Consider the solution of the diffusion equation by separation of variables in the linear geometry of the plasma slab indicated in Fig. 4. Show that the solutions of the equation

that satisfy the boundary condition S = 0 at x = ±L, are

and

Explain why the solution as a sine series is not a physically acceptable solution for this case. Consequently, from n(x, t) = S(x) T(t), show that the number density can be expressed as

Therefore, the decay time constant for the m^th mode is

This result shows that the higher modes decay faster than the lower ones. How are the coefficients am determined in terms of n₀(x)?

Figure 4.

Transcribed Image Text:

d² S(x) dx² +k² S(x) = 0