*2.2. Discrete Symmetry Group*

Hydon introduced a technique [7] by which all discrete point symmetries of the partial differential equations can be found on basis of the results [6] that each continuous symmetry generator of a Lie algebra of a differential equation brings an automorphism that preserve the following commutator relation

$$[X\_l, X\_k] = c\_{lk}^m X\_m \dots$$

Hydon's method categorizes and factor out all those automorphisms of a Lie algebra that are equivalent under the action of a Lie symmetry in a Lie group that is generated by the Lie algebra and provide the most general realization of these automorphisms as point transformations. Finally, by using these point transformations, an entire list of discrete point symmetries of a partial differential equation, is obtained.

The discrete symmetry group of Equation (2) has already been obtained in [8], which is isomorphic to *Z*<sup>4</sup> = {group of residues modulo 4} and is generated by

$$(\mathbf{x}, t, w) \to \left(\frac{\mathbf{x}}{2t}, \frac{-1}{4t}, \sqrt{2\mu} \exp\left(\frac{\mathbf{x}^2}{4t}\right) w\right),\tag{3}$$

where *<sup>ι</sup>* <sup>=</sup> √−1.

### **3. Finite Difference Schemes for the Heat Equation**

Some finite difference schemes are available in the literature, which help us to find the numerical solutions of the partial differential equations. In this section, the backward difference scheme, forward difference scheme, and the Crank Nicolson method for the heat equation along their stability conditions, are given [22].
