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Introduction

A Partial Differential Equation (PDE) is an equation relating a function of two or more independent variables and its partial derivatives. Consider the general equation
\begin{displaymath}
a{\partial^{2}u\over \partial x^{2}} + b{\partial^{2}u\over...
...\over
\partial x} + e{\partial u \over \partial t} + fu = g,
\end{displaymath} (1.1)

where $a$, $b$, $c$, $d$, $e$ and $f$ may be functions of $x$, $t$ and even $u$. The order of the equation is given by the order of the highest derivative. Thus, if one of the functions, $a$, $b$ or $c$ are non-zero, then the PDE is second order. If $ a = b =
c = 0$ but $d$ or $e$ are non-zero, then the PDE is first order.

If the functions $a$, $b$, $c$, $d$, $e$ and $f$ do not depend on the dependent variable $u$, then Equation (1.1) is linear otherwise it is non-linear. We are mainly concerned with linear PDEs in this course.

If $g = 0$, then Equation (1.1) is honogeneous otherwise it is inhomogeneous.

Example 1. .1First order Partial Differential Equations.


  1. \begin{displaymath}
{\partial u\over \partial x} - {\partial u\over \partial t} = 0,
\end{displaymath}

            linear PDE.


  2. \begin{displaymath}
u{\partial u\over \partial x} - {\partial u\over \partial t} = 0,
\end{displaymath}

            non-linear PDE.


  3. \begin{displaymath}
e^{x}{\partial u\over \partial x} + 4{\partial u\over \partial t} =
t,
\end{displaymath}

            linear, inhomogeneous PDE.

Example 1. .2Second order Partial Differential Equations.


  1. \begin{displaymath}
{\partial^{2} u\over \partial x^{2}} + 4{\partial u\over \partial t} = 0,
\end{displaymath}

            linear PDE.


  2. \begin{displaymath}
{1\over x}{\partial \partial x}\left (x{\partial u\over \pa...
...
x}\right ) - {\partial^{2} u\over \partial t^{2}} = x^{2},
\end{displaymath}

            linear, inhomogeneous PDE.


  3. \begin{displaymath}
{\partial^{2} u\over \partial x^{2}} - e^{2x}{\partial^{2}
u\over \partial t^{2}} = u^{3},
\end{displaymath}

            non-linear PDE.


next up previous
Next: Revision of Partial Differentiation Up: Partial Differential Equations of Previous: Partial Differential Equations of
Prof. Alan Hood
2000-10-30