u(x,t) = [* H(x − y, t)f(y) dy = e-(x-y)²/4t -f(y) dy, xER, t> 0. (23.6) With this explicit formula for H, we can prove the convergence of this integral with two different hypotheses on f. 23.5 Problem. Prove that the (second) integral in (23.6) converges in each of the following cases, assuming x = R and t>0. [Hint: use the comparison test-the function that you "compare the integrand to" will be different in each case.] (i) f is bounded in the sense that there exists M > 0 such that |f(y)|≤ M for all y Є R. is integrable. (ii)
u(x,t) = [* H(x − y, t)f(y) dy = e-(x-y)²/4t -f(y) dy, xER, t> 0. (23.6) With this explicit formula for H, we can prove the convergence of this integral with two different hypotheses on f. 23.5 Problem. Prove that the (second) integral in (23.6) converges in each of the following cases, assuming x = R and t>0. [Hint: use the comparison test-the function that you "compare the integrand to" will be different in each case.] (i) f is bounded in the sense that there exists M > 0 such that |f(y)|≤ M for all y Є R. is integrable. (ii)
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
![u(x,t) = [* H(x − y, t)f(y) dy =
e-(x-y)²/4t
-f(y) dy, xER, t> 0. (23.6)
With this explicit formula for H, we can prove the convergence of this integral with two
different hypotheses on f.
23.5 Problem. Prove that the (second) integral in (23.6) converges in each of the following
cases, assuming x = R and t>0. [Hint: use the comparison test-the function that you
"compare the integrand to" will be different in each case.]
(i) f is bounded in the sense that there exists M > 0 such that |f(y)|≤ M for all y Є R.
is integrable.
(ii)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F690bc708-737a-4036-8bde-cd8ee17ec8dd%2Ff10fc374-3f07-4699-b7de-7ca0ca99c32c%2Fio92q59_processed.jpeg&w=3840&q=75)
Transcribed Image Text:u(x,t) = [* H(x − y, t)f(y) dy =
e-(x-y)²/4t
-f(y) dy, xER, t> 0. (23.6)
With this explicit formula for H, we can prove the convergence of this integral with two
different hypotheses on f.
23.5 Problem. Prove that the (second) integral in (23.6) converges in each of the following
cases, assuming x = R and t>0. [Hint: use the comparison test-the function that you
"compare the integrand to" will be different in each case.]
(i) f is bounded in the sense that there exists M > 0 such that |f(y)|≤ M for all y Є R.
is integrable.
(ii)
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