Page 351 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes 2 2
 
x 2 y 2 .
2
A Parabolic equation has ac = b , for example the diffusion equation
2  
K 0 ...(here y = t)
x 2 y

2
A hyperbolic equation has ac < b , for example the wave equation
2 2
 1  0 ...(here y is time)
x 2 c 2 y 2

21.2 Canonical Form

Any equation of the form (1) can be written in Canonical form by choosing the canonical co-
ordinate system in terms of which the second derivative appear in the simplest possible way.

Hyperbolic Equation ac < b 2

In this case we can factorize A and C to give

A = a 2 x 2b  c 2 y (p  x q  y )(p  x q  y )
x y
1
2
1
2
C = a 2 x 2b  c 2 y (p  x q  y )(p  x q  y )
2
2
1
x y
1
with the two factors not multiples of each other. We can then choose and so that
p + q = p + q = 0
1 x 1 y 2 x 2 y
and hence A = C = 0. This means that
dy q 1
is constant on curves with dx p , is constant
1
dy q 2
on curves with
dx p 2
we can therefore write
p dy q dx p dy q dx = 0
1 1 2 2
and hence
(p dy q dx) (p dy q dx) = 0
1 1 2 2
which gives
2
2
ad y 2b dxdy + cdx = 0 ...(7)
As we shall see, this is the easiest equation to use to determine ( , ). We call ( , ) the characteristic
co-ordinate system in terms of which (1) takes its Canonical form
+ b ( , ) + b ( , ) + b = g( , ) ...(8)
1 2 3
The curves where is constant and the curves where is constant are called characteristic curves
or simply characteristics. As we shall see it is the existence or non-existence of characteristic
curves for the three types of equations that determines the distinctive properties of their solutions.

344 LOVELY PROFESSIONAL UNIVERSITY