Temperature distribution - PDE's

In summary, the problem asks to find the temperature distribution in a square with insulated sides and a known temperature distribution along one side. The equilibrium distribution of temperature on a plate satisfies the Laplace equation. The boundary conditions are given as partial derivatives with respect to x and y, and can be solved using the separation of variables method.
  • #1
Anabelle37
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0

Homework Statement



Consider a square of unit length which is insulated on the sides x=0, x=1 and y=0 and has the temperature distribution u(x,1)=x along the side y=1. find the temperature distribution u in the square.

Homework Equations





The Attempt at a Solution



So I have the boundary conditions:
du(0,y)/dx = 0; du(1,y)/dx = 0; du(x,0)/dx = 0; u(x,1)=x (where these are partial derivatives)

I am unsure where to go from here...do i solve laplaces equation? if so I don't know how to put the boundary conditions in terms of the separation of variables method.

please help!
 
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  • #2
Yes, the equilibrium distribution of temperature on a plate satisfies [itex]\nabla^2 u(x, y)= 0[/itex].

If you write u(x, y)= X(x)Y(y), then
[tex]\frac{\partial u}{\partial x}= Y(y)\frac{dX}{dx}[/tex]
so
[tex]\frac{\partial u(0, y)}{\partial x}[/tex][tex]= Y(y)\frac{dX(0)}{dx}= 0[/tex]
so either Y(y)= 0 for all y, which gives only the trivial solution, u= 0 for all x and y or
[tex]\frac{dX(0)}{dx}= 0[/tex]
That's your boundary condition on X.

Similarly for the other two.
 
  • #3
ok thanks,
do i partial du/dx for the other 2? or du/dy?
 

1. What is a PDE?

A PDE, or partial differential equation, is a mathematical equation that involves functions of multiple variables and their partial derivatives. It is commonly used to describe physical phenomena such as temperature distribution.

2. How is temperature distribution modeled using PDE's?

Temperature distribution is commonly modeled using the heat equation, which is a type of PDE that describes how temperature changes over time and space. It takes into account factors such as heat sources, heat conductivity, and boundary conditions.

3. What are some practical applications of temperature distribution modeling using PDE's?

PDE's are widely used in fields such as physics, engineering, and environmental sciences to study temperature distribution in various systems. Some practical applications include predicting the temperature distribution in buildings, analyzing heat transfer in manufacturing processes, and studying the effects of climate change on global temperature patterns.

4. What are some numerical methods used to solve PDE's for temperature distribution?

Some commonly used methods for solving PDE's for temperature distribution include finite difference methods, finite element methods, and spectral methods. These methods involve discretizing the PDE into a set of algebraic equations that can be solved using computational techniques.

5. How accurate are PDE models of temperature distribution?

The accuracy of PDE models depends on various factors such as the complexity of the system being studied, the quality of data used, and the numerical methods employed. With proper calibration and validation, PDE models can provide accurate predictions of temperature distribution in various systems.

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