Solving the Neumann boundary value problem

The paper is devoted to solving the Neumann boundary value problem for the Poisson equation in an elliptic body. In this case, heat transfer occurs under boundary conditions of the second kind. Based on the methods of differentiation and integration, a solution was obtained to the problem of the distribution of the temperature field of the body under study. The resulting solution has an analytical form containing hypergeometric and hyperbolic functions. The reliability of the obtained result is confirmed by the fact that the general solution of the problem coincides with the solution obtained in one of the author’s works for the case of a temperature field distribution in a body with an elliptical cross section of infinite length under boundary conditions of the third kind.