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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

Characteristic Vector Cauchy Problem Characteristic Point Characteristic System Implicit Function Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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