## 3 Mathematical Statement and Methodology  
The object of the study was a wooden structure of thickness I and height h, located next to a forest fire. The front of fire spreads at a low speed in the absence of wind. The enclosing structures of this building are presented in the form of a layer of wood material. Heat from the forest fire front is transferred to the wall by thermal radiation. The main safety parameter of a wooden structure is the fact of ignition (or non-ignition) of the enclosing structure of a wooden building when a sufficient surface temperature is reached. The following assumptions were made:  
* In the enclosing structures, heat exchange is carried out by the heat conduction mechanism;
* Two-dimensional setting;
* The shape of the fire front is a parabola;
* Thermophysical properties of building materials do not depend on temperature;
* A catastrophic scenario of fire weather is assumed when there is no moisture in the surface layer of the wall;
* Disregard wood pyrolysis;
* The main mechanism of heat transfer from the line of fire to the building is heat radiation;
* The temperature of the forest fire front is taken into account using the Stefan-Boltzmann law;
* The impact of the forest fire front on the wall is determined by \(q_{\mathit{ff}}\) and \(T_{\mathit{ff}}\). The wooden structure is under the influence of the heat flow from the forest fire front \(q_{\mathit{ff}}\). On the surface of the fence exposed to the heat flow from the forest fire front, the material itself becomes a radiation source. The physical model diagram is shown in Figure 1.  
For the numerical solution of the differential equation of heat conduction, the finite difference method described in [64; 65] was used:  
\[\rho c\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\left(\lambda \frac{\partial T}{\partial x}\right)+\frac{\partial}{\partial y}\left(\lambda \frac{\partial T}{\partial y}\right)+\frac{\partial}{\partial z}\left(\lambda \frac{\partial T}{\partial z}\right)+Qw(x,y,z,T) \tag{1}\]  
where \(\rho\), \(c\), \(\lambda\), \(T\) are the density, heat capacity, heat conduction coefficient and temperature; \(Q_{w}\) is the heat source inside the decision area; \(t\) is the temporal coordinate; \(x\), \(y\), \(z\) are the spatial coordinates. An implicit difference scheme was used for the solution. The left boundary condition was used to determine the coefficients \(\alpha\) and \(\beta\) from the relation  
\[T_{1}=\alpha_{1}T_{2}+\beta_{1}, \tag{2}\]  
Figure 1: Scheme of the physical model. where \(T_{1}\), \(T_{2}\) are the temperature in computational mesh nodes number 1 and 2; \(\alpha_{1}\) is the first running coefficient number 1; \(\beta_{1}\) is the second running coefficient number 1. The right boundary condition is used to determine the temperature at the boundary \(x=L\); for the number of nodes \(N\), the temperature at the boundary will have the form \(T_{N}(K)\). The boundary conditions have been discredited to the first order of approximation. It is assumed that a forest fire front is approaching the structure from the left side. Convection-radiation heat transfer at the boundary of the enclosing structure is calculated as an approximation on the left and right boundaries. On the left border:  
\[-\lambda_{1}\frac{\partial T_{1}}{\partial x}|_{x=0}=\alpha\Big{(}T_{ff}-T_{1 }\Big{)}+\varepsilon\sigma\Big{(}T_{ff}^{4}-T_{1}^{4}\Big{)}+q_{ff} \tag{3}\]  
where \(T_{1}\) is the temperature in the first node of computational mesh; \(T_{ff}\) is the temperature in the fire front; \(\lambda_{1}\) is heat conduction coefficient of first layer of the enclosing construction of the wall; \(\alpha\) is the heat transfer coefficient; \(\varepsilon\) is the emissivity factor; \(\sigma\) is the Stefan-Boltzmann constant; \(q_{ff}\) is the heat flux from fire line; \(x\) is the spatial coordinate. Then:  
\[\left\{ \begin{array}{c}\alpha_{1}=\frac{\lambda_{1}}{\lambda_{1}+h\alpha}\\ \beta_{1}=\frac{h\alpha T_{ff}+coth\big{(}T_{ff}^{4}-T_{1}^{4}\big{)}+hq_{ff} }{\lambda_{1}+h\alpha}\end{array} \right.