The level of water in a levee between two ponds can with som

The level of water in a levee between two ponds can (with some approximation) be modelled by the equation ht = hxx + R, where h(x, t) is the level of water and R is a constant rate of “recharge” (e.g. water coming in through the top as rainfall). Suppose the water level in the levee is originally h(x, 0) = 1 and the levels at the two ends are h(0, t) = 1 and h(1, t) = 0.5

a) A steady-state solution is one that is no longer changing with time. By leaving out the u/t term, you can obtain an ordinary differential equation for u_s(x), the steady state solution as a function of x only. Letting t    in the boundary conditions, solve for the steady-state (long term) solution.

Please show all workings. Thanks.

Solution

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