Abstract
2.1. The problem is analyzed in Section 2.3.3. Its solution is given by (2.34):
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Notes
- 1.
The general solution for the linear first order equation \( \dot{y}=a(t)y(t)+\beta (t) \) is
$$ y(t)={e}^{{\displaystyle \int a\; dt}}\left(C+{\displaystyle \int \beta {e}^{-{\displaystyle \int a\; dt}} dt}\right). $$ - 2.
Appendix A.
- 3.
Adiabatic extremes, considering the heat conduction in the bar.
- 4.
It can be proved that ∫ L0 cos(k n x)cos(k m x)dx = 0 if m ≠ n. Furthermore, the functions \( {\varphi}_m(x)= \cos \left({k}_mx\right)\Big/\sqrt{a_m} \) are a base of the space L 2 (0, L) of square integrable functions (Chapter 7).
- 5.
Appendx D.
- 6.
See equation (10.2).
- 7.
Appendix D.
- 8.
It is a standard substitution for Euler equations.
- 9.
The condition ψ′ > 0 ensures the local invertibility of the transformation.
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Salsa, S., Vegni, F.M.G., Zaretti, A., Zunino, P. (2013). Solutions of selected exercises. In: A Primer on PDEs. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2862-3_10
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DOI: https://doi.org/10.1007/978-88-470-2862-3_10
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