18.2.1.3 Solution stanexl-b1

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

$$u = (y^2 + 1) x \qquad\mbox{for}\qquad x^2+y^2=1$$

To find the value of $u$ at the point (0.1,0.2), can I just plug in the coordinates of that point into the boundary condition? If not, what is the correct value of u at the point, and what would I get from the boundary condition?

Also answer the above questions for the following problem:

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

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:

First verify that the given boundary condition expression, $u$ $\vphantom0\raisebox{1.5pt}{$=$}$ $(y^2+ 1)x$ does not satisfy the Laplace equation. So this expression is not valid for $u$ inside the circle.

Of course, the value for $u$ at the given point might still be right by coincidence. To check that, first verify that the correct solution to the problem is

$$u={\textstyle\frac{5}{4}} x + \frac 34 x y^2 - {\textstyle\frac{1}{4}} x^3$$

Do so by plugging it into the partial differential equation and boundary condition. Then compute the value of $u$ at the given point and compare with what you would get from the given boundary condition.

Next verify that in the second case, the function given for $u$ on the boundary does satisfy the Laplace equation. So it must be the correct solution $u$ at all $x$ and $y$.

So, now just plug in the given coordinates.