Problem 1 (Smooth convergence of the Dirichlet heat equation) Let u be a smooth solution to the heat equation on [a, b] × [0, ∞) with Dirichlet boundary condition, u(a) = 0 = u(b), and initial condition u(x,0) = (x). (a) Show that u(x, t) = 0 at the lateral boundary x € {a,b} for all even k. (b) Show (using part (a)) that, for even k, |au(x, t)| ≤ max 2kp(x)| x= [a,b] (c) Show (using the interpolation inequalities listed below) that u(·,t) → 0 in the C sense as t→∞. -2 Hint: recall from the lectures that ||u(·,t)|| ₁² ≤ ||u(·, 0)||₁²€¯¯(b-a) ²² Interpolation inequalities: (i) If f is smooth on [a, b] and satisfies f(a) = 0 = f(b), then ||fx|max ≤2||f||max || fxx || max, where ||f||max := maxx=[a,b] |ƒ(x)|. (ii) If f is smooth on [a, b] and satisfies f(a) = 0 = f(b), then ||f|| max ≤ 10|| f(x)||L²||fx(x) ||max, where ||f||L2 := √√So f²(x) dx.
Problem 1 (Smooth convergence of the Dirichlet heat equation) Let u be a smooth solution to the heat equation on [a, b] × [0, ∞) with Dirichlet boundary condition, u(a) = 0 = u(b), and initial condition u(x,0) = (x). (a) Show that u(x, t) = 0 at the lateral boundary x € {a,b} for all even k. (b) Show (using part (a)) that, for even k, |au(x, t)| ≤ max 2kp(x)| x= [a,b] (c) Show (using the interpolation inequalities listed below) that u(·,t) → 0 in the C sense as t→∞. -2 Hint: recall from the lectures that ||u(·,t)|| ₁² ≤ ||u(·, 0)||₁²€¯¯(b-a) ²² Interpolation inequalities: (i) If f is smooth on [a, b] and satisfies f(a) = 0 = f(b), then ||fx|max ≤2||f||max || fxx || max, where ||f||max := maxx=[a,b] |ƒ(x)|. (ii) If f is smooth on [a, b] and satisfies f(a) = 0 = f(b), then ||f|| max ≤ 10|| f(x)||L²||fx(x) ||max, where ||f||L2 := √√So f²(x) dx.
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
![Problem 1 (Smooth convergence of the Dirichlet heat equation)
Let u be a smooth solution to the heat equation on [a, b] × [0, ∞) with Dirichlet boundary
condition, u(a) = 0 = u(b), and initial condition u(x,0) = (x).
(a) Show that u(x, t) = 0 at the lateral boundary x = {a,b} for all even k.
(b) Show (using part (a)) that, for even k,
|au(x, t)| ≤ max |(x)|
x= [a,b]
(c) Show (using the interpolation inequalities listed below) that u(·,t) → 0 in the Co sense
as t→∞.
Hint: recall from the lectures that ||u(·,t)||₁² ≤ ||u(·, 0)||₁²€¯¯ (b-a) ²²
Interpolation inequalities:
(i) If f is smooth on [a, b] and satisfies f(a) = 0 = f(b), then
||fx|max ≤ 2||f||max ||fxx||max,
where ||f||max := maxx=[a,b] |ƒ(x)\.
(ii) If f is smooth on [a, b] and satisfies f(a) = 0 = f(b), then
||f||max ≤ 10|| f(x)||L²||fx(x) ||max,
where ||f||L² := √√√å f²(x) dx.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9dd77f94-a5e7-47d3-a2ba-eff6d53f0d48%2Fa99a0f85-3c4b-4f12-a8eb-ef7d7688495e%2F1iy6pf_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 1 (Smooth convergence of the Dirichlet heat equation)
Let u be a smooth solution to the heat equation on [a, b] × [0, ∞) with Dirichlet boundary
condition, u(a) = 0 = u(b), and initial condition u(x,0) = (x).
(a) Show that u(x, t) = 0 at the lateral boundary x = {a,b} for all even k.
(b) Show (using part (a)) that, for even k,
|au(x, t)| ≤ max |(x)|
x= [a,b]
(c) Show (using the interpolation inequalities listed below) that u(·,t) → 0 in the Co sense
as t→∞.
Hint: recall from the lectures that ||u(·,t)||₁² ≤ ||u(·, 0)||₁²€¯¯ (b-a) ²²
Interpolation inequalities:
(i) If f is smooth on [a, b] and satisfies f(a) = 0 = f(b), then
||fx|max ≤ 2||f||max ||fxx||max,
where ||f||max := maxx=[a,b] |ƒ(x)\.
(ii) If f is smooth on [a, b] and satisfies f(a) = 0 = f(b), then
||f||max ≤ 10|| f(x)||L²||fx(x) ||max,
where ||f||L² := √√√å f²(x) dx.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
