Heat Diffusion Equation - Using BCs to model as an orthonormal system

In summary, the conversation discusses using the sine Fourier series on a given interval to show that ##\sqrt\frac{2}{a} = \frac{1}{a} \int_0^{2a} Sin(q_kx)g_k dx##, which leads to the conclusion that it is an orthonormal basis. The conversation also mentions a potential method for parts c and d, which involves showing that ##\mathrm{d}^2/\mathrm{d} x^2## is a self-adjoint operator on the Hilbert space and clarifies that the sum should be over k, not n.
  • #1
physconomic
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1
Homework Statement
Consider heat equation
##\kappa \frac {\partial^2 \psi} {\partial x^2} = \frac{\partial \psi}{\partial t}##


## \kappa ## is positive, ## x \subset [0,a] ##, ##\psi## is real

For ##t>0##: ##\psi(t,0) = \psi_0## at ##x=0##

##\frac{\partial \psi}{\partial x}(t,a) = 0## at ##x=a##

##\psi(0,x) = 0##



We introduce ##g_k(x) = \sqrt\frac{2}{a} sin(q_k x)##

where ## q_k = \frac{\pi}{a}(k + \frac{1}{2}) ##

##k = 0, 1, ...##



b) Argue that the functions ##g_k## form an ortho-normal basis of the space ##L_b ^2 ([0, a])##, of square integrable functions ##f## on ##[0, a]## with a Dirichlet boundary condition ##f(0) = 0 ## at ## x = 0## and a von Neumann condition ##f'(a) = 0## at ##x = a##.



c) Based on the results in (a) (I've done this part - it's ##\frac{d^2}{dx^2}g_k = -q_k^2g_k##) and (b) argue that the most general ##\psi## with the correct boundary conditions can be written as ##\psi(t, x) = \psi_0+ \Sigma_0^\inf T_k(t)g_k(x)##. Find the solutions for the functions ##T_k##.



d) Fix the remaining constants in your solution by imposing the initial condition. Compute the average value ##\psi_{avg}(t)## of ##\psi(x, t)## by averaging over x ∈ [0, a] and find an approximate equation for the time as a function of## r := (\psi_0 − \psi_{avg}(t))/\psi_0##.
Relevant Equations
Fourier series, Dirichlet, Von Neumann
I've tried to show b) by using the sine Fourier series on ##[0,2a]##, to get ##g_k = \Sigma_{n=0}^{2a} \sqrt\frac{2}{a} Sin(q_k x)##

Therefore ##\sqrt\frac{2}{a} = \frac{1}{a} \int_0^{2a} Sin(q_kx)g_k dx##

These are equal therefore it is an orthonomal basis.

I'm not sure if this is correct so it would be great if somebody could help me by checking it and also letting me know how I could go about doing parts c and d.

Thank you
 
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  • #2
For (b) I'd rather argue by showing that ##\mathrm{d}^2/\mathrm{d} x^2## is a self-adjoint operator on the said Hilbert space. I'm also not sure what you mean by the sum over ##n##.
 
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  • #3
vanhees71 said:
For (b) I'd rather argue by showing that ##\mathrm{d}^2/\mathrm{d} x^2## is a self-adjoint operator on the said Hilbert space. I'm also not sure what you mean by the sum over ##n##.

Thanks for your reply. Can I ask how I would do this? I meant sum over k.
 

Related to Heat Diffusion Equation - Using BCs to model as an orthonormal system

1. What is the heat diffusion equation?

The heat diffusion equation is a mathematical model that describes the flow of heat through a medium over time. It is based on the principle of conservation of energy and is commonly used in physics and engineering to analyze heat transfer processes.

2. How is the heat diffusion equation solved?

The heat diffusion equation can be solved using various methods, including numerical methods such as finite difference, finite element, and boundary element methods. Analytical solutions can also be obtained for simple geometries and boundary conditions.

3. What are boundary conditions in the heat diffusion equation?

Boundary conditions in the heat diffusion equation refer to the conditions that are specified at the boundaries of the system being modeled. These conditions can include the initial temperature distribution, the temperature at the boundaries, and the rate of heat transfer at the boundaries.

4. How can boundary conditions be used to model a system as an orthonormal system?

Boundary conditions can be used to model a system as an orthonormal system by specifying the temperature distribution and heat transfer rates at the boundaries in terms of orthogonal functions. This allows for the use of Fourier series or other orthogonal expansions to solve the heat diffusion equation.

5. What are some applications of the heat diffusion equation?

The heat diffusion equation has many applications in various fields, including heat transfer in buildings, thermal analysis of electronic devices, and geothermal energy systems. It is also used in materials science to study the behavior of materials under different thermal conditions.

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