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Part CCSIR NET June 2023transport-equation-data-on-a-line

Transport equation data on a line

Let solve on with u(x, y) = sin x on the line y = 3x + 1, and let solve with v(x, 0) = sin x. Let S = [0,1] × [0,1]. Which of the following statements are true?

  1. A.u changes sign in the interior of S.
  2. B.u(x, y) = v(x, y) along a line in S.✓
  3. C.v changes sign in the interior of S.✓
  4. D.v vanishes along a line in S.✓

Solution

Both are constant on lines y − 2x = c. v = sin(x − y/2), which vanishes on x = y/2 and changes sign across it inside S. For u, the data line y = 3x + 1 meets the characteristic through (x, y) at , so u = sin(y − 2x − 1); on S the argument lies in , so u ≤ 0 and does not change sign. u = v where y − 2x − 1 = x − y/2, i.e. on the line y = 2x + 2/3, which crosses S.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: A first-order quasilinear Cauchy problem with smooth data has a global smooth solution

More on this topic

From Partial Differential Equations › First-order PDE: Lagrange, Charpit, characteristics

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