I am working on this question for my Partial Differential Equation seminar and would like some help in understanding how to create boundary conditions.

An endothermic chemical reaction takes place within the region x [0,L], consuming

heat energy at a rate R = x(L − x). The ends of the region are perfectly insulated

and the region is initially heated to A degrees.

(a) Write down a partial differential equation, together with boundary and initial

conditions governing the temperature u(x, t) in the region.

[Hints: The rate of heat leaving the region at x = 0,L is assumed to be propor-

tional to du/dx (partial, curly d) . Your equation should contain an inhomogeneity corresponding to the heat sink, R, which is independent of u.]

So I worked out that the initial condition will be u(x,0)=A and from there on I am a little stuck. I believe that because of the hint that the boundary conditions will be Neumann conditions (ie u'(0,t)=? and u'(L,t)=? ) but I have looked at all kinds of examples and cannot work out how to find what they are equal to. I think either they would equal 0 but my instinct tells me they should maybe be a function of t since surely the temperature would depend on what time it is?

For the equation i have u_t = Du_xx -x(L-x) but i am not sure whether i need the D at the front or not.

Any help would be greatly appreciated, i would really like to understand what I am doing

Thanks