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Use characteristics to find the strong solution u(x, t) to the inviscid Burgers equation Ut + UU x = 0 (4) = on the half-line x ≥ 0 with initial data u(x, 0) x². (This involves solving a quadratic equation, you need to pick the root that is consistent with the initial conditions.) Make a sketch that illustrates some of the characteristics in the xt- plane.

Use characteristics to find the strong solution u(x, t) to the inviscid Burgers equation Ut + UU x = 0 (4) = on the half-line x ≥ 0 with initial data u(x, 0) x². (This involves solving a quadratic equation, you need to pick the root that is consistent with the initial conditions.) Make a sketch that illustrates some of the characteristics in the xt- plane.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. Use characteristics to find the strong solution u(x, t) to the inviscid Burgers equation = 0 ut + uux - on the half-line x ≥ 0 with initial data u(x, 0) x². (This involves solving a quadratic equation, you need to pick the root that is consistent with the initial conditions.) Make a sketch that illustrates some of the characteristics in the xt- plane. = For the general initial-value problem u(x, 0) = A(x) use the strong solution to derive an expression for u, in terms of A'(x – ut) (use the chain rule diligently). Use this to show that shock formation is inevitable if the initial slope u(x, 0) < 0 at any x.
Transcribed Image Text:2. Use characteristics to find the strong solution u(x, t) to the inviscid Burgers equation = 0 ut + uux - on the half-line x ≥ 0 with initial data u(x, 0) x². (This involves solving a quadratic equation, you need to pick the root that is consistent with the initial conditions.) Make a sketch that illustrates some of the characteristics in the xt- plane. = For the general initial-value problem u(x, 0) = A(x) use the strong solution to derive an expression for u, in terms of A'(x – ut) (use the chain rule diligently). Use this to show that shock formation is inevitable if the initial slope u(x, 0) < 0 at any x.
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