On the exact analytical solution of some families of equilibrium critical thickness transcendental equations

The problem of finding an exact analytical closed-form solution of some families of transcendental equations, which describe the equilibrium critical thickness of misfit dislocation generation in epitaxial thin films, is studied in some detail by the Special Trans Functions Theory (STFT). A novel STFT mathematical approach with an analytical closed-form solution is presented. Structure of the STFT exact solutions, numerical results and graphical simulations confirm the validity of the basic principle of the STFT. The proposed STFT analytical approach shows qualitative improvement in theoretical sense (a novel gradient coefficient genesis), and, in accuracy when compared to the conventional analytical and numerical methods.