1.2.1.4 Solution stanexl-b2

Question:

Consider the Laplace equation within a unit circle:

$$u_{xx} + u_{yy} = 0 \qquad\mbox{for}\qquad x^2+y^2<1$$

The boundary condition on the perimeter of the circle is

$$u = 2 + 3x + 5 y \qquad\mbox{for}\qquad x^2+y^2=1$$

Find the value of $u$ at the point (0.1,0.2). Fully defend your solution.

Answer:

Verify that in this case, the function given for $u$ on the boundary also satisfies the Laplace equation. So it is the solution $u$ at all $x$ and $y$.

So, just plug in the given coordinates.