DD choose difference approximations for the derivatives. In Chapter 3, we have already introduced a second-order formula for the first derivative, the central difference approximation: du =(x) = (x+1)(x-1)+ +0(h²) 2h For the second derivative, there is a similar second-order formula: d²u _ u(x;+1) − 2u(x;) + u(x¡_1) +0(h²) dx² h² (4.23) (4.24) G3-use the original grid GO but apply second-order unsymmetric difference approximation for the BCs, e.g.. dx −u(2h) + 4u(h) — 3u(0) 2h and, at the other end of the interval, du dx +0(4²) (4.34a) 3u(1) – 4u(1 − h) + u(1 − 2h) 2/ +0(h²) (4.34b) With this grid and these difference approximations, no points outside the inter- val [0, 1] are needed. Hence, we can use the grid GO. h² h³ 2u(x)+u(xh) = u(xi)+hu' (xi)+ h3 u" (xi h² xi) - hu' (xi))+ u"(xi) -- -u!" (xi)+O(h*) = hu" (2)
DD choose difference approximations for the derivatives. In Chapter 3, we have already introduced a second-order formula for the first derivative, the central difference approximation: du =(x) = (x+1)(x-1)+ +0(h²) 2h For the second derivative, there is a similar second-order formula: d²u _ u(x;+1) − 2u(x;) + u(x¡_1) +0(h²) dx² h² (4.23) (4.24) G3-use the original grid GO but apply second-order unsymmetric difference approximation for the BCs, e.g.. dx −u(2h) + 4u(h) — 3u(0) 2h and, at the other end of the interval, du dx +0(4²) (4.34a) 3u(1) – 4u(1 − h) + u(1 − 2h) 2/ +0(h²) (4.34b) With this grid and these difference approximations, no points outside the inter- val [0, 1] are needed. Hence, we can use the grid GO. h² h³ 2u(x)+u(xh) = u(xi)+hu' (xi)+ h3 u" (xi h² xi) - hu' (xi))+ u"(xi) -- -u!" (xi)+O(h*) = hu" (2)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 92E
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![DD choose difference approximations for the derivatives. In Chapter 3, we have
already introduced a second-order formula for the first derivative, the central
difference approximation:
du
=(x) =
(x+1)(x-1)+
+0(h²)
2h
For the second derivative, there is a similar second-order formula:
d²u
_ u(x;+1) − 2u(x;) + u(x¡_1)
+0(h²)
dx²
h²
(4.23)
(4.24)
G3-use the original grid GO but apply second-order unsymmetric difference
approximation for the BCs, e.g..
dx
−u(2h) + 4u(h) — 3u(0)
2h
and, at the other end of the interval,
du
dx
+0(4²)
(4.34a)
3u(1) – 4u(1 − h) + u(1 − 2h)
2/
+0(h²)
(4.34b)
With this grid and these difference approximations, no points outside the inter-
val [0, 1] are needed. Hence, we can use the grid GO.
h²
h³
2u(x)+u(xh) = u(xi)+hu' (xi)+
h3
u" (xi
h²
xi) - hu' (xi))+
u"(xi)
--
-u!" (xi)+O(h*) = hu" (2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7e060b9-dd7f-43a4-bb74-d053d88e38b9%2F552b46c5-ca12-4118-9934-12d3b43502b0%2Fkdyrq3_processed.png&w=3840&q=75)
Transcribed Image Text:DD choose difference approximations for the derivatives. In Chapter 3, we have
already introduced a second-order formula for the first derivative, the central
difference approximation:
du
=(x) =
(x+1)(x-1)+
+0(h²)
2h
For the second derivative, there is a similar second-order formula:
d²u
_ u(x;+1) − 2u(x;) + u(x¡_1)
+0(h²)
dx²
h²
(4.23)
(4.24)
G3-use the original grid GO but apply second-order unsymmetric difference
approximation for the BCs, e.g..
dx
−u(2h) + 4u(h) — 3u(0)
2h
and, at the other end of the interval,
du
dx
+0(4²)
(4.34a)
3u(1) – 4u(1 − h) + u(1 − 2h)
2/
+0(h²)
(4.34b)
With this grid and these difference approximations, no points outside the inter-
val [0, 1] are needed. Hence, we can use the grid GO.
h²
h³
2u(x)+u(xh) = u(xi)+hu' (xi)+
h3
u" (xi
h²
xi) - hu' (xi))+
u"(xi)
--
-u!" (xi)+O(h*) = hu" (2)
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