Mtg 3:
NOTE: - page numbering 3-1 defined as meeting number 3, page 1
Fig.p.1-1
Eq.(3)P.2-1 : "Ordinary" Differential Equation (ODE)
y
1
(
t
)
{\displaystyle y^{1}(t)\ }
u
1
(
s
,
t
)
{\displaystyle u^{1}(s,t)\ }
u
2
(
s
,
t
)
{\displaystyle u^{2}(s,t)\ }
≡
{\displaystyle \equiv \ }
⇒
{\displaystyle \Rightarrow \ \ }
Ref:VQ&O 1989
→
{\displaystyle \rightarrow \ \ }
→
{\displaystyle \rightarrow \ \ }
→
{\displaystyle \rightarrow \ \ }
Note: Math structure of coefficient
c
i
(
Y
′
,
t
)
{\displaystyle c_{i}(Y',t)\ }
i
=
0
,
1
,
.
.
.3
{\displaystyle i=0,1,...3\ }
u
1
{\displaystyle u^{1}\ }
u
2
{\displaystyle u^{2}\ }
P
(
x
)
y
″
+
Q
(
x
)
y
′
+
R
(
x
)
y
=
F
(
x
)
{\displaystyle \displaystyle {\begin{aligned}P(x)y''+Q(x)y'+R(x)y=F(x)\end{aligned}}}
(1)
where
y
″
=
d
2
y
d
x
2
{\displaystyle y''={\frac {d^{2}y}{dx^{2}}}\ }
x
=
{\displaystyle x=\ }
y
(
x
)
=
{\displaystyle y(x)=\ }
Δ
X
+
k
2
X
=
0
{\displaystyle \Delta \ X+k^{2}X=0\ }
Δ
X
=
0
{\displaystyle \Delta \ X=0\ }
Ref F09 Mtg.28 , Ref F09 Mtg.29
, Ref F09 Mtg.30
In 3_D,
x
=
(
x
1
,
x
2
,
x
3
)
{\displaystyle x=(x_{1},x_{2},x_{3})\ }
X
(
x
)
=
X
1
(
x
1
)
X
2
(
x
2
)
X
3
(
x
3
)
{\displaystyle \displaystyle {\begin{aligned}X(x)=X_{1}(x_{1})X_{2}(x_{2})X_{3}(x_{3})\end{aligned}}}
(1)
Where the lowercase
x
{\displaystyle x\ }
X
(
x
)
{\displaystyle X(x)\ }
x
=
(
x
1
,
x
2
,
x
3
)
{\displaystyle x=(x_{1},x_{2},x_{3})\ }
X
1
(
x
1
)
X
2
(
x
2
)
X
3
(
x
3
)
{\displaystyle X_{1}(x_{1})X_{2}(x_{2})X_{3}(x_{3})\ }
X
(
ξ
)
=
X
1
(
ξ
1
)
X
2
(
ξ
2
)
X
3
(
ξ
3
)
{\displaystyle \displaystyle {\begin{aligned}X(\xi \ )=X_{1}(\xi \ _{1})X_{2}(\xi \ _{2})X_{3}(\xi \ _{3})\end{aligned}}}
(2)
Where
ξ
{\displaystyle \xi \ \ }
X
(
ξ
)
{\displaystyle X(\xi \ )\ }
ξ
=
(
ξ
1
,
ξ
2
,
ξ
3
)
{\displaystyle \xi \ =(\xi \ _{1},\xi \ _{2},\xi \ _{3})\ }
X
1
(
ξ
1
)
X
2
(
ξ
2
)
X
3
(
ξ
3
)
{\displaystyle X_{1}(\xi \ _{1})X_{2}(\xi \ _{2})X_{3}(\xi \ _{3})\ }
i
=
1
,
2
,
3
{\displaystyle i=1,2,3\ }
1
g
i
(
ξ
i
)
d
d
ξ
i
[
g
i
(
ξ
i
)
d
X
i
(
ξ
i
)
d
ξ
i
]
+
f
i
(
ξ
i
)
X
i
(
ξ
i
)
=
0
{\displaystyle \displaystyle {\begin{aligned}{\frac {1}{g_{i}(\xi \ _{i})}}{\frac {d}{d\xi \ _{i}}}\left[g_{i}(\xi \ _{i}){\frac {dX_{i}(\xi \ _{i})}{d\xi \ _{i}}}\right]+f_{i}(\xi \ _{i})X_{i}(\xi \ _{i})=0\end{aligned}}}
(3)
Simplify:
ξ
i
→
x
{\displaystyle \xi \ _{i}\rightarrow \ x\ }
X
i
(
ξ
i
)
→
y
(
x
)
{\displaystyle X_{i}(\xi \ _{i})\rightarrow \ y(x)\ }
g
i
(
ξ
i
)
→
g
(
x
)
{\displaystyle g_{i}(\xi \ _{i})\rightarrow \ g(x)\ }
f
i
(
ξ
i
)
→
a
0
(
x
)
{\displaystyle f_{i}(\xi \ _{i})\rightarrow \ a_{0}(x)\ }
Eq.(3)p.3-3 :
y
″
+
g
′
(
x
)
g
(
x
)
y
′
+
a
0
(
x
)
y
=
0
{\displaystyle \displaystyle {\begin{aligned}y''+{\frac {g'(x)}{g(x)}}y'+a_{0}(x)y=0\end{aligned}}}
(1)
Where
g
′
(
x
)
g
(
x
)
=
a
1
(
x
)
{\displaystyle {\frac {g'(x)}{g(x)}}=a_{1}(x)\ }
Eq.(1)p.3-2
F
(
.
)
{\displaystyle F(.)\ }
u
{\displaystyle u\ }
v
{\displaystyle v\ }
F
(
.
)
{\displaystyle F(.)\ }
F
(
α
u
+
β
v
=
α
F
(
u
)
+
β
F
(
v
)
{\displaystyle F(\alpha \ u+\beta \ v=\alpha \ F(u)+\beta \ F(v)\ }
α
{\displaystyle \alpha \ \ }
β
{\displaystyle \beta \ \ }
References
edit
![{\displaystyle \displaystyle {\begin{aligned}{\frac {1}{g_{i}(\xi \ _{i})}}{\frac {d}{d\xi \ _{i}}}\left[g_{i}(\xi \ _{i}){\frac {dX_{i}(\xi \ _{i})}{d\xi \ _{i}}}\right]+f_{i}(\xi \ _{i})X_{i}(\xi \ _{i})=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bfae4828e6b610b743c2f795dbdcbd3740ca918)
