## 1 Introduction  
Constant demand exists for permanent accuracy improvement in machine tool manufacturing technology, aiming to eliminate errors caused by different deformations. There are two approaches to solving this problem. First, all sources of erroneous movements can be considered to produce perfect equipment with a reasonable assignment of stiffness, and optimal addition of damping, material selection, and symmetrical structure design [1, 2]. In this case, regardless of how well a machine may be designed, the accuracy is limited because of thermal or other deformations that are not accounted for [3]. Alternatively, it is possible to analyze erroneous movements and eliminate them with compensation [4]. This technique allows the accuracy to be increased even for a moderately accurate machine tool. The causes of errors were analyzed and classified in [5]. The authors distinguished three sources of errors: geometric and kinematic errors, thermal errors, and cutting force errors. The most significant factor is the thermal error, which causes 40-70% of machine tool errors [6]. It is known that up to 75% of overall geometrical errors can be induced by thermal effects [7]. The sources of thermal influence are listed in [6]: heat caused by the cutting process; heat generated in the machine system due to friction, engines, etc. ; heating or cooling provided by the cooling system; and thermal memory from the previous state. The machine's moving parts generate heat, which results in an inaccurate position of the cutting tool end in the thermoplastic machine system. These erroneous movements are unavoidable and significantly affect the accuracy [5]. Error compensation consists of choosing a model, determining the optimal position of the sensors, measuring each of the error components, determining the current position of the instrument and temperature, calculating corrective actions, and reducing the deviation. The first step in this list is choosing a thermal model and quantitatively estimating the machine part's thermal characteristics. Ref. [8] reviews the actual methods of machine tool modeling: linear polynomial models, neural network models, approaches based on linear regression analysis, probabilistic methods, frequency domain analysis, and others. The authors propose the linear state-space representation of the lumped mathematical model. A promising approach to predicting temperature distribution and estimating thermal errors is to use models of systems with distributed parameters. Distributed parameter thermal models have been found to predict temperature distribution and estimate thermal errors with high accuracy. The heating and cooling devices also may be arranged for temperature stabilization. The calculation of the temperature error can also be performed based on measurements and a dynamic model, and used to calculate corrective movements by applying the actuators. In most cases, we are interested in improving the main assemblies, for example, spindle or feed, so we can use the models of these assemblies, which is much less time consuming [7]. The models of assemblies in PDE form consider their distributed nature and allow them to be described with higher accuracy. Heat transfer coefficients may vary with time, and their variations can be random or periodic [9]. The factors that influence the heat transfer coefficient are the temperature distribution of the structure, ambient effects, and temperature-dependent material properties [10]. In the case of preliminary assessment of system behavior based on analytical solutions, time-varying parameters make the PDE nonstationary and difficult for analytical analysis. In [11], the influence of the transfer coefficient values on the machine tool thermal behavior is investigated, and the changing convection regime is observed using finite element method simulation. It is shown that the machine error depends on the transfer coefficient. Different methods of distributed systems analysis exist.