A general boundary value problem for heat and mass transfer equations with high order normal derivatives in boundary conditions

A boundary value problem for the equations of heat and mass transfer in a half-space is considered, which contains in the boundary conditions derivatives of a higher order than the order of the derivative of a system of equations. The solution of the boundary problem is found in the form of a double layer potential. The lemma on finding the limit of the normal derivative of the finite order of the function in the neighborhood of the hyperplane x_n = 0 is given. Using the boundary conditions, a system of integro-differential equations (SIDE) is obtained. The characteristic part of the SIDE is solved by the method of Fourier-Laplace integral transforms. The conditions for the correctness and incorrectness of the problem, expressed in terms of the given constants and boundary conditions, are found. Using the regularization method, the SIDE is reduced to a system of Volterra-Fredholm integral equations. A theorem on the solvability of the boundary value problem for a parabolic system of heat and mass transfer equations is presented.