21.11 Problem. Let a > 0 and let f(x) = ea². Show that f is integrable. [Hint: if |x|≥ 1, then |f(x)|≤e-all. How does this help? u(x,t) = √ √ √ û(k, t)e³k² dk 1 ikx √2π La F (k)e-1² teikz dk. (22.1) 2πT This may well be a valid candidate for a solution formula! 22.4 Problem. (i) Suppose that f: RC is continuous with |f| integrable. Fix t > 0 and x E R. Prove that g(k) = e-k²teik is integrable, so the integral on the right in (22.1) converges. [Hint: Problem 21.11.] =e (ii) Assume that we may differentiate under the integral on the right in (22.1) with respect to x and t as much as we want for x R and t> 0. Show that u as defined by (22.1) satisfies ut = Uxx' (iii) Show that u as defined by (22.1) meets u(x, 0) = f(x). [Hint: Untheorem 22.2.] 22.2 Untheorem. Let f: R→C be "nice." Then f(x)= = 1 √2πT Lo ƒ (k)eikz dk.
21.11 Problem. Let a > 0 and let f(x) = ea². Show that f is integrable. [Hint: if |x|≥ 1, then |f(x)|≤e-all. How does this help? u(x,t) = √ √ √ û(k, t)e³k² dk 1 ikx √2π La F (k)e-1² teikz dk. (22.1) 2πT This may well be a valid candidate for a solution formula! 22.4 Problem. (i) Suppose that f: RC is continuous with |f| integrable. Fix t > 0 and x E R. Prove that g(k) = e-k²teik is integrable, so the integral on the right in (22.1) converges. [Hint: Problem 21.11.] =e (ii) Assume that we may differentiate under the integral on the right in (22.1) with respect to x and t as much as we want for x R and t> 0. Show that u as defined by (22.1) satisfies ut = Uxx' (iii) Show that u as defined by (22.1) meets u(x, 0) = f(x). [Hint: Untheorem 22.2.] 22.2 Untheorem. Let f: R→C be "nice." Then f(x)= = 1 √2πT Lo ƒ (k)eikz dk.
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
![21.11 Problem. Let a > 0 and let f(x) = ea². Show that f is integrable. [Hint: if
|x|≥ 1, then |f(x)|≤e-all. How does this help?
u(x,t) = √ √ √ û(k, t)e³k² dk
1
ikx
√2π La F (k)e-1² teikz dk.
(22.1)
2πT
This may well be a valid candidate for a solution formula!
22.4 Problem. (i) Suppose that f: RC is continuous with |f| integrable. Fix t > 0
and x E R. Prove that g(k) = e-k²teik is integrable, so the integral on the right in (22.1)
converges. [Hint: Problem 21.11.]
=e
(ii) Assume that we may differentiate under the integral on the right in (22.1) with respect
to x and t as much as we want for x R and t> 0. Show that u as defined by (22.1)
satisfies ut = Uxx'
(iii) Show that u as defined by (22.1) meets u(x, 0) = f(x). [Hint: Untheorem 22.2.]
22.2 Untheorem. Let f: R→C be "nice." Then
f(x)=
=
1
√2πT
Lo ƒ (k)eikz dk.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F690bc708-737a-4036-8bde-cd8ee17ec8dd%2F8de31d86-378d-4ee1-bd33-2f733b844735%2Fm6ax98_processed.jpeg&w=3840&q=75)
Transcribed Image Text:21.11 Problem. Let a > 0 and let f(x) = ea². Show that f is integrable. [Hint: if
|x|≥ 1, then |f(x)|≤e-all. How does this help?
u(x,t) = √ √ √ û(k, t)e³k² dk
1
ikx
√2π La F (k)e-1² teikz dk.
(22.1)
2πT
This may well be a valid candidate for a solution formula!
22.4 Problem. (i) Suppose that f: RC is continuous with |f| integrable. Fix t > 0
and x E R. Prove that g(k) = e-k²teik is integrable, so the integral on the right in (22.1)
converges. [Hint: Problem 21.11.]
=e
(ii) Assume that we may differentiate under the integral on the right in (22.1) with respect
to x and t as much as we want for x R and t> 0. Show that u as defined by (22.1)
satisfies ut = Uxx'
(iii) Show that u as defined by (22.1) meets u(x, 0) = f(x). [Hint: Untheorem 22.2.]
22.2 Untheorem. Let f: R→C be "nice." Then
f(x)=
=
1
√2πT
Lo ƒ (k)eikz dk.
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