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0 y u 1 y 0 4 solve using separation of variables u

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(0 , y ) = u (1 , y ) = 0 . 4. Solve using separation of variables u tt ( r, θ, t ) = 16∆ u ( r, θ, t ) , 0 r 1 , t 0 , all θ, u (1 , θ, t ) = 0 , t 0 , all θ, u ( r, θ, 0) = 1 - r, u t ( r, θ, 0) = 0 . Use the fact that if the initial conditions are radial (do not depend on θ ), then the solution will also be radial.Write out the solution as explicitly as possible as it can be done without the aid of a calculator or computer; explain what all the symbols are. 5. Solve using separation of variables u tt ( r, θ, t ) = 16∆ u ( r, θ, t ) , 0 r 1 , t 0 , all θ, u (1 , θ, t ) = 0 , t 0 , all θ, u ( r, θ, 0) = J 0 ( ξ 01 r ) + 3 J 0 ( ξ 02 r ) , u t ( r, θ, 0) = 0 where ξ 0 m denotes the m -th positive zero of J 0 . Use the fact that if the initial conditions are radial (do not depend on θ ), then the solution will also be radial.Write out the solution as explicitly as possible as it can be done without the aid of a calculator or computer; explain what all the symbols are. 6. Consider the ICBC problem u t ( x, t ) = ku xx ( x, t ) + q ( x ) , 0 < x < L, t > 0 , u (0 , t ) = α, u ( L, t ) = β, t > 0 , u ( x, 0) = f ( x ) , 0 < x < L, where α, β are constants. I will call this Problem A.

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2 (a) Let u ( x ) satisfy u ′′ ( x ) = - 1 k k ( x ) , 0 x L, u (0) = α, u ( L ) = β. Assuming u solves the first ICBC problem, show that the function v defined by v ( x, t ) = u ( x, t ) - u ( x ) solves the following Problem B: v t ( x, t ) = kv xx ( x, t ) , 0 < x < L, t > 0 , v (0 , t ) = 0 , v ( L, t ) = 0 , t > 0 , v ( x, 0) = f ( x ) - u ( x ) , 0 < x < L,
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