Exact analytical solution for the stationary two-dimensional heat conduction problem with a heat source

The exact analytic solution for the stationary two-dimensional heat conduction problem with a heat source for an infinite square bar was obtained. It was based on the Bubnov–Galyorkin orthogonal method using trigonometric systems of coordinate functions. The infinite system of ordinary differential equations obtained by the Bubnov–Galyorkin method is divided and reduced by the orthogonality property of trigonometric coordinate functions to the solution of a generalized equation which provides the exact analytical solution in a simple form, i.e. in the form of an infinite series. In view of the symmetry of the problem, only a quarter of the cross-section of the bar is considered for the boundary conditions of the adiabatic wall (the absence of heat transfer) along the cut lines, which allows (in contrast to the well-known classical exact analytical solution) to significantly simplify the process of the solution and the final equation.