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Question 3. (25 marks) Bjork likes baths, but she doesn’t like them to get cold. She has therefore constructed an ingenious mechanism
whereby hot lava from deepest darkest Iceland drips periodically into one end of her bath. Unfortunately, whilst
the end with the lava drip is well-insulated, probably because of all the congealed lava, demons have been eating
the insulation at the other end and heat keeps leaking out at a constant rate. We can model this bathtime adventure using a one-dimensional inhomogeneous heat equation in which the source
term represents the periodic dripping of lava (once per time unit) at one end (x = 1): 2 DO
a&quot; — 3—“ + h61'(x) 2 51a).
j=1 5—613 Here, it is a constant representing how much heat each drip imparts to the bath and 61‘ is a one-sided Dirac delta function (so [01 (SI—(x) dx = 1). Assuming that the temperature distribution of Bjork’s bath is initially uniform at 1
unit of temperature, we impose the following initial-boundary conditions: u(x,0) = I, draw, it) = 2, 61140, t) = 0. (a) Try to cope with the inhomogeneities by writing a = um, + acres, and forcing ulin(x, t) = A(t) + B(t)x to satisfy
the boundary conditions. Show that this fails miserably. (2 marks) (b) Recover spectacularly by replacing um, by upoly (x, t) = A(t)x + B(t)x2, determining A(t) and B(t) explicitly, and
showing that this leads to homogeneous boundary conditions for urest. (3 marks) (0) Determine an inhomogeneous PDE satisﬁed by a:rest and explain carefully why the Ansatz
l 00
umoc, z) = 5%“) + ; an(t) cos(mrx). is justifed. What is the initial condition satisﬁed by uresl(x, t)? (4 marks) (cl) Substitute this Ansatz into the IBVP to obtain an IVP for each an. Solve them and thereby write down the formal
solutions for tires, that satisfy the homogenous boundary conditions and the inhomogeneous PDE. [You might like to recall that an antiderivative of 51.0) is H j(t). Please also feel free to conveniently forget that
our pointwise convergence theorem doesn’t technically apply to delta functions!] (9 marks) (e) Find a formal solution that also satisﬁes the initial condition. Use this to determine a formal solution for u(x, t).
(4 marks) (f) B jork prefers to bathe in the middle (x = é). Get a computer to draw graphs of your solution, for 0 a I Q 5
and h = 1.5, 1.8, 2, 2.2 and 2.5. Based on these graphs, do you think she will be happy to ignore the demons’
vandalism or will she be incandescent with rage? (3 marks)

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