Modelling the heating of a lake. When the sun starts to shine on to a lake, the lake starts to warm up. The warming sunlight penetrates the lake, warming it from within. The temperature of the lake at depth z metres below the surface is governed by

T /t = ²T/ z²+ 20e^-z

T(0, t) = 15 and T(4, t) = 15

T(z, 0) = 15

where T(z, t) is the temperature of the lake at depth z (with z [0, 4]) at time t seconds.

(a) State what the boundary and initial conditions mean physically and sketch the solution domain with these conditions included.

(b) Write a numerical scheme to solve this system for t [0, 5] and implement it on a computer (Excel is good for this). Use z = 0.5 and choose t to ensure a stable scheme.

(c) From your numerical results, plot (on the same set of axes) temperature profiles (as a function of z) for t = 0, 1.0, 2.0, 3.0, 4.0 and 5.0 (so seven profiles in total).