2. Present H(x, t) in terms of Fourier coefficients

of variables 3. Solve the resulting ODE for F(x) 4. Solve the resulting ODE for G(t) 5. Full
solution
First step
(define parameters):
Initial condition has been given in Eq.1
Boundary conditions have been given in Eq.3
Governing equation is shown in Eq.2
Second step
(separation of variables);
(4)
H
(
x ,t
)
=
F
(
x
)
G
(
t
)
According to the partial derivative, we could both get the first derivative function response to
t and the second derivative function response to x. Then we could get two equations below:
(5)
H
t
=
FG
(
t
)
H
xx
=
G F
xx
By substituting these two equations back to Eq.2, we will get a new relation:
FG
t
=
DG F
xx
G
t
DG
=
F
xx
F
=−
P
2
(let it be a constant)
Third step
(ODE for F(x))
F
xx
F
=−
P
2
F
xx
+
P
2
F
=
0
F
(
x
)
=
acos
(
px
)
+
bsin
(
px
)
Then apply boundary condition (Eq.3
①
):

)

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