u = 2 uzz) 0 < < 6, t >0 with boundary/initial conditions: u(0, t) = 0, u(6, t) = 0, S 2, 0
u = 2 uzz) 0 < < 6, t >0 with boundary/initial conditions: u(0, t) = 0, u(6, t) = 0, S 2, 0
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.3: Euler's Method
Problem 1YT: Use Eulers method to approximate the solution of dydtx2y2=1, with y(0)=2, for [0,1]. Use h=0.2.
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Question
![We will solve the heat equation
=2ur
2 Usc
0<r<6, t> 0
with boundary/initial conditions:
(0, t) = 0,
2(6, t) = 0,
2, 0<x<3
and u(x, 0) = {
0, 3<x < 6
This models temperature in a thin rod of length L = 6 with thermal diffusivity a- 2 where the temperature at the ends is fixed at 0 and the initial temperature
distribution is u(x, 0).
For extra practice we will solve this problem from scratch.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe25202bd-019b-4fcf-bac1-345353ef246b%2F9424f578-b5fe-404d-97ac-f8e3db82c592%2F5o9o8jr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:We will solve the heat equation
=2ur
2 Usc
0<r<6, t> 0
with boundary/initial conditions:
(0, t) = 0,
2(6, t) = 0,
2, 0<x<3
and u(x, 0) = {
0, 3<x < 6
This models temperature in a thin rod of length L = 6 with thermal diffusivity a- 2 where the temperature at the ends is fixed at 0 and the initial temperature
distribution is u(x, 0).
For extra practice we will solve this problem from scratch.
![Find Eigenfunctions for ).
The problem splits into cases based on the sign of A
(Notation: For the cases below, use constants a and b)
• Case 1: A = 0
X(x)
Plugging the boundary values into this formula gives
0 = X(0)
%3D
0 = X(6)
So X(x) =
which means u(x, t) =
We can ignore this case.
• Case 2: A =-y² <0
(In your answers below use gamma instead of lambda)
X(x) =
Plugging the boundary values into this formula gives
0 – X(0)
0 = X(6) =
So X(x) =
which means u(x, t) =
We can ingore this case.
• Case 3: A= >
X(x) =
(In your answers below use gamma instead of lambda)
Plugging in the boundary values into this formula gives
0 = X(0) =
%3D
0 = X(6)
Which leads us to the eigenvalues A, = Yn where Yn =
and eigenfunctions X,(z) =
(Notation: Eigenfunctions should not include any constants a or b.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe25202bd-019b-4fcf-bac1-345353ef246b%2F9424f578-b5fe-404d-97ac-f8e3db82c592%2F0hozkp88_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Find Eigenfunctions for ).
The problem splits into cases based on the sign of A
(Notation: For the cases below, use constants a and b)
• Case 1: A = 0
X(x)
Plugging the boundary values into this formula gives
0 = X(0)
%3D
0 = X(6)
So X(x) =
which means u(x, t) =
We can ignore this case.
• Case 2: A =-y² <0
(In your answers below use gamma instead of lambda)
X(x) =
Plugging the boundary values into this formula gives
0 – X(0)
0 = X(6) =
So X(x) =
which means u(x, t) =
We can ingore this case.
• Case 3: A= >
X(x) =
(In your answers below use gamma instead of lambda)
Plugging in the boundary values into this formula gives
0 = X(0) =
%3D
0 = X(6)
Which leads us to the eigenvalues A, = Yn where Yn =
and eigenfunctions X,(z) =
(Notation: Eigenfunctions should not include any constants a or b.)
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