Introduction to Partial Differential Equations
Exercise Sheet 12
Ross Ogilvie 18th November,
2024
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Find the solution of the initial and boundary value problem: (6 points)
Find the solution of the initial and boundary value problem:
Further, show that your solution obeys (7 points)
and therefore that
as claimed in the script. (2 points)
Show that the functions in vanish at and that contains all continuous odd and periodic functions with period . (1 point)
and part (a), show the relation
where the left hand side the heat kernel in terms of theta functions as given in the lecture script. Thus the method of images gives the same heat kernel as the Fourier series method (of course, the heat kernel is unique). (2 bonus points)