Spectral Numerical Simulation and Generalized Solution of the Regularized n -Dimensional Nonlinear Heat Conduction Equation

This paper investigates a regularized nonlinear heat conduction equation with Robin boundary conditions in a bounded n-dimensional domain, modeling thermal transport with interface resistance. We theoretically formulate the concept of a weak solution, deriving energy inequalities and a priori estimates. Using Galerkin approximations and the Aubin-Lions-Simon compactness theorem, we prove the global existence and uniqueness of the solution. Furthermore, we develop a highly accurate numerical scheme based on the Fibonacci collocation method with equidistant Riemann points for 1D, 2D, and 3D domains. The method achieves exponential spectral convergence, confirming the computational tractability of the regularized model.