Finding Temperature in a Rectangular Plate: $\nabla^2u=0$

In summary, the problem involves finding the steady state temperature $u(x,y)$ in a rectangular plate with dimensions $0\leq x\leq L$, $0\leq y\leq M$, where the boundary conditions are $u(0,y) = 0$, $u(L,y) = y$, and the edges $y = 0$ and $y = M$ are insulated. The appropriate differential equation for this problem is $\nabla^2u = 0$. It should be noted that the insulated boundaries do not necessarily mean that the Fourier coefficients are 0, but rather that the derivatives at these boundaries are 0.
  • #1
Dustinsfl
2,281
5
The steady state temperature $u(x,y)$ in a rectangular plate $0\leq x\leq L$, $0\leq y\leq M$, is sought, under the condition that the edge $x = 0$ is maintained at zero degrees, $x = L$ is kept at $u(L,y) = y$ degrees, and the edges $y = 0$ and $y = M$ are insulated. The appropriate differential equation $\nabla^2u = 0$.

Since the vertical boundary conditions are insulated, wouldn't this be the same as just dealing with $u_t=u_{xx}$ since Fourier coefficients for those boundaries will be 0?
 
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  • #2
Saying that the boundaries are insulated does NOT mean that the Fourier coefficients are 0. It means that the derivatives there are 0 and so the coefficients of the sine terms are 0.
 
  • #3
Wasn't thinking figured out this post.
 
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FAQ: Finding Temperature in a Rectangular Plate: $\nabla^2u=0$

1. What is the mathematical equation for finding temperature in a rectangular plate?

The mathematical equation for finding temperature in a rectangular plate is given by $\nabla^2u=0$, where $\nabla^2$ is the Laplace operator and $u$ represents the temperature function.

2. What does the Laplace operator represent in the equation for finding temperature in a rectangular plate?

The Laplace operator, denoted by $\nabla^2$, represents the sum of the second-order partial derivatives with respect to the spatial coordinates. In this case, it represents the rate of change of temperature in the rectangular plate.

3. How is the solution to the equation $\nabla^2u=0$ found?

The solution to the equation $\nabla^2u=0$ is found by using separation of variables and solving for the temperature function $u(x,y)$ in terms of the boundary conditions of the rectangular plate.

4. What are the boundary conditions for solving the equation $\nabla^2u=0$ in a rectangular plate?

The boundary conditions for solving the equation $\nabla^2u=0$ in a rectangular plate typically include the temperature values at the edges of the plate, as well as any heat sources or sinks present in the system.

5. Can the equation $\nabla^2u=0$ be applied to non-rectangular plates?

Yes, the equation $\nabla^2u=0$ can be applied to non-rectangular plates as long as the appropriate boundary conditions are specified for the specific shape of the plate.

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