Analytic Modelling of 2-D Transient Heat Conduction with Heat Source Under Mixed Boundary Constraints by Symplectic Superposition

Abstract

Many studies have been conducted on 2-D transient heat conduction, but analytic modelling is still uncommon for the cases with complex boundary constraints due to the mathematical challenge. With an unusual symplectic superposition method, this paper reports new analytic solutions to 2-D isotropic transient heat conduction problems with heat source over a rectangular region under mixed boundary constraints at an edge. With the Laplace transform, the Hamiltonian governing equation is derived. The applicable mathematical treatments, e.g., the variable separation and the symplectic eigenvector expansion in the symplectic space, are implemented for the fundamental solutions whose superposition yields the ultimate solutions. Benchmark results obtained by the present method are tabulated, with verification by the finite element solutions. Instead of the conventional Euclidean space, the present symplectic-space solution framework has the superiority on rigorous derivations without pre-determining solution forms, which may be extended to more issues with the complexity caused by mixed boundary constraints.