Functional analytic methods have been applied to the analysis of the extended Graetz problem with prescribed wall flux and viscous dissipation between parallel plates. First, the non‐self‐adjoint elliptic energy equation is decomposed into a set of first order partial differential equations to obtain a self‐adjoint formulism. Next, the induced eigenvalue problems are solved by applying an approximation method in a Hilbert space, and an algebraic characteristic equation is obtained. In addition, the expansion coefficients of the solutions on upstream and downstream regions can be explicitly obtained and unnecessary to match at the entrance.