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The inviscid Burgers' equation:
is a classical example of nonlinear hyperbolic equations.
This equation has unique solution for increasing initial data.
If not, solution becomes discontinuous and even uniqueness is not assured.
The physically correct solution can be obtained by solving
when  tends to zero.
The following example shows that when initial data is decreasing solution becomes discontinuous.
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Initial condition |
Solution at time t = 4.5
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