According to of Section 9.7, the temperature u.r; t / in a
Chapter , Problem 16(choose chapter or problem)
According to of Section 9.7, the temperature u.r; t / in a uniform solid spherical ball of radius a satisfies the partial differential equation .ru/t D k.ru/rr . Suppose that the ball has initial temperature u.r; 0/ D f .r/ and that its surface r D a is insulated, so that ur .a; t / D 0. Substitute v.r; t / D ru.r; t / to derive the solution u.r; t / D c0 C X1 nD1 cn r exp 2 nkt a2 ! sin nr a ; where fng 1 1 are the positive roots of the equation tan x D x and c0 D 3 a3 Z a 0 r2f .r/ dr; cn D 2 a sin2 n Z a 0 rf .r/sin nr a
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