A Mini-course on EllipticâHyperbolic Equations - ICMS
A Mini-course on EllipticâHyperbolic Equations - ICMS
A Mini-course on EllipticâHyperbolic Equations - ICMS
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Example of the use of the hodograph method: [Magnanini &Talenti, in: Ill-posed and Inverse Problems, VSP, (2002)]( )( )|∇θ| 4 − θy2 θ xx + 2θ x θ y θ xy + |∇θ| 4 − θx2 θ yy = 0. (7)This quasilinear elliptic–hyperbolic equati<strong>on</strong> arises in the c<strong>on</strong>text ofc<strong>on</strong>structing uniform asymptotic expansi<strong>on</strong>s of the Helmholtzequati<strong>on</strong> near a smooth, c<strong>on</strong>vex caustic.We can write this equati<strong>on</strong> as a homogeneous first-order system inp (x, y) and q (x, y) , with quasilinear term depending <strong>on</strong>ly <strong>on</strong> pand q.