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Differential Equations Lecture Work Solutions 47

# Differential Equations Lecture Work Solutions 47 - φ = −...

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3.7 Full solution of Several Problems Problems 1. Solve the heat equation u t = ku xx , 0 < x < L, t > 0 , subject to the boundary conditions u (0 , t ) = u ( L, t ) = 0 . Solve the problem subject to the initial value: a. u ( x, 0) = 6 sin 9 π L x. b. u ( x, 0) = 2 cos 3 π L x. 2. Solve the heat equation u t = ku xx , 0 < x < L, t > 0 , subject to u x (0 , t ) = 0 , t > 0 u x ( L, t ) = 0 , t > 0 a. u ( x, 0) = 0 x < L 2 1 x > L 2 b. u ( x, 0) = 6 + 4 cos 3 π L x. 3. Solve the eigenvalue problem
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Unformatted text preview: φ = − λφ subject to φ (0) = φ (2 π ) φ (0) = φ (2 π ) 4. Solve Laplace’s equation inside a wedge of radius a and angle α , ∇ 2 u = 1 r ∂ ∂r r ∂u ∂r ! + 1 r 2 ∂ 2 u ∂θ 2 = 0 subject to u ( a, θ ) = f ( θ ) , u ( r, 0) = u θ ( r, α ) = 0 . 47...
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