Method of Characteristics in Smooth Problems

  • Arik Melikyan

Abstract

Consider general nonlinear first order partial differential equation (PDE):
$$ F\left({x,u\left( x \right),p\left( x \right)} \right) = 0,x \in D \subset {\mathbb{R}^n}\left( {p = {{\partial u} \mathord{\left/ {\vphantom {{\partial u} {\partial x = {u_x}}}} \right. \kern-\nulldelimiterspace} {\partial x = {u_x}}}} \right) $$
(1.1)
Here x = (x 1 , x n ) is n-dimensional vector of the space \( {\mathbb{R}^n} \), D is an open neighborhood of a reference point x* \( {\mathbb{R}^n} \) u is the scalar unknown function, u: D→ \( {\mathbb{R}^n} \) 1 , and p = (P1,, p n ) is the vector of its gradient, pi =\( \partial \)u/\( \partial \)x i , i = 1,…, n. The scalar function F will be called the Hamiltonian, F: N→ \( {\mathbb{R}^1} \) , where \( N = D \times {\mathbb{R}^1} \times {\mathbb{R}^n} \) is a domain in (2n + 1)-dimensional space of (x, u, p) \( \in {\mathbb{R}^{2n + 1}} \)

Keywords

Manifold Guaran 

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Arik Melikyan
    • 1
  1. 1.Russian Academy of ScienceInstitute for Problems in MechanicsMoscowRussia

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