The function \(c(x, \tau)\) satisfies the following problem in the region \(00 \end{gathered}$$ Show that there is a similarity solution with \(c(x, \tau)=\tau c^{*}(x / \sqrt{\tau})\), \(x_{f}(\tau)=\xi_{0} \sqrt{\tau}\), where \(\xi_{0}\) satisfies the transcendental equation $$ \frac{1}{2} \xi_{0} e^{\xi_{0}^{2} / 4} \int_{0}^{\xi_{0}} e^{-s^{2} / 4} d s=1 $$

Boundary conditions are constraints necessary for solving PDEs, providing information about the solution at the borders of the domain of interest. In this problem, the conditions are:

• \(c(x_{f}(\tau), \tau)=0\): the function value is zero at \(x_{f}(\tau)\), the moving boundary.
• \(\frac{\partial c}{\partial x}(x_{f}(\tau), \tau) = 0\): the gradient of the function with respect to space is zero at \(x_{f}(\tau)\), indicating a maximum or minimum point.
• \(c(0, \tau)=\tau\): the function value when \(x=0\) grows linearly with time.

These conditions guide the behavior of the solution at the edges of the region of interest, leading to a well-defined problem.

Transcendental equations involve transcendental functions, such as exponential, logarithmic, trigonometric, or integrals, which make them more complex than polynomial equations. The given transcendental equation in the exercise is:
\[\begin{equation}\frac{1}{2} \xi_{0} e^{\xi_{0}^{2} / 4} \int_{0}^{\xi_{0}} e^{-s^{2} / 4} ds=1\end{equation}\]This equation determines the value of \(\xi_{0}\), a parameter that scales the boundary condition in relation to time for the similarity solution. It is often solved using numerical methods due to its complexity.

Similarity transformation is a method used to reduce PDEs to ordinary differential equations (ODEs), which are easier to solve. By introducing a similarity variable, such as \(\xi = \frac{x}{\sqrt{\tau}}\) in this problem, the PDE can be transformed into an ODE with fewer independent variables. It's a powerful technique because it exploits the inherent symmetry of the problem, leveraging scale invariance to simplify the system under study.
The similarity solution assumes a form that remains invariant under scaling of time and space, essentially 'collapsing' the multi-dimensional problem into a simpler, one-dimensional one. Identifying the correct form of similarity transformation, as was done with \(c(x, \tau) = \tau c^{*}(x / \sqrt{\tau})\) and \(x_{f}(\tau) = \xi_{0} \sqrt{\tau}\) in the exercise, is pivotal in effectively solving many PDEs.