A Finite Difference Method for Any Partial Differential Equation,

The most general heat diffusion equation possesses not only first- and second-order derivatives in space but also first-and second-order derivatives in time. Therefore, the governing equation can be parabolic, elliptic or even hyperbolic depending upon the parameters chosen. The model includes various physical problems, such as, steady and unsteady classical heat conduction usually known as classical Fourier model, heat pulse Non-Fourier model, abrasive cut-off and surface grinding operations in machining of metal components. An explicit and unconditionally stable finite difference scheme is developed for the general purpose governing equation. The heat transfer example is included to discuss the accuracy and stability of this numerical scheme. Author