MiniLogo

EqWorld

The World of Mathematical Equations

IPM Logo

Exact Solutions Methods Software For Authors Math Forums

EqArchive: Add Equation/Solution > View Equation

 English only

View Equation

The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:2. First-Order Partial Differential Equations
Subcategory:2.3. Nonlinear Equations
Equation(s):\noindent
$\displaystyle f_1(x,y,z)\frac{\partial w}{\partial x}+
f_2(x,y,z)\frac{\partial w}{\partial y}
+f_3(x,y,z)\frac{\partial w}{\partial z}+\sqrt{
\Bigl(\frac{\partial w}{\partial x}\Bigr)^{\!2}+
\Bigl(\frac{\partial w}{\partial y}\Bigr)^{\!2}+
\Bigl(\frac{\partial w}{\partial z}\Bigr)^{\!2}}=1$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
The following condition is assumed to hold:
$$
\frac 1r\bl[xf_1(x,y,z)+yf_2(x,y,z)+zf_3(x,y,z)\br]=
\varphi(r),\quad \ \ r=\sqrt{x^2+y^2+z^2}.
$$
The Cauchy problem with the initial condition
$$
w=0\quad\hbox{at}\quad r=0\qquad
(w>0\quad\hbox{for}\quad r>0)
$$
has the solution
$$
w=\int^r_0\frac{d\xi}{1+\varphi(\xi)}\qquad (1+\varphi>0).
$$
Remarks:\noindent This is a Hamilton--Jacobi--Belamn type equation. 
It arises in variational calculus and optimal control.
Novelty:Material has been fully published elsewhere
References:Akulenko, L.D. (1987) Asymptotic Methods of Optimal Cotrol. Nauka, Moscow.
Author/Contributor's Details
Last name:Zhurov
First name:Alexei
Country:Russia
Statistic information
Submission date:Sun 17 Dec 2006 06:13
Edits by author:0

Edit (Only for author/contributor)


The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.

Copyright © 2006-2011 Andrei D. Polyanin, Alexei I. Zhurov and Alexander L. Levitin