Contents 1. Introduction. 2. Models of PDE 's in the 18th and 19th century. 3. Methods of calculating solutions in the 19th century. 4. Developments of rigorous theories of solvability in the last decades of the 19th century. 5. The period 18901900: the beginning of modern PDE and the work of Poincare. 6. The Hilbert programs. 7. S. Bernstein and the beginning of a priori estimates. 8. Solvabil...

This article is motivated by the practical problem of highway traffic estimation using velocity measurements from GPS enabled mobile devices such as cell phones. In order to simplify the estimation procedure, a velocity model for highway traffic is constructed, which results in a dynamical system in which observation the operator is linear. This article presents a new scalar hyperbolic partial ...

This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily com...

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The principal goal of this work is to solvability the mild solution for second‐order hyperbolic PDE with initial/boundary‐value problem nonlocal condition form where open. Our analysis relies on technique measure noncompactness.

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