Application of Symmetry Analysis to a PDE Arising in the Car Windshield Design

Abstract

A new approach to parameter identification problems from the point of view of symmetry analysis theory is given. A mathematical model that arises in the design of car windshield represented by a linear second order mixed type PDE is considered. Following a particular case of the direct method (due to Clarkson and Kruskal), we introduce a method to study the group invariance between the parameter and the data. The equivalence transformations associated with this inverse problem are also found. As a consequence, the symmetry reductions relate the inverse and the direct problem and lead us to a reduced order model.

Response solutions for elliptic-hyperbolic equations with nonlinearities and periodic external forces

“…Elliptic-hyperbolic equations emerge in a diverse range of real-world physical phenomena. Examples include the dynamics of plasmas under the influence of small-amplitude electro-magnetic waves with phase velocities significantly greater than thermal velocities [26,45,47,49], the behavior of light near a caustic [43], extremal surfaces in the context of special relativity, the formation of rapids, transonic and multi-phase fluid flow [17,22,36], the dynamics of certain models for elastic structures [51], the shaping of industrial surfaces like windshields and airfoils [9], issues related to traffic flow [4], and harmonic fields in extended projective space [48]. Additionally, these equations find applications in models connected with the early Universe [30], cosmic acceleration, and the possible violation of causality within the interiors of specific compact stars.…”