Solving Eigenvalue & Eigenfunction of 1D Heat Equation

In summary, the conversation is about deriving the eigenvalue of the heat equation using complex analysis. The variables are separated into u(x,t) = X(x)T(t) = XT and the equations T' + λkT = 0 and X'' + λX = 0 are obtained. It is shown that λ cannot equal 0 as it only yields elementary solutions. The Auxiliary Equation (AE) is r = ± i*√(λ) and the solution for X is Acosh(rx) + Bsinh(rx) and for X' is Asinh(rx) + Bcosh(rx). The solution is questioned and other methods are suggested, such as using laplace transforms.
  • #1
Powertravel
9
0
Hi,
I am struggling with the heat equation
ut = kuxx
with the boundary conditions
u(0,t) = u'(L,t) = 0
and initial condition
u(x,0) = f(x)
0 ≤ x ≤ L
0 ≤ t

I want to derive it's eigenvalue using complex analysis.

After separating the variables into u(x,t) = X(x)T(t) = XT and getting
T' + λkT = 0 (1)
X'' + λX = 0 (2)

I start with (2);

It is easily shown that λ ≠ 0 because it only yields elementary solutions.

Equation (2)s Auxiliary Equation (AE) is
r = ± i*√(λ)

So
X = Acosh(rx) + Bsinh(rx)
and
X' = Asinh(rx) + Bcosh(rx)

Using the Boundrary Conditions I get
X(0)=0 [itex]\Rightarrow[/itex] A = 0 and B ≠ 0
X'(L) = 0 [itex]\Rightarrow[/itex] cosh(rx) = 0
so
X = Bsinh(rx)

Now comes my first question:

My textbook says that cosh(rx) only is zero when
λ = (((2n -1)[itex]\pi[/itex])/(2*L))2 n = 1,2,3,... (3)
Can't it also be
λ = (((1 + 2n)[itex]\pi[/itex])/(2*L))2 n = 0,1,2,... ? (4)
Why is it as (3) instead of (4) and will (4) cause problems if I want to expand u(x,t) in a sine series?

Now for my second and primary question.

Using (3) I get
i*λL = n*[itex]\pi[/itex] - 0.5[itex]\pi[/itex] [itex]\Rightarrow[/itex]
√(λ) = ((2n -1)[itex]\pi[/itex]) / (2Li) = -i((2n -1)[itex]\pi[/itex])/(2L) [itex]\Rightarrow[/itex]
λ = (-((2n -1)[itex]\pi[/itex])/(2L))2
That means λ < 0 so √(-λ) is real.
How can I get the eigen function of λ equal Csin(√(-λ)x) when X = sinh(√(-λ)x) ?

Thanks in advance.
 
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  • #2


Powertravel said:
Hi,
I am struggling with the heat equation
ut = kuxx
with the boundary conditions
u(0,t) = u'(L,t) = 0
and initial condition
u(x,0) = f(x)
0 ≤ x ≤ L
0 ≤ t

I want to derive it's eigenvalue using complex analysis.

After separating the variables into u(x,t) = X(x)T(t) = XT and getting
T' + λkT = 0 (1)
X'' + λX = 0 (2)

I start with (2);

It is easily shown that λ ≠ 0 because it only yields elementary solutions.
"
Equation (2)s Auxiliary Equation (AE) is
r = ± i*√(λ)

So
X = Acosh(rx) + Bsinh(rx)
and
X' = Asinh(rx) + Bcosh(rx)"

Are you sure about this solution? the solutions for ODE's are in the form of e^(rx), so e^(ix)... is correlating to cos(x)? correct? Check that

Alternatively you could solving this using laplace transforms, which much easier, and evaluate the poles accordingly, s, functions, using the residual formulature and going from there.

YS
 

Related to Solving Eigenvalue & Eigenfunction of 1D Heat Equation

1. What is the 1D heat equation?

The 1D heat equation is a partial differential equation that describes the flow of heat in a one-dimensional system, such as a rod or wire. It takes into account the temperature distribution along the length of the system and the rate of change of temperature over time.

2. What are eigenvalues and eigenfunctions?

Eigenvalues and eigenfunctions are important concepts in linear algebra and differential equations. Eigenvalues are the values that satisfy a specific equation, while eigenfunctions are the corresponding functions that satisfy that equation. In the case of the 1D heat equation, the eigenvalues and eigenfunctions describe the possible temperature distributions and how they change over time.

3. Why is solving for eigenvalues and eigenfunctions important in the 1D heat equation?

Solving for eigenvalues and eigenfunctions allows us to find the specific solutions to the 1D heat equation for a given system. This is important because it helps us understand how the temperature in the system will change over time, and can also help us predict and control the behavior of the system.

4. What methods can be used to solve for eigenvalues and eigenfunctions in the 1D heat equation?

There are several methods that can be used to solve for eigenvalues and eigenfunctions in the 1D heat equation, including separation of variables, Fourier series, and the method of eigenfunction expansion. Each method has its own advantages and limitations, and the choice of method often depends on the specific problem at hand.

5. Can the 1D heat equation be applied to real-world systems?

Yes, the 1D heat equation is commonly used in various fields such as physics, engineering, and materials science to model and analyze the behavior of real-world systems. It has been successfully applied to problems such as heat transfer in pipes, thermal conductivity in materials, and temperature distribution in electronic circuits.

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