th Runge–Kutta method finding a good agreement between both methods.
