Abstract
Remarkably, the theory of linear and quasi-linear first-order PDEs can be entirely reduced to finding the integral curves of a vector field associated with the coefficients defining the PDE. This idea is the basis for a solution technique known as the method of characteristics. It can be used for both theoretical and numerical considerations. Quasi-linear equations are particularly interesting in that their solution, even when starting from perfectly smooth initial conditions, may break up. The physical meaning of this phenomenon can be often interpreted in terms of the emergence of shock waves.
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Notes
- 1.
This point is made most forcefully by Arnold in [1].
- 2.
This visualization has nothing to do with the more abstract geometric interpretation given in Box 3.1, which we will not pursue.
- 3.
- 4.
Later, however, we will allow certain types of discontinuities of the solution.
- 5.
But see Box 3.2.
- 6.
Some authors reserve the name of initial value problem for the particular case in which the data are specified on one of the coordinate axes (usually at t=0).
- 7.
The classical reference work for this kind of problem is [2]. The title is suggestive of the importance of the content.
References
Arnold VI (2004) Lectures on partial differential equations. Springer, Berlin
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Courant R, Hilbert D (1962) Methods of mathematical physics, vol II. Interscience, Wiley, New York
Duff GFD (1956) Partial differential equations. University of Toronto Press, Toronto
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John F (1982) Partial differential equations. Springer, Berlin
Sneddon IN (1957) Elements of partial differential equations. McGraw-Hill, Maidenheach Republished by Dover 2006
Zauderer E (1998) Partial differential equations of applied mathematics, 2nd edn. Interscience, Wiley, New York
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Epstein, M. (2017). The Single First-Order Quasi-linear PDE. In: Partial Differential Equations. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-55212-5_3
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DOI: https://doi.org/10.1007/978-3-319-55212-5_3
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