The Fourier-based dimensional reduction method for solving a nonlinear inverse heat conduction problem with limited boundary data

The objective of this article is to introduce a novel technique for computing numerical solutions to the nonlinear inverse heat conduction problem. This involves solving nonlinear parabolic equations with Cauchy data provided on one side Gamma of the boundary of the computational domain Omega. The key step of our proposed method is the truncation of the Fourier series of the solution to the governing equation. The truncation technique enables us to derive a system of 1D ordinary differential equations. Then, we employ the well-known Runge-Kutta method to solve this system, which aids in addressing the nonlinearity and the lack of data on partial derivative Omega\Gamma. This new approach is called the Fourier-based dimensional reduction method. By converting the high-dimensional problem into a 1D problem, we achieve exceptional computational speed. Numerical results are provided to support the effectiveness of our approach.