In this paper, we introduce two tempered linear and non-linear time series models, namely, an autoregressive tempered fractionally integrated moving average (ARTFIMA) with α-stable noise and ARTFIMA with generalized autoregressive conditional heteroskedasticity (GARCH) noise (ARTFIMA-GARCH). We provide estimation procedures for the processes and explain the connection between ARTFIMA and their tempered continuous-time counterparts. Next, we demonstrate an application of the processes to modeling of heavy-tailed data from solar flare soft x-ray emissions. To this end, we study the solar flare data during a period of solar minimum, which occurred most recently in July, August, and September 2017. We use a two-state hidden Markov model to classify the data into two states (lower and higher activity) and to extract stationary trajectories. We do an end-to-end analysis and modeling of the solar flare data using both ARTFIMA and ARTFIMA-GARCH models and their non-tempered counterparts. We show through visual inspection and statistical tests that the ARTFIMA and ARTFIMA-GARCH models describe the data better than the ARFIMA and ARFIMA-GARCH, especially in the second state, which justifies that tempered processes can serve as the state-of-the-art approach to model signals originating from a power-law source with long memory effects.