View source for Partial differential equation

==First-Order Partial Differential Equations==

===General Form of First-Order Partial Differential Equation===

A first-order partial differential equation with <math>n</math>independent variables
has the general form
:<math>
F\biggl(x_1,x_2,\dots, x_n,w,\frac{\partial w}{\partial x_1},
\frac{\partial w}{\partial x_2},\dots,\frac{\partial w}{\partial x_n}\biggl)=0,
</math>
where <math>w=w(x_1,x_2,\dots, x_n)</math> is the unknown function and
<math>F(\dots)</math> is a given function.

===Quasilinear Equations. Characteristic System. General Solution===

====General form of first-order quasilinear PDE====

A ''first-order quasilinear partial differential equation with two independent variables'' has the general form
:<math>\label{eq1}
f(x,y,w)\frac{\partial w}{\partial x}+g(x,y,w)\frac{\partial w}{\partial y}=h(x,y,w).
</math>

Such equations are encountered in various applications (continuum
mechanics, gas dynamics, hydrodynamics, heat and mass transfer, wave
theory, acoustics, multiphase flows, chemical engineering, etc.).

If the functions <math>f</math>, <math>g</math>, and <math>h</math> are independent of the unknown <math>w</math>, then equation \eqref{eq1} is called ''linear''.

====Characteristic system. General solution====

The system of [[ordinary differential equations]]
:<math>\label{eq3}
\frac{dx}{f(x,y,w)}=\frac{dy}{g(x,y,w)}=\frac{dw}{h(x,y,w)}
</math>

is known as the ''characteristic system'' of equation \eqref{eq1}.
Suppose that two independent particular solutions of this system have been found in the form
:<math>\label{eq2}
u_{1}(x, y, w)=C_{1},\qquad u_{2}(x,y,w)=C_{2},
</math>
where <math>C_1</math> and <math>C_2</math> are arbitrary constants; such particular solutions are known as ''integrals'' of system \eqref{eq3}. Then the general solution to equation \eqref{eq1} can be written as
:<math>\label{eq4}
\Phi(u_{1},u_{2})=0,
</math>
where <math>\Phi</math> is an arbitrary function of two variables. With equation \eqref{eq4}
solved for <math>u_2</math>, one often specifies the general solution in the form
<math>u_2=\Psi(u_1)</math>, where <math>\Psi(u)</math> is an arbitrary function of one variable.

''Remark''. If <math>h(x,y,w)\equiv 0</math>, then <math>w=C_2</math> can be used as the second
integral in \eqref{eq2}.

''Example''. Consider the linear equation
:<math>
\frac{\partial w}{\partial x}+a\frac{\partial w}{\partial y}=b.
</math>
The associated characteristic system of ordinary differential equations
:<math>
\frac{dx}{1}=\frac{dy}{a}=\frac{dw}{b}
</math>
has two integrals
:<math>
y-ax=C_1,\quad \ w-bx=C_2.
</math>
Therefore, the general solution to this PDE can be written as <math>w-bx=\Psi(y-ax)</math>, or
:<math>
w=bx+\Psi(y-ax),
</math>
where <math>\Psi(z)</math> is an arbitrary function.

===Cauchy Problem: Two Formulations. Solving the Cauchy Problem===

====Generalized Cauchy problem====

''Generalized Cauchy problem'': find a solution <math>w=w(x,y)</math> to
equation \eqref{eq1} satisfying the initial conditions
:<math>\label{eq5}
x=\varphi_1(\xi ),\quad y=\varphi_2(\xi ),\quad w=\varphi_3(\xi ),
</math>
where <math>\xi</math> is a parameter <math>(\alpha\le \xi  \le \beta)</math> and the
<math>\varphi_k(\xi)</math> are given functions.

''Geometric interpretation'': find an integral surface of equation \eqref{eq1}
passing through the line defined parametrically by equation \eqref{eq5}.

====Classical Cauchy problem====

''Classical Cauchy problem'': find a solution <math>w=w(x,y)</math> of
equation \eqref{eq1} satisfying the initial condition
:<math>\label{eq6}
w=\varphi(y)\quad \hbox{at}\quad x=0,
</math>
where <math>\varphi(y)</math> is a given function.

It is often convenient to represent the classical Cauchy problem as a
generalized Cauchy problem by rewriting condition \eqref{eq6} in the parametric
form
:<math>
x=0,\quad y=\xi,\quad w=\varphi(\xi).
</math>

====Existence and uniqueness theorem====

If the coefficients <math>f</math>, <math>g</math>, and <math>h</math> of equation \eqref{eq1} and the functions
<math>\varphi_k</math> in \eqref{eq5} are continuously differentiable with respect to each of
their arguments and if the inequalities <math>f\varphi'_2-g\varphi'_1\not=0</math> and
<math>(\varphi'_1)^2+(\varphi'_2)^2\not=0</math> hold along the curve \eqref{eq5}, then there is
a unique solution to the Cauchy problem (in a neighborhood of the
curve \eqref{eq5}).

====Procedure of solving the Cauchy problem====

The procedure for solving the Cauchy problem \eqref{eq1}, \eqref{eq5} involves several
steps. First, two independent integrals \eqref{eq2} of the characteristic
system \eqref{eq3} are determined. Then, to find the constants of integration <math>C_1</math>
and <math>C_2</math>, the initial data \eqref{eq5} must be substituted into the
integrals \eqref{eq2} to obtain
:<math>\label{eq7}
u_{1}\bigl(\varphi_1(\xi),\varphi_2(\xi),\varphi_3(\xi)\bigr)=C_{1},\qquad
u_{2}\bigl(\varphi_1(\xi),\varphi_2(\xi),\varphi_3(\xi)\bigr)=C_{2}.
</math>

Eliminating <math>C_1</math> and <math>C_2</math> from \eqref{eq2} and \eqref{eq7} yields
:<math>\label{eq8}
\begin{array}{rcl}
&&u_{1}(x,y,w)=u_{1}\bigl(\varphi_1(\xi),\varphi_2(\xi),\varphi_3(\xi)\bigr),\\
&&u_{2}(x,y,w)=u_{2}\bigl(\varphi_1(\xi),\varphi_2(\xi),\varphi_3(\xi)\bigr).
\end{array}
</math>

Formulas \eqref{eq8} are a parametric form of the solution to the Cauchy problem
\eqref{eq1}, \eqref{eq5}. In some cases, one may succeed in eliminating the parameter&nbsp;<math>\xi</math>
from relations \eqref{eq8}, thus obtaining the solution in an explicit form.

In the cases where first integrals \eqref{eq2} of the characteristic system \eqref{eq3}
cannot be found using analytical methods, one should employ numerical
methods to solve the Cauchy problem \eqref{eq1}, \eqref{eq5} (or \eqref{eq1}, \eqref{eq6}).