Using generating functions to convert an implicit \((3,3)\) finite difference method to an explicit form on diffusion equation with different boundary conditions (Q281472)

The authors solve the time-dependent diffusion equation \(u_t=\alpha u_{xx}\) using the well-known finite difference approximations. They propose to convert the well-known \(\theta\) scheme into a new function involving infinite summation and use this new function to obtain the numerical solution. The most direct method has been known to be the solution of a tridiagonal linear system for the case when \(\theta\) is nonzero. Stability of the \(\theta\) scheme is well known as well as the fact that \(\theta = 0.5\) provides the Crank-Nicolson method with truncation errors of second order. Stability of this method is well known for \(\theta \geq 0.5\) and the authors re-derive many of the results that have been available in the literature for more than 5 decades. The authors state nothing about the implementation of their method. Converting a standard problem into an infinite series and extracting solutions from that series is not a novelty.