Abstract:
Di erential equation is an equation which involves derivatives of unknown function
as well as the function it self.The two basic DEs are ODE and PDE. ODE is
an equation that the unknown function depends only on a single independent
variable,while a PDE is an equation involving one or more partial derivative of
function u, that depends on two or more variables.The 1D wave equation is one
of the PDE that can be modeled by vibrating string, and solved analytically
through ODE by method of separation variables.The statement of the problem
we consider in this thesis is to determine the vibrations of string at any point x
and any time t> 0 to model 1D wave equation
@
2
u
@t
2 = c
2 @
2
u
@x
2
. The objective of
thesis is to provide the basic derivation of 1D wave equation and its solution
by using method of data analysis gathered from secondary sources would be
studied ,analyzed and interpreted using examples of real world.The solution of
1D wave equation solved using Fourier series separation method in three steps
is u
n
(x; t) =
P
1
n=1
(Cn
cos
n
t + Dn
sin
n
t) sin
n x
L
n=1,2,... Finally the shape
of a string of length L plucked in the middle that has initial shape given in
f (x) =
8
>
<
>
:
0:1x; if 0 x 1
0:1(2 x); if 1 x 2:
would be drawn using matlab.
