# Boundary Value Problem from Laplace's eq (Thermal)

lanan

## Homework Statement

A rectangular plate extends to infinity along the y-axis and has a width of 20 cm. At all faces except y=0, T= 0°C. Solve the semi-infinite plate problem if the bottom edge is held at
T = {0°C when, 0 < x < 10,
T = {100°C when, 10 < x < 20.

2T=0

## The Attempt at a Solution

I generalized Laplace's equation for two dimensions, then used separation of variables to find solutions of the form T(x,y)=X(x)Y(x) such that
Y = {eky , e-ky }
X = {sinkx, coskx }

Then, by applying two of the initial conditions required to satisfy T (y→∞ T=0 and x=0 T=0) I was able to eliminate one of the possible choices for X and Y each, leaving a T of:

T=e-kysinkx.

Here is where I'm having trouble. In other problems from this section I could continue to apply the boundary conditions and solve for k then use a fourier series approximation to solve the remaining condition where T=100°C. However, with x broken into several more cases I am unsure how to solve for a k that works for all of the conditions (for which there are a confusing six, rather than the previous 3 I solved). Any guidance would be much appreciated.

Mentor
Hi lanan. Welcome to Physics Forums.

I don't consider myself an expert on this type of problem, but the first thing I would do would be to shift the origin to x = 10, so that the region of interest is from x = -10 to x = + 10. The base temperature would then be 0 from x = -10 to x = 0, and 100 from x = 0 to x = +10. I would then represent the solution as the linear superposition of two solutions. In solution 1, T = 50 over the entire base from x = -10 to x = +10. This could be described by a cosine series. In solution 2, the base temperature would be T = -50 from x = -10 to x = 0, and T =+50 from x = 0 to x = +10. This could be described by a sine series. This approach should simplify things considerably (I think).

Chet

lanan
In solution 1, T = 50 over the entire base from x = -10 to x = +10. This could be described by a cosine series. In solution 2, the base temperature would be T = -50 from x = -10 to x = 0, and T =+50 from x = 0 to x = +10. This could be described by a sine series.

Chet

Thank you, this is helping a little. However, this is my first time actually solving a physical problem with difeq, so I'm not sure how to apply multiple boundary conditions over x for these two circumstances (do I just use 20?). Also, I'm not sure why you used T=50 over the entire base for solution one. Also, do I construct the same T for each solution? I'm really quite lost.

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Staff Emeritus
$$T(x,y) = \sum_n c_nT_n(x,y) = \sum_n c_ne^{-k_n y}\sin k_n x.$$ You now apply the boundary condition at y=0 to solve for the ##c_n##'s.