Numerical Spectral Approximation and Stability Analysis of the Dirichlet Problem for the n -Dimensional Heat Equation with Taylor-Series-Type Harmonic Source

We examine an initial-boundary value problem with homogeneous Dirichlet boundary conditions to model diffusion within rectangular parallelepipeds of arbitrary dimension. While the classical Fourier method typically constructs eigenfunction sets to find analytical solutions for such setups in one, two, or three dimensions, extending these techniques to higher dimensions presents unique challenges. Here, we explore the stability and spectral convergence of solutions—expressed via Taylor series—in bounded multidimensional spaces subjected to a harmonic source. By applying a spectral collocation approach, the numerical approximations reach near machine-precision accuracy as the polynomial degree grows. Furthermore, we provide rigorous mathematical proofs, including specific lemmas and theorems, to confirm the orthogonality, completeness, and uniqueness of the analytical solution within the L 2 (Ω) framework.