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**Extra resources for Analytical and Numerical Aspects of Partial Differential Equations: Notes of a Lecture Series**

**Sample text**

Thus the construction produces a piecewise constant solution with the zero initial datum and the two intermediate states α, β . 6. 14). , 0 < α < β , then the analogous construction yields a non-trivial generalized solution with the initial datum u0 (x) ≡ α. The above construction breaks down in the case where such non-aligned points on the graph of f = f (u) cannot be found. , f (u) = au + b, a, b ∈ R. In the latter case, our quasilinear problem is in fact linear: ut + aux = 0, u|t=0 = u0 (x).

19′ ) means that the chord Ch with the endpoints (u− , f (u−)), (u+, f (u+ )) has a smaller slope (the slope is measured as the inclination of the chord with respect to the positive direction of the u-axis) than the slope of the segment joining the point (u− , f (u− )) with the point (u, f (u)), where u runs over the interval (u− , u+ )). Consequently, the point (u, f (u)) and thus the whole graph of f = f (u) on the interval (u− , u+ ) lies above the chord Ch. 20′ ) signifies that the graph of f = f (u) for u ∈ (u+ , u− ) is situated below the chord Ch.

Then u(t, x(t)) ≡ v (t, x(t)). 10) with respect to t, we obtain dx dx = vt (t, x(t)) + vx (t, x(t)) · dt dt Here and in the sequel, ux , vx , ut , vt denote the corresponding limits of the derivatives as the point (t, x) tends to the weak discontinuity curve Γ. 2), we have ut (t, x(t)) + ux (t, x(t)) · dx dx − f ′ (u(t, x(t)))ux = vx (t, x(t)) · − f ′ (v (t, x(t)))vx. 10), we obtain ux (t, x(t)) · ux (t, x(t)) − vx (t, x(t)) dx − f ′ (u(t, x(t)) = 0. 9) follows. 26 Gregory A. Chechkin and Andrey Yu.