Separation of variables The lateral radiation of heat from a bar of length L insulted at both ends can be modelled with the linear partial differential equation (PDE) Ou 8²u <= D -au for 00, 02² (1) Ot with the conditions ₂ (0.t)=₂(L,t)=0 and u(,0) = f(x). where Lo, DER are positive constants. 1. Seek a separation solution of the form u(x, t)= X(r)T(t) to show T 0 = k₁ DT + D X where & denotes the separation constant. ] 2. Use equation (2) to derive two ordinary differential equations (ODEs), one in spacer and one in time t. 3. Determine the boundary conditions for the ODE that depends on z. 4. Find the non-trivial solutions of X(r)= X₁ (r) and corresponding values of the separation constant k=k, for n=0, 1, 2, 3..... Hint: you may need to consider cach of the three cases k=p². k=0, k=p² (p+0) to find all the non-trivial solutions 5. Find the solutions T(t) = T(t), for n=0,1,2,3.... 6. Use the superposition principle for linear PDEs to write down the solution for u that satisfies the boundary conditions u. (0,t)=u₂(L,t) = 0. 7. Determine expressions for the unknown coefficients in your solution for that satisfies the initial condition u(,0) = f(x). 8. Find the solution when f(x)=1+ cos rr, for values of L=1.0=1 and D = 1. 19. In a single figure, plot the solution found in Question 8 at times t=0,0.25, 0.5, 2. Briefly describe the behaviour as t increases. What is the constant value temperature of the environment that surrounds the bar? 19
Separation of variables The lateral radiation of heat from a bar of length L insulted at both ends can be modelled with the linear partial differential equation (PDE) Ou 8²u <= D -au for 00, 02² (1) Ot with the conditions ₂ (0.t)=₂(L,t)=0 and u(,0) = f(x). where Lo, DER are positive constants. 1. Seek a separation solution of the form u(x, t)= X(r)T(t) to show T 0 = k₁ DT + D X where & denotes the separation constant. ] 2. Use equation (2) to derive two ordinary differential equations (ODEs), one in spacer and one in time t. 3. Determine the boundary conditions for the ODE that depends on z. 4. Find the non-trivial solutions of X(r)= X₁ (r) and corresponding values of the separation constant k=k, for n=0, 1, 2, 3..... Hint: you may need to consider cach of the three cases k=p². k=0, k=p² (p+0) to find all the non-trivial solutions 5. Find the solutions T(t) = T(t), for n=0,1,2,3.... 6. Use the superposition principle for linear PDEs to write down the solution for u that satisfies the boundary conditions u. (0,t)=u₂(L,t) = 0. 7. Determine expressions for the unknown coefficients in your solution for that satisfies the initial condition u(,0) = f(x). 8. Find the solution when f(x)=1+ cos rr, for values of L=1.0=1 and D = 1. 19. In a single figure, plot the solution found in Question 8 at times t=0,0.25, 0.5, 2. Briefly describe the behaviour as t increases. What is the constant value temperature of the environment that surrounds the bar? 19
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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![Separation of variables
The lateral radiation of heat from a bar of length L insulted at both ends can be modelled with
the linear partial differential equation (PDE)
Ou
8²u
<= D -au for 0<x<L, t>0,
02²
(1)
Ot
with the conditions
u, (0.1)=₂(L,t)=0 and u(,0) = f(x).
where Lo, DER are positive constants.
1. Seek a separation solution of the form u(x, t) = X(z)T(t) to show
T 0
DTD X
= k₁
where & denotes the separation constant.
] 2. Use equation (2) to derive two ordinary differential equations (ODEs), one in spacer and
one in time t.
3. Determine the boundary conditions for the ODE that depends on z.
4. Find the non-trivial solutions of X(r)= X(r) and corresponding values of the separation
constant k=k, for n= 0, 1, 2, 3.....
Hint: you may need to consider cach of the three causes k=-p², k = 0, k=p² (p0) to
find all the non-trivial solutions
5. Find the solutions T(t)= T(t), for n=0,1,2,3....
6. Use the superposition principle for linear PDEs to write down the solution for u that satisfies
the boundary conditions (0,t)=u₂(L,t) = 0.
7. Determine expressions for the unknown coefficients in your solution for that satisfies the
initial condition u(,0) = f(x).
8. Find the solution when f(x)=1+ cos rr, for values of L=1.0=1 and D = 1.
¹9. In a single figure, plot the solution found in Question 8 at times t=0,0.25, 0.5, 2. Briefly
describe the behaviour as t increases. What is the constant value temperature of the
environment that surrounds the bar?
19](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdadc9f0b-c3fb-49f0-b9ae-54c754c65aad%2F87811cc2-2da7-4311-8b1c-8e964710aafd%2Fixfmrb5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Separation of variables
The lateral radiation of heat from a bar of length L insulted at both ends can be modelled with
the linear partial differential equation (PDE)
Ou
8²u
<= D -au for 0<x<L, t>0,
02²
(1)
Ot
with the conditions
u, (0.1)=₂(L,t)=0 and u(,0) = f(x).
where Lo, DER are positive constants.
1. Seek a separation solution of the form u(x, t) = X(z)T(t) to show
T 0
DTD X
= k₁
where & denotes the separation constant.
] 2. Use equation (2) to derive two ordinary differential equations (ODEs), one in spacer and
one in time t.
3. Determine the boundary conditions for the ODE that depends on z.
4. Find the non-trivial solutions of X(r)= X(r) and corresponding values of the separation
constant k=k, for n= 0, 1, 2, 3.....
Hint: you may need to consider cach of the three causes k=-p², k = 0, k=p² (p0) to
find all the non-trivial solutions
5. Find the solutions T(t)= T(t), for n=0,1,2,3....
6. Use the superposition principle for linear PDEs to write down the solution for u that satisfies
the boundary conditions (0,t)=u₂(L,t) = 0.
7. Determine expressions for the unknown coefficients in your solution for that satisfies the
initial condition u(,0) = f(x).
8. Find the solution when f(x)=1+ cos rr, for values of L=1.0=1 and D = 1.
¹9. In a single figure, plot the solution found in Question 8 at times t=0,0.25, 0.5, 2. Briefly
describe the behaviour as t increases. What is the constant value temperature of the
environment that surrounds the bar?
19
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