Alternate directions implicit scheme for a non-linear heat equation

This third Note is dedicated to the discussion of application of the 3D Douglas – Rachford ADI scheme to the solution of a non-linear heat equation. We will discuss the Newton – Raphson method and the “method of frozen coefficients”.

Before starting discussion of how to approach solution of a non-linear equation, let us define what will be called a non-linear heat equation.
Below, we will consider a heat equation with temperature dependent thermal conductivity and heat capacity. Thus, the coefficients of the heat equation appear to be temperature dependent:

, (1)
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