at point Q: a;(4)² + b;(4) + c, = 2 ------ egn 1 az(4)² + ba(4) + C2 = 2 ------- egn 2 at point R: az(7)° + b2(7) + c2 = 8 as(7)° + b,(7) + C3 = 8 ---- egn 3 ------- egn 4 Condition #2: From the exterior knots: P, S At point P: a;(3)² + b;(3) + c, = 5 ------- eqn 5 At point S: as(9)? + ba(9) + C3 = 6 ------- egn 6 Condition #3: First derivatives of the interior knots are equal 2a,x + b, = 2a,x + b2 2a,x + b2 = 2a3x + b3 egn 7 egn 8
at point Q: a;(4)² + b;(4) + c, = 2 ------ egn 1 az(4)² + ba(4) + C2 = 2 ------- egn 2 at point R: az(7)° + b2(7) + c2 = 8 as(7)° + b,(7) + C3 = 8 ---- egn 3 ------- egn 4 Condition #2: From the exterior knots: P, S At point P: a;(3)² + b;(3) + c, = 5 ------- eqn 5 At point S: as(9)? + ba(9) + C3 = 6 ------- egn 6 Condition #3: First derivatives of the interior knots are equal 2a,x + b, = 2a,x + b2 2a,x + b2 = 2a3x + b3 egn 7 egn 8
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Quadratic Spline.
Show how to obtain the constants below (a1,b1,c1 etc...) from the equations above through algebra particularly substitution or elimination. Please show complete solution and and also neat handwriting.
Transcribed Image Text:Condition #1: From the Interior knots: Q, R
at point Q:
a,(4)? + b;(4) + c, = 2
egn 1
az(4)? + b2(4) + C2 = 2
egn 2
at point R:
az(7)² + b2(7) + C2 = 8
as(7)? + b3(7) + C3 = 8
eqn
3
egn 4
-------
Condition #2: From the exterior knots: P, S
At point P:
a;(3)² + b;(3) + c, = 5
At point S:
as(9)? + b3(9) + C3 = 6
egn 5
egn 6
Condition #3: First derivatives of the interior knots are equal
2a,x + bị = 2a,x + b2
2a,x + b2 = 2a,x + b3
----- eqn 7
egn 8
Condition # 4:
a, = 0
eqn 9
From the 9 equations, by solving for the unknowns algebraically, we will get
the following:
a1 = 0
a2 = 15/9
a3 = -4
b1 = -3
b2 = -49/3
b3 = 63
c1 = 14
c2 = 122/3
c3 = -237
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