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The Qualitative Analysis and the Critical Hypersurfaces of Elliptic and Hyperbolic PDEs

Many boundary value problems of PDEs of the applied mathematics lead to the solving of equivalent elliptic and hyperbolic quadratic algebraic equations (QAEs) with variable coefficients. The qualitative analysis of elliptic and hyperbolic QAEs is started here by the determination of their behaviors by systematical variation of their free and linear terms, from −∞ to +∞ and by their visualization. It comes out that, for these variations of their coefficients, the elliptic and hyperbolic QAEs have critical hypersurfaces, which are obtained by cancellation of their great determinant as in [1], [2]. The critical hypersurface can be considered as a limit of existence of real solutions of an elliptic QAE. The hyperbolic QAE degenerates jumps and breaks along its critical hypersurface, which is also its asymptote. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)