Advanced Mathematical Models & Applications

In the paper a mixed problem with time derivative in the boundary conditions for the second order inhomogeneous parabolic equation with complex coefficients is considered. It is impossible to apply the Fourier decomposition method to find a solution to problems of this type. Despite the fact that the boundary conditions of the considered problem seem to be local, in fact they do not contain the property of locality, since the boundary conditions involve a derivative with respect to time. The spectral problem in the complex plane corresponding to this problem is formulated and studied. Then the solution of the corresponding Cauchy problem is constructed. The solution of the mixed problem is found in the form of an infinite sequence of deductions using the residue method proposed by M.L. Rasulov.