Deriving and Proving the Heat Equation's Integral Product

In summary, the conversation discusses using separation of variables to derive Xn(x) = cos((n+1/2)\pix) for a heat equation with boundary conditions. It also explores using the product to sum formula and the identity \cos^2(\theta) = \frac {1 + \cos(2\theta)}{2} to show that \int_{0}^1 Xn(x)Xm(x) dx = 1/2 if m = n and 0 if m \neq n. The conversation ends with the advice to use the fact that n and m are integers in order to integrate the remaining term.
  • #1
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Consider a heat equation for the temperature u of a rod of length 1:
ut = uxx, 0 < x < 1, t > 0 with boundary conditions ux(0,t) = 0 & u(1,t) = 0. I derived Xn(x) = cos((n+1/2)[tex]\pi[/tex]x) using separation of variables.
How do I show that [tex]\int_{0}^1[/tex] Xn(x)Xm(x) dx = 1/2 if m = n and 0 if m [tex]\neq[/tex] n.
I used the product to sum formula: cos(A)cos(B) = cos(A+B)/2 + cos(A-B)/2 to get 1/2cos((n+m+1)[tex]\pi[/tex]x) 1/2cos((n-m)[tex]\pi[/tex]x) but I am stuck after that. Someone help, am I even on the right track.
 
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  • #2
Yes, you are on the right track. When m = n use the formula

[tex]\cos^2(\theta) = \frac {1 + \cos(2\theta)}{2}[/tex]

which is easy to integrate. I'm not sure why you are stuck on the others.

[tex]\frac 1 2 \cos((n+m+1)\pi x)[/tex]

is just as easy to integrate is [itex]\cos(kx)[/itex].

n and m are integers, you know.
 

1. What is the heat equation's integral product?

The heat equation's integral product is a mathematical expression that describes how heat is transferred and distributed in a given physical system. It is derived from the fundamental laws of thermodynamics and can be used to model and predict the behavior of heat in various systems.

2. What is the process for deriving the heat equation's integral product?

The process for deriving the heat equation's integral product involves using the laws of thermodynamics, specifically the first and second laws, to derive an expression for the rate of change of heat in a system. This expression is then integrated over time to obtain the integral product.

3. How is the heat equation's integral product used in real-world applications?

The heat equation's integral product has many practical applications, such as in the fields of engineering, physics, and chemistry. It is used to model heat transfer in various systems, such as in materials, fluids, and biological systems. This allows for the prediction and optimization of heat-related processes, such as heating and cooling systems, thermal insulation, and chemical reactions.

4. What is the significance of proving the heat equation's integral product?

Proving the heat equation's integral product is important because it verifies the validity of the mathematical expression and its derivation. It also allows for a deeper understanding of the underlying principles and laws that govern heat transfer. In addition, it provides a basis for further research and developments in the field of thermodynamics.

5. Are there any limitations to the heat equation's integral product?

Like any mathematical model, the heat equation's integral product has its limitations. It assumes certain idealized conditions, such as constant temperature and uniform properties, which may not always be true in real-world situations. It also does not take into account other factors that may affect heat transfer, such as radiation and convection. Therefore, it should be used with caution and its predictions should be verified through experimentation.

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