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Hodograph and Partial Hodograph Methods

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Elliptic–Hyperbolic Partial Differential Equations

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

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Abstract

The geometry of the hodograph and partial hodograph transformations is reviewed. We show how the hodograph method can introduce new geometric interpretations into the study of an elliptic–hyperbolic equation. The partial hodograph transformation dates from the 1930s, but its application to elliptic–hyperbolic equations is relatively recent; our treatment follows the work of S.-X. Chen. In addition, various methods are given for constructing explicit solutions to both linear and quasilinear elliptic–hyperbolic equations. In particular, a recently introduced method is given for the direct construction of explicit solutions to a large and useful class of quasilinear elliptic–hyperbolic systems on both sides of the sonic transition. An example shows how the individual solutions on the elliptic and hyperbolic regions can be pasted together to produce a solution which crosses the sonic line with continuity.

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  1. Department of Mathematical Sciences, Yeshiva University, New York, NY, USA

    Thomas H. Otway

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Otway, T.H. (2015). Hodograph and Partial Hodograph Methods. In: Elliptic–Hyperbolic Partial Differential Equations. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-19761-6_3

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