Variational Approach to Two-Dimensional and Three-Dimentional Heat Conduction Modeling

Abstract

The paper proposes an approximate solution to the classical (parabolic) multidimensional 2D and 3D heat conduction equation for a 5 X 5 cm aluminium plate and a 5 X 5 X 5 cm aluminum cube. One-dimensional heat conduction problems with Dirichlet boundary conditions can be directly solved by the Fourier method of separation of variables. The application of this method in engineering practice to solving 2D and 3D heat conduction problems is quite complex. An approximate solution of the generalized (hyperbolic) 2D and 3D equation for the considered plate and cube is also proposed. By using the Euler-Lagrange equations and calculus of variations, approximate solutions were found. The proposed approximate solutions were compared with the exact solutions of the parabolic and hyperbolic equations to confirm their correctness. The paper also presents the research on the influence of time parameters \(\tau\) as well as the relaxation times \(\tau^*\) to the variation of the profile of the temperature field for the considered aluminum plate and cube. Further considerations should be devoted to the study of heat conduction with variable thermophysical coefficients because they depend on changes in their temperature fields, which is required and dictated by engineering practice.