Differential Equations Lecture Work Solutions 230

Differential Equations Lecture Work Solutions 230 - 7...

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Unformatted text preview: 7 Method of Characteristics 7.1 Advection Equation (first order wave equation) Problems 1. Solve ∂w ∂w −3 =0 ∂t ∂x subject to w (x, 0) = sin x 2. Solve using the method of characteristics ∂u ∂u +c = e2x ∂t ∂x ∂u ∂u b. +x =1 ∂t ∂x ∂u ∂u + 3t =u c. ∂t ∂x ∂u ∂u d. −2 = e2x ∂t ∂x ∂u ∂u e. − t2 = −u ∂t ∂x a. subject to u(x, 0) = f (x) subject to u(x, 0) = f (x) subject to u(x, 0) = f (x) subject to u(x, 0) = cos x subject to u(x, 0) = 3ex 3. Show that the characteristics of ∂u ∂u + 2u =0 ∂t ∂x u(x, 0) = f (x) are straight lines. 4. Consider the problem ∂u ∂u + 2u =0 ∂t ∂x 1 x<0 x u(x, 0) = f (x) = 1 + L 0 < x < L 2 L<x a. Determine equations for the characteristics b. Determine the solution u(x, t) c. Sketch the characteristic curves. 230 ...
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