Differential Equations Lecture Work Solutions 49

Differential Equations Lecture Work Solutions 49 - ∂u...

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subject to ∂u ∂x (0 ,t )= ∂u ∂x ( L, t )=0 , u ( x, 0) = f ( x ) . Hint : Look for a solution as a Fourier cosine series. Assume k 6 = 2 L 2 9 π 2 . 12. Solve the wave equation by the method of separation of variables u tt c 2 u xx =0 , 0 <x<L , u (0 ,t )=0 , u ( L, t )=0 , u ( x, 0) = f ( x ) , u t ( x, 0) = g ( x ) . 13. Solve the heat equation u t =2 u xx , 0 <x<L , subject to the boundary conditions u (0 ,t )= u x ( L, t )=0 , and the initial condition u ( x, 0) = sin 3 2 π L
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Unformatted text preview: ∂u ∂t = k 1 r ∂ ∂r r ∂u ∂r ! + 1 r 2 ∂ 2 u ∂θ 2 ! inside a disk of radius a subject to the boundary condition ∂u ∂r ( a, θ, t ) = 0 , and the initial condition u ( r, θ, 0) = f ( r, θ ) where f ( r, θ ) is a given function. 15. Determine which of the following equations are separable: 49...
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