Inverse problem for a system of heat and moisture transfer equations to determine coefficients in the third-order boundary condition (in Ukrainian)

There is considered the problem to define the functions $u_i(x,t)$, $u_2(x,t)$, $\alpha_i(t)$ і $\beta_i(t)$ from the conditions $$ \frac{1}{a_1^2} \frac{\partial u_1}{\partial t} = \frac{\partial^2 u_1}{\partial x^2} - b_1^2 \frac{\partial^2 u_2}{\partial x^2}, \quad 0 < x < h, \quad t > -\infty, \quad (1) $$ $$ \frac{1}{a_2^2} \frac{\partial u_2}{\partial t} = \frac{\partial^2 u_2}{\partial x^2} - b_2^2 \frac{\partial^2 u_1}{\partial x^2}, \quad 0 < x < h, \quad t > -\infty, $$ $$ \left.\left( \frac{\partial u_1}{\partial x} + \alpha_1(t)u_1 + \alpha_2(t)u_2 \right)\right|_{x=0} = \mu_1(t), \quad t > -\infty, \quad (2) $$ $$ \left.\left( \frac{\partial u_2}{\partial x} + \beta_1(t)u_1 + \beta_2(t)u_2 \right)\right|_{x=0} = \mu_2(t), \quad t > -\infty, $$ $$ u_1|_{x=h} = \nu_1(t), \quad u_2|_{x=h} = \nu_2(t), \quad t > -\infty, \quad (3) $$ $$ \frac{\partial u_1}{\partial x}\bigg|_{x=h} = \gamma_1(t), \quad \frac{\partial u_2}{\partial x}\bigg|_{x=h} = \gamma_2(t), \quad t > -\infty. $$ The constant coefficients $a_i^2$, $b_i^2$, and also the functions $\beta_1(t)$, $\alpha_2(t)$, $\nu_i(t)$, $\mu_i(t)$ ($i=1,2$) are given. First, we find the solution to the Cauchy problem for the system of equations with the specified initial conditions using the symbolic method [1] for this purpose.