**2. Heat Equation**

The homogeneous heat (diffusion) [20] equation

$$\frac{\partial w}{\partial t} - D \frac{\partial^2 w}{\partial x^2} = 0,\tag{1}$$

plays a vital role in the study of heat conduction and other diffusion processes, in which a thin metal rod of length *L*, whose sides are insulated, is considered. The temperature of the bar at the point *x* and at the time *t* is represented by *w*(*x*, *t*). The parameter *D* is called the thermal conductivity and it depends only upon the material from which the rod is made. For simplicity, we take the parameter *D* = 1 then the Equation (1) becomes

$$
\frac{\partial w}{\partial t} - \frac{\partial^2 w}{\partial x^2} = 0.\tag{2}
$$
