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Differential Equations Lecture Work Solutions 75

# Differential Equations Lecture Work Solutions 75 - t 4...

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10. u t = u xx + q ( x, t ) 0 < x < L subject to BC u (0 , t ) = u ( L, t ) = 0 Assume: q ( x, t ) piecewise smooth for each positive t. u and u x continuous u xx and u t piecewise smooth. Thus, u ( x, t ) = n =1 b n ( t ) sin n π L x (a). Write the ODE satisfied by b n ( t ), and (b). Solve this heat equation. STEPS: 1. Compute q n ( t ), the known heat source coeﬃcient 2. Plug u and q series expansions into PDE. 3. Solve for b n ( t ) - the homogeneous and particular solutions, b H n
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Unformatted text preview: t ) 4. Apply initial condition, b n (0), to Fnd coeﬃcient A n in the b n ( t ) solution. Assume u ( x, 0) = f ( x ) 1. q ( x, t ) = ∞ X n =1 q n ( t ) sin n π L x q n ( t ) = 2 L Z L q ( x, t ) sin n π L x dx 2. u t = ∞ X n =1 b n ( t ) sin n π L x u xx = ∞ X n =1 b n ( t ) " − n π L ± 2 # sin n π L x 75...
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