(5) Fourier series techniques to prove the existence of solutions to the heat equation with Dirichlet boundary values on [0, 7]. This problem is given explicitly by; In class we used Ju(0, t) = 0 lu(7, t) = 0 du with boundary data (B) u(x, 0) = f(x) %3D You might ask whether this solution is unique. To see that the solution is unique do the following things: (a) Show that if u1, u2 are solutions of (B) then w = u1 – u2 solves the heat equation with Dirichlet boundary data, and satisfies w(x, 0) = 0. %3D (b) Now show that if u solves (B) then d | (u(x, t))²dx < 0 dt (c) Using part (b) argue that if w(x, t) solves (B) with w(x,0) = 0 then w(x, t) = 0 for all t >0 (Hint: Consider (w(x, t))²dx)
(5) Fourier series techniques to prove the existence of solutions to the heat equation with Dirichlet boundary values on [0, 7]. This problem is given explicitly by; In class we used Ju(0, t) = 0 lu(7, t) = 0 du with boundary data (B) u(x, 0) = f(x) %3D You might ask whether this solution is unique. To see that the solution is unique do the following things: (a) Show that if u1, u2 are solutions of (B) then w = u1 – u2 solves the heat equation with Dirichlet boundary data, and satisfies w(x, 0) = 0. %3D (b) Now show that if u solves (B) then d | (u(x, t))²dx < 0 dt (c) Using part (b) argue that if w(x, t) solves (B) with w(x,0) = 0 then w(x, t) = 0 for all t >0 (Hint: Consider (w(x, t))²dx)
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
![(5)
Fourier series techniques to prove the existence of solutions to the heat equation with
Dirichlet boundary values on [0, ]. This problem is given explicitly by;
In class we used
Su(0, t) = 0
lu(7, t) = 0
with boundary data
(B)
u(x, 0) = f(x)
%3D
You might ask whether this solution is unique. To see that the solution is unique do
the following things:
(a) Show that if u1, u2 are solutions of (B) then w = u1 – u2 solves the heat equation
with Dirichlet boundary data, and satisfies w(x, 0) = 0.
(b) Now show that if u solves (B) then
d
| (u(x, t))²dx < 0
dt
(c) Using part (b) argue that if w(x, t) solves (B) with w(x,0) = 0 then w(x, t) = 0
for all t >0 (Hint: Consider (w(x, t))²dx)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab955650-0f06-4aeb-a2fc-da4dfdfa73f2%2F4f29484b-f759-47f8-8c13-5a81ac01a9c9%2Fqv7q846_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(5)
Fourier series techniques to prove the existence of solutions to the heat equation with
Dirichlet boundary values on [0, ]. This problem is given explicitly by;
In class we used
Su(0, t) = 0
lu(7, t) = 0
with boundary data
(B)
u(x, 0) = f(x)
%3D
You might ask whether this solution is unique. To see that the solution is unique do
the following things:
(a) Show that if u1, u2 are solutions of (B) then w = u1 – u2 solves the heat equation
with Dirichlet boundary data, and satisfies w(x, 0) = 0.
(b) Now show that if u solves (B) then
d
| (u(x, t))²dx < 0
dt
(c) Using part (b) argue that if w(x, t) solves (B) with w(x,0) = 0 then w(x, t) = 0
for all t >0 (Hint: Consider (w(x, t))²dx)
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