Page 227 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes up to second order partial differential equations i.e.
2 z 2 z 2 z z z
a 1 2 a 2 a 3 2 a 4 a 5 z = f ( , )
y
x
x x y y x y
y
x
or a r a s a t a p a q z = f ( , )
3
5
4
2
1
(a) Depending upon the values of a , a and a we can have:
1 2 3
1. Hyperbolic type of partial differential equations in which 4a a a 2 .
1 3 2
Such equations are found in wave motion as well as in vibration of strings etc.
The example is wave motion
2 2
V 1 V
, here y is replaced by time variable
x 2 c 2 t 2
2. Parabolic type: Partial differential equations in which

a 2 2 4a a 3 0
1
Examples of such type of equations are diffusion problems as well as conduction of heat
problems i.e.
2 V V
K , here y is replaced by time t.
x 2 t

3. Elliptic type partial differential equation in which
a 2 2 4a a 3 0.
1
We come across such differential equations in electrostatics or gravitational potential
problems. Such equations are Laplace equations i.e.

2 2
V V
0
x 2 y 2
The signification of these equations is that if we transform from x, y co-ordinate to another
co-ordinate system by canonical transformation these three properties do not change.

(b) Homogeneous Partial Differential Equations

In these equations the coefficients of differential equations of any order is a constant multiple of
the variables of the same degree i.e.
z z 2 2 z 2 2 z
x y x 2 y 2 0
x y x y

(c) Linear Partial Differential Equations with Constant Coefficients

In these equations the coefficients of the partial derivatives are constant i.e.
c r c s c t c p c q c z f ( , )
y
x
5
3
4
6
2
1
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