##
**A numerical experiment on a second-order partial differential equation of mixed type.**
*(English)*
Zbl 0836.35102

In an earlier article in this journal one of us examined a problem of linear, electromagnetic wave propagation in a plasma based on a particular plasma model [H. Weitzner, Commun. Pure Appl. Math. 38, 919-932 (1985; Zbl 0608.76115)] and commented on the mathematically rather nonstandard system of linear partial differential equations for the problem. Since then our attempts to solve the system numerically have not led to entirely satisfactory results. In this note we report on numerical experiments on a much simpler model partial differential equation which can be studied in much greater detail and which clarifies the numerical and mathematical difficulties inherent in the original problem. The model equation is
\[
\bigl[ (x - y^2 + i \varepsilon) u_x \bigr]_x = \pm u_{yy}, \tag{1}
\]
and we are interested in the properties of the solution as \(\varepsilon\), which is positive, tends to zero. In (1), \(u(x,y)\) is complex, so that (1) represents two real, coupled second-order equations, which constitute an elliptic system, for which the Dirichlet problem is plausibly well-posed. Further, we consider the equation
\[
(x - y^2) u_{xx} + u_{yy} + {1 \over 2} u_x = 0
\]
and we show that in an appropriately shaped domain the Dirichlet problem is ill-posed.

### MSC:

35M10 | PDEs of mixed type |

65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

65N99 | Numerical methods for partial differential equations, boundary value problems |

76X05 | Ionized gas flow in electromagnetic fields; plasmic flow |

### Keywords:

electromagnetic wave propagation; numerical experiments; elliptic system; Dirichlet problem### Citations:

Zbl 0608.76115
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XMLCite

\textit{C. S. Morawetz} et al., Commun. Pure Appl. Math. 44, No. 8--9, 1091--1106 (1991; Zbl 0836.35102)

Full Text:
DOI

### References:

[1] | Weitzner, Comm. Pure Appl. Math. 38 pp 919– (1985) |

[2] | Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley & Sons, New York, 1958. · Zbl 0083.20501 |

[3] | and , Methods of Mathematical Physics, Vol. II, Interscience, New York, 1962, pp. 154–170. |

[4] | Chu, Phys. Fluids 5 pp 550– (1962) |

[5] | Grossman, Phys. Fluids 27 pp 1699– (1984) |

[6] | Piliya, Sov. Phys. JETP 33 pp 210– (1971) |

[7] | Morawetz, Rend. di Mat. 25 pp 1– (1966) |

[8] | Morawetz, Comm. Pure Appl. Math. 23 pp 587– (1970) |

[9] | Morawetz, Comm. Pure Appl. Math. 17 pp 357– (1964) |

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