*2.1. Continuous Symmetry Groups*

Equation (2) is a linear and homogenous partial differential equation with one dependent variable *w* and two independent variables *x* and *t*. The point transformation of Equation (2) defines the local diffeomorphism

$$
\Gamma \colon (\mathfrak{x}, \mathfrak{t}, w) \mapsto (\hat{\mathfrak{x}}(\mathfrak{x}, \mathfrak{t}, w), \hat{\mathfrak{f}}(\mathfrak{x}, \mathfrak{t}, w), \hat{w}(\mathfrak{x}, \mathfrak{t}, w)) \dots
$$

This transformation maps any surface *w*= *f*(*x*, *t*) to the following surface

$$
\hat{\mathfrak{x}} = \hat{\mathfrak{x}}(\mathbf{x}, t, f(\mathbf{x}, t)),
$$

$$
\hat{\mathfrak{f}} = \hat{\mathfrak{f}}(\mathbf{x}, t, f(\mathbf{x}, t)),
$$

$$
\hat{\mathfrak{w}} = \hat{\mathfrak{w}}(\mathbf{x}, t, f(\mathbf{x}, t)).
$$

The heat Equation (2) has infinite dimensional Lie algebra. The infinitesimal generators and the continuous symmetry groups of the Equation (2) [21] are presented in Table 1.

**Table 1.** Continuous symmetry transformations of (2).


*Symmetry* **2020**, *12*, 359

Due to the arbitrariness of the function appearing in the last generator of Table 1, Lie algebra of Equation (2) is infinite dimensional. Each of the groups given in Table 1 has the property of mapping solutions of heat Equation (2) to the other solutions. For example, consider the explicit form of the projection

$$g(\mathbf{x}, t, w) \mapsto \left(\frac{\mathbf{x}}{1 - 4\epsilon t}, \frac{t}{1 - 4\epsilon t}, w\sqrt{1 - 4\epsilon t} \exp(\frac{-\epsilon \mathbf{x}^2}{1 - 4\epsilon t})\right),$$

where is the group parameter. Computing the induced action on graphs of functions, we conclude that if *w* = *f*(*x*, *t*) is any solution to heat Equation (2), so is

$$w = \frac{1}{\sqrt{1 - 4\epsilon t}} \exp\left(\frac{\epsilon \chi^2}{1 - 4\epsilon t}\right) f\left(\frac{\chi}{1 - 4\epsilon t}, \frac{t}{1 - 4\epsilon t}\right).$$
