A novel Hamiltonian-based method for two-dimensional transient heat conduction in a rectangle with specific mixed boundary conditions

Abstract

A novel Hamiltonian-based method is introduced to the two-dimensional (2-D) transient heat conduction in a rectangular domain with partial temperature and partial heat flux density on one boundary. This boundary condition is very difficult to deal with in the classical Lagrangian solving system. Because of this, a total unknown vector consisting of both temperature and heat flux density is regarded as the primary unknown so that the problem is converted to the Hamiltonian form. By using the Laplace transform and method of separation of variables, the total unknown vector is solved and expressed in terms of symplectic eigensolutions in the complex frequency domain (s-domain). The undetermined coefficients of the symplectic series are obtained according to a generalized adjoint symplectic orthogonality. In this manner, analytical expressions for the rectangular domain with specific mixed boundary conditions are achieved in the s-domain. Highly accurate numerical results in the time domain (t-domain) are then obtained by using inverse Laplace transform. Numerical examples are given to demonstrate the efficiency and accuracy of the proposed method.