A Ritz-like approximation method for solving two-point boundary value problems with fractal derivatives

Abstract

Two-point boundary value problems are prevalent in thermal engineering, yet conventional integer-order derivative models are inadequate in accurately characterizing the non-linear, non-local thermal behaviors of discontinuous media (e.g., fractal porous materials). Existing analytical methods (variational iteration, homotopy perturbation) have limitations in solving fractal-derivative models. The present study proposes a Ritz-like approximation method. The method under discussion retains the traditional Ritz method’s advantage of determining unknown constants via approximate solutions but eliminates the need for problem-specific variational formulations. The study elucidates the fractal derivative’s definition, introduces a two-scale transform to simplify equations and boundary conditions, and develops the method systematically. The efficacy of the method was validated using a benchmark problem, and the results demonstrated a high degree of accuracy (relative error as low as 3.12%) and strong adaptability to fractal thermal systems. This study serves as a practical tool for solving problems in thermal engineering that are relevant to industry.