Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Show me the steps of determine green and all information is here
![Bxz
g(x, y, z) = ax +
(3)
VY – dz
Then,
ag(x, y, z)
Bz
(4)
ag(x, y, z)
ду
ag(x, y, z)
a +
VY – 8z'
Byæz
(ry – dz)² '
Byxy
(YY – dz)²"
(5)
(6)
dz
Now, we evaluate Eqs. (4), (5) and Eq. (6) at ū. That is
dg(ū, ū, ū)
Bū
= a +
a +
:=
-91,
Yū – dū
Byu?
(yū – dū)2
дд(й, и,
u)
By
:=-42,
ду
(y – 8)2
dg(ū, ū, ū)
Byu?
(7й — би)?
By
93·
dz
(y – 8)2
Thus, the linearised equation of Eq. (1) about the equilibrium point is given by
Un+1 + 41Vn-1+ 92Vn-3 + 93Vn-5 = 0.
Theorem 1 Assume that
|a(y – 8) + B||y – 8| < (y – 8)² – 2By.
Then, the equilibrium point ū = 0 is locally asymptotically stable.
Proof.
Theorem A in [20] guarantees that the stability of the equilibrium point
occurs if
|41|+ |42| + |g3| < 1.
(7)
Plugging qi, i = 1,2, 3, into Eq. (7) leads to
BY
By
<1.
(y – 8)2 |
a +
+
(y – 8)2
Hence,
23y
+
< 1,
a +
(7 – 8)2
which can be written as
|a(y – 8) + B||y – 8| < (y – 8)² – 2,3y.](https://content.bartleby.com/qna-images/question/5d564931-7cab-47e3-baee-f646b4270efc/2f2e5a38-a66b-4cd0-8957-e648f7a8ee57/k0tkrn6_processed.jpeg)
Transcribed Image Text:Bxz
g(x, y, z) = ax +
(3)
VY – dz
Then,
ag(x, y, z)
Bz
(4)
ag(x, y, z)
ду
ag(x, y, z)
a +
VY – 8z'
Byæz
(ry – dz)² '
Byxy
(YY – dz)²"
(5)
(6)
dz
Now, we evaluate Eqs. (4), (5) and Eq. (6) at ū. That is
dg(ū, ū, ū)
Bū
= a +
a +
:=
-91,
Yū – dū
Byu?
(yū – dū)2
дд(й, и,
u)
By
:=-42,
ду
(y – 8)2
dg(ū, ū, ū)
Byu?
(7й — би)?
By
93·
dz
(y – 8)2
Thus, the linearised equation of Eq. (1) about the equilibrium point is given by
Un+1 + 41Vn-1+ 92Vn-3 + 93Vn-5 = 0.
Theorem 1 Assume that
|a(y – 8) + B||y – 8| < (y – 8)² – 2By.
Then, the equilibrium point ū = 0 is locally asymptotically stable.
Proof.
Theorem A in [20] guarantees that the stability of the equilibrium point
occurs if
|41|+ |42| + |g3| < 1.
(7)
Plugging qi, i = 1,2, 3, into Eq. (7) leads to
BY
By
<1.
(y – 8)2 |
a +
+
(y – 8)2
Hence,
23y
+
< 1,
a +
(7 – 8)2
which can be written as
|a(y – 8) + B||y – 8| < (y – 8)² – 2,3y.
Transcribed Image Text:Bun-1un-5
Ип+1 — QИп-1 +
п %3D 0, 1,...,
(1)
Yun-3 – dun–5
Bun-1un-5
Ип+1 — QUn-1
п 3D 0, 1, ...,
(2)
YUn-3 + dun-5
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