Transforms Chapter 3

LAPLACE
TRANSFORMS

CHAPTER 3$

, Value Problems
Boundary

Laplace transforms
↳
So far differential equations involved the derivative with respect to
one variable It A BVP is differential equation that involves the
.




derivative with respect to two variables x , y or x. t or
y ,
t .




We call it a
partial differential equation .




When solve these pole 's get constants and find
.


we we we

these constants conditions often called
using given
BOUNDARY conditions .




-



Heat .
wave , Laplace

'



Suppose u is a function of exit ) and we are told to solve the

using Laplace transforms I
BVP an
-




L Iu Hit ) ) U Cx s )
=
,




L ( If but ) ) )
=



sulks ) -

u Ix ,
o




L ( 2¥ lait ) ) UH Et la )
o
5 s) su be ol ,
= -

-


, ,




L ( but ) ) Ex Ex = U bust



L ( 3¥ Hit ) ) =


II .
UH . s )



find UH s ) o I as before )
-

-



,

, example :




* just treat x as a constant

SUCKS ) ULK.si
kf÷
-

UH ,
a) =




,




* I
D :=
SUCKS ) -
( I
-

x ) =
kD2ulx , s ) 2x




(k s )
'
D -
Ulk , s ) =
a
-

I




( D2 I ) complementary
*
-


UH , s )
=
O
function


ID Fr ) ( -




Dt Fa ) UH ,ss=0

" FE "

A B Css e-
yo
:
esse
=

+
-




T.se functions of S




particular integral :




ik
F.
)eFk"




Its
' '
Acs Base
-




pz÷
U l " )
'



s =
I ) Ucas ) -
t
La
.




x
.



.
-
t




but lim UGC , s ) =D
I
)
=

(
th
K
-
t
s sa
-
-

-




-




I ( I
-




Foz ) '




Als ) =D
.
-




= 1- .

11¥
that ulo.tt O t o
given
-
>
¥132 S ,
I
-




UCO 's ) o



)
=
2

D2 ) ( x )
I [ (
I
EDY
= -




It ( t Es



)
-

E ' " i

=
I i D= Ot Bcs e

S
: .



Bls ) = -




Is