## 3 Mathematical Modeling
The system of equations describing the heating and evaporation of the IDI surface material and the discharge electrodes under the action of an incident and completely absorbed thermal radiation flux with density \(q_{z}\), without taking into account hydrodynamic processes in a condensed medium, consists of a quasi-one-dimensional heat conduction equation in a moving (associated with the evaporation wave front) coordinate system with an \(0z\) axis perpendicular to the surface and an \(0r\) axis parallel to the surface:  
\[\frac{\partial T_{s}}{\partial t}=a_{M}\frac{\partial^{2}T_{s}}{\partial z^{2}}+V _{0}\frac{\partial T_{s}}{\partial z}, \tag{5}\]  
with boundary and initial conditions:  
\[k_{m}\frac{\partial T_{s}}{\partial z}(0,\ \text{r},\ \text{t})=q_{z}(0,\ \text{r},\ \text{t})-L_{ \rm{r}}\rho(0,\ \text{r},\ \text{t})\
u(0,\ \text{r},\ \text{t}),\]and systems of equations that determine the kinetics of evaporation of the surface of condensed matter in the framework of the model with a Knudsen layer [21]:  
\[\frac{T(0,\ \mathrm{t})}{T_{s}(0,\ \mathrm{t})}=\left[\sqrt{1+\pi\bigg{(}\frac{( \gamma-1)m}{(\gamma+1)2}\bigg{)}^{2}}-\sqrt{\pi}\frac{\gamma-1}{\gamma+1}\frac{m }{2}\right]^{2},\]  
\[\frac{\rho(0,\ \mathrm{t})}{\rho_{s}(0,\ \mathrm{t})}=\sqrt{\frac{T_{s}(0,\ \mathrm{t})}{T(0,\ \mathrm{t})}} \bigg{[}\bigg{(}m^{2}+\frac{1}{2}\bigg{)}e^{m^{2}}erfc(m)-\frac{m}{\sqrt{\pi}} \bigg{]}+\frac{1}{2}\frac{T_{s}(0,\ \mathrm{t})}{T(0,\ \mathrm{t})}\bigg{[}1-\sqrt{\pi}me^{m^{2}} erfc(m)\bigg{]},\]  
where \(T_{s}\) is the temperature of the condensed medium at the moment of time \(t,\ \mathrm{a_{M},\ \mathrm{k_{M}},\ \rho_{m}}\) are the coefficients of thermal diffusivity, thermal conductivity and density of the material, respectively, \(V_{0}\) is the speed of the evaporation wave, \(p_{s},\ \rho_{s}\) are the pressure and density of the saturated vapor of condensed matter at surface temperature \(T_{s}\), \(R\) is the universal gas constant, \(T_{1}\) is the value of the surface temperature of the condensed medium, corresponding to the pressure of saturated vapor \(p_{1}\), \(L_{v}\) is the latent heat of vaporization, \(\Im\) is the molar mass of vapor, \(T(0,\ \mathrm{t}),\ \rho(0,\ \mathrm{t}),\ \mathrm{v}(0,\ \mathrm{t})\) are the temperature, density, and velocity of the plasma at the outer boundary of the Knudsen layer, and \(\gamma\) is the adiabatic exponent of condensed matter vapors. The condition for applying the quasi-one-dimensional approximation is determined by the possibility of neglecting the term \(\partial^{2}T_{s}/\partial\tau^{2}\) in the two-dimensional heat equation, which is justified by its smallness with respect to the derivative with respect to the normal component, i.e., \(\left(\partial^{2}T_{s}/\partial\tau^{2}\right)/\left(\partial^{2}T_{s}/ \partial z^{2}\right)\approx 10^{-3}\div 10^{-4}\). The electrical equation of the equivalent discharge circuit for the case when a capacitor bank with a capacitance \(C\) is used as a power energy storage device has the form [22]:  
\[\frac{1}{c^{2}}\frac{dL_{C}f}{dt}+R_{C}J=U_{k},\ \frac{dU_{k}}{dt}=-\frac{J}{C},\]  
and initial conditions  
\[t=0,\ J=0,\ L_{k}(t=0)=U_{0}, \tag{6}\]  
where \(J\) is the full discharge current; \(R_{C}=R_{0}+R_{pl}-\frac{1}{2c^{2}}\frac{dL_{pl}}{dt}\), \(L_{C}=L_{0}+L_{pl}\), \(L_{0}\), \(R_{0}\) are the resistance and inductance of the external circuit, \(U_{0},\ U_{k}(t)\) are the initial and current voltage across the capacitor bank (\"\(\kappa\)\" indicates the number of the frequency band), \(L_{pl}=\frac{c^{2}}{4\pi l_{V}^{2}}\int H^{2}dV,\ R_{pl}=\frac{1}{J}\frac{\beta} {\sigma}dV\) are the inductance and Ohmic resistance of the discharge plasma, in the calculation of which the integration is carried out over the volume of the computational domain, at the boundaries of which the value of the magnetic field strength tends to zero.