Download Partial differential equations Objective: Partial

Partial differential equations
Objective: Partial differential equation arise quite frequently in various physical
and engineering problems when the function involved depend on two or more
independent variables. Most of the problems in fluid and solid mechanics, heat
transfer, vibrations, electromagnetic theory and other areas of engineering lead
to partial differential equations.
 Definition: The differential equations which involve partial derivatives with
respect to two or more independent variables are called partial differential
equations.
 What is Lagrange’s Linear Partial differential equation?
z
z
An equation of the form P
Q
 R , where P, Q, R are functions of x, y, z
x
y
is known as Lagrange’s Linear Partial differential equation
 What is Homogeneous Linear Partial differential equation with constant
coefficients?
An equation of the form
n z
n z
n z
n z

a

a

..........
...

a
 F ( x, y)
1
2
n
x n
x n1y
x n2 y 2
y n
Where a1 , a2, ………. an are constants is called Homogeneous Linear Partial
differential equation with constant coefficients.
 What is method of separation of variables? Explain.
In this method we assume the solution to be the product of two functions, each of
which involves only one of the variables.
This method is also known as Product method.
 STANDARD PARTIAL DIFFERENTIAL EQUATIONS
2
2 y
2  y
c
a) One dimensional wave equation:
t 2
x 2
u
 2u
 c2 2
b) One dimensional Heat equation:
t
x
c) Laplace equation (Two dimensional heat equation) :
 2u  2u
 2 0
x 2
y
These equations can be solved by method of separation of variables