“Hand-in question" (only 6(a)): Use, Theorem 4.6 in the full set of lecture notes to obtain zeroth order approximations, valid for t € [0, 1], to the following boundary-value problems in which 0 < € < 1. (a) ɛï+ (t+1)x+x=0, x(0) = 0, x(1) = 1
“Hand-in question" (only 6(a)): Use, Theorem 4.6 in the full set of lecture notes to obtain zeroth order approximations, valid for t € [0, 1], to the following boundary-value problems in which 0 < € < 1. (a) ɛï+ (t+1)x+x=0, x(0) = 0, x(1) = 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Theorem 4.6. (A Theorem on the Method of Singular Perturbations) Given a linear boundary-
value problem of the form
e#(t) + p(t)*(t)+ q(t)x(t) = 0, te (0,1), 0 < ɛ « 1,
r(0) = a,
(4.13)
r(1) = b,
where p, q are continuous on [0, 1] with p(t) > 0 for t e [0, 1], there erists a boundary layer at t = 0 with
outer and innет аррrолітations:
q(T)
xouter(t) = bexp
Гimner (t) — А + (a — А) еxp
where
A = besp L)
q(T)
p(T)
The approrimate solution valid on [0, 1] is given by
Ta(t) = xinner(t) + xouter(t) – A.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9bfeb01-5657-47f2-90f5-8be14b613af0%2Fde029ea7-31e9-41b4-b76a-8f233b2578d7%2F6jka3td_processed.png&w=3840&q=75)
Transcribed Image Text:Theorem 4.6. (A Theorem on the Method of Singular Perturbations) Given a linear boundary-
value problem of the form
e#(t) + p(t)*(t)+ q(t)x(t) = 0, te (0,1), 0 < ɛ « 1,
r(0) = a,
(4.13)
r(1) = b,
where p, q are continuous on [0, 1] with p(t) > 0 for t e [0, 1], there erists a boundary layer at t = 0 with
outer and innет аррrолітations:
q(T)
xouter(t) = bexp
Гimner (t) — А + (a — А) еxp
where
A = besp L)
q(T)
p(T)
The approrimate solution valid on [0, 1] is given by
Ta(t) = xinner(t) + xouter(t) – A.
!["Hand-in question" (only 6(a)): Use, Theorem 4.6 in the full set of lecture notes to obtain
zeroth order approximations, valid for t e [0, 1], to the following boundary-value problems in which
0 < ɛ «1.
(a)
eä + (t+ 1)i + x = 0, x(0) = 0, x(1) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9bfeb01-5657-47f2-90f5-8be14b613af0%2Fde029ea7-31e9-41b4-b76a-8f233b2578d7%2F6ydcad9_processed.png&w=3840&q=75)
Transcribed Image Text:"Hand-in question" (only 6(a)): Use, Theorem 4.6 in the full set of lecture notes to obtain
zeroth order approximations, valid for t e [0, 1], to the following boundary-value problems in which
0 < ɛ «1.
(a)
eä + (t+ 1)i + x = 0, x(0) = 0, x(1) =
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