Differential Equations Lecture Work Solutions 263

Differential Equations Lecture Work Solutions 263 - ρ = ρ...

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7.2.2 Graphical Solution 7.2.3 Numerical Solution 7.2.4 Fan-like Characteristics 7.2.5 Shock Waves Problems 1. Consider Burgers’ equation ∂ρ ∂t + u max " 1 2 ρ ρ max # ∂ρ ∂x = ν 2 ρ ∂x 2 Suppose that a solution exists as a density wave moving without change of shape at a velocity V , ρ ( x, t )= f ( x Vt ). a. What ordinary diFerential equation is satis±ed by f b. Show that the velocity of wave propagation, V , is the same as the shock velocity separating ρ = ρ 1 from
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Unformatted text preview: ρ = ρ 2 (occuring if ν = 0). 2. Solve ∂ρ ∂t + ρ 2 ∂ρ ∂x = 0 subject to ρ ( x, 0) = 4 x < 3 x > 3. Solve ∂u ∂t + 4 u ∂u ∂x = 0 subject to u ( x, 0) = 3 x < 1 2 x > 1 4. Solve the above equation subject to u ( x, 0) = 2 x < − 1 3 x > − 1 5. Solve the quasilinear equation ∂u ∂t + u ∂u ∂x = 0 subject to u ( x, 0) = 2 x < 2 3 x > 2 263...
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.

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