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

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SUMMARY

The discussion focuses on solving the steady state temperature distribution $u(x,y)$ in a rectangular plate defined by the Laplace equation $\nabla^2u = 0$. The boundary conditions include a fixed temperature of 0 degrees at $x = 0$, a linear temperature profile $u(L,y) = y$ at $x = L$, and insulated edges at $y = 0$ and $y = M$. It is clarified that insulated boundaries imply zero derivatives, affecting the Fourier coefficients of sine terms, rather than the coefficients themselves being zero.

PREREQUISITES
  • Understanding of Laplace's equation and its applications in heat distribution.
  • Familiarity with boundary value problems in partial differential equations.
  • Knowledge of Fourier series and their role in solving differential equations.
  • Basic concepts of thermal insulation and its implications in mathematical modeling.
NEXT STEPS
  • Study the derivation and solutions of Laplace's equation in rectangular domains.
  • Learn about boundary value problems and their significance in physics and engineering.
  • Explore Fourier series, particularly focusing on sine and cosine expansions.
  • Investigate the implications of insulated boundaries in heat transfer problems.
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Mathematicians, physicists, and engineers involved in thermal analysis, particularly those working with heat distribution in solid bodies and solving partial differential equations.

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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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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.
 
Wasn't thinking figured out this post.
 
Last edited:

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