**Unformatted text preview: **( y, t ) solves the 2D heat equation w t = k ( w xx + w yy ) 5. Show explicitly that each of the following is a solution to the 1D heat equation: (a) the heat kernel S ( x, t ) = 1 √ 4 πkt exp(-x 2 / 4 kt ) (ignore the singularity at t = 0) (b) the separation of variables solution u n ( x, t ) = sin( nx ) exp(-n 2 kt ) Note that t appears in two diﬀerent places in the two exponentials! 6. Strauss 3.1 #1 -heat equation on the half line 7. Strauss 3.2 #5 -wave equation on the half line...

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