Methods of
Mathematical Physics
Integral Transformation
Integral Transformation
? General concept of the integral
transformation
? Fundamental ideas of The method of
Integral Transformation
? Fourier Transforms of wave problems
? Fourier Transforms of heat problems
? Fourier Transforms of steady
problems
? Conclusion of this charter
General Concepts
? Integral transformation
? Definition,
? Turn the oragl function into one respect to new
coefficients
? General form
? f(x) → F(k) = ∫f(x) K(x,k) dx
? General properties：
? linearity,? f(x) + ? g(x) → ? F(k) + ? G(k)
? Inverse transformation,f(x) ←→ F(k)
? derivative：
f’(x) → f(x) K(x,k)|bound - ∫f(x) Kx(x,k) dx
General Concepts
? Typical integral transformations：
? Fourier
Transfor
mation dkekFxf
dxexfkF
ik x
ik x
?
?
?
??
?
??
?
?
?
)()(
)(
2
1
)(
?
? Laplace
Transfor
mation ?
?
??
??
?
?
?
?
i
i
px
px
dpepF
i
xf
dxexfpF
?
??
)(
2
1
)(
)()(
0
Fundamental Idea
? Programmer：
? Turn the original equation into an ordinary differential
equation.
? Solve the ordinary differential equation.
? Charge the solution of the ordinary differential equation
into the solution of the original.
? Applying range：
? Fourier Transformation,
? Unbounded region
? Laplace Transformation,
? Semi- unbounded region 。
Fourier Transform for the
wave problem
?
?
?
?
?
??
??
??
?
?
0
0
22
|
|
0
tt
t
tt
U
U
UkaU
?
?
?
?
????
atx
atx
dss
a
atxatxu
)(
2
1
)]()([21
?
??
Solving the wave problems
?
?
?
?
?
?
?
??
?
?
)(|
)(|
0
0
0
2
xu
xu
uau
tt
t
xxtt
?
?
k a t
k a tU
ka sin
c o s
1 ??
??
)()(
)()(
),(),(
kx
kx
tkUtxu
??
??
?
?
?
?? dketkUu ik x),(
Fourier Transform for the
heat problems
??
???
??
??
? 0
22
|
0
t
t
U
UkaU
ta
dssu ta
sx
?
?
2
]e x p [
)( 2
2
4
)( ??
? ?
Solving the heat problems
??
?
?
?
?
??
? )(|
0
0
2
xu
uau
t
xxt
?
)e x p ( 22 takU ???
)()(
),(),(
kx
tkUtxu
??
?
?
?? dketkUu ik x),(
Fourier Transform for the
steady problems
?
?
?
??
?
?
?
??
??
??
?
0|
|
0
0
2
y
y
yy
U
U
UkU
])[()( 22 ysx
y d ssu
??? ? ??
Solving the steady problems
?
?
?
?
?
?
?
???
??
?
0|
)(|
0,0
0
y
y
yyxx
u
xu
yuu
?
)||e x p ( ykU ???
)()(
),(),(
kx
ykUyxu
??
?
?
?? dkeykUu ik x),(
Conclusion for this charter
? In principle,Fourier Transform can be
applied to any unbounded problems
? The steady problem
? The heat problem
? The wave problem
? The key to solving the Eq,by use of
Fourier Transform method is inverse
transfrom
? Generally,the results from Fourier
Transform method are integral formulas.