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Equation data
Category:2. First-Order Partial Differential Equations
Subcategory:2.3. Nonlinear Equations
$\displaystyle f_1(x,y,z)\frac{\partial w}{\partial x}+
f_2(x,y,z)\frac{\partial w}{\partial y}
+f_3(x,y,z)\frac{\partial w}{\partial z}+\sqrt{
\Bigl(\frac{\partial w}{\partial x}\Bigr)^{\!2}+
\Bigl(\frac{\partial w}{\partial y}\Bigr)^{\!2}+
\Bigl(\frac{\partial w}{\partial z}\Bigr)^{\!2}}=1$.
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
Statistic information
Submission date:Sun 17 Dec 2006 06:13
Edits by author:0

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