The square problem can also be fitted using boundary collocation. For this we need functions that satisfy the Laplace equation in an exact manner. The solution can then be expanded in terms of these functions. Let us see whether we find suitable functions.

Prove that the real and imaginary parts of \(F(z)\), where \(z\) is a complex variable, satisfy the Laplace equation.

We seek polynomial functions for simplicity. Thus we seek \(z^{n}\), where \(z=x+i y\), as the trial functions. Generate a set of functions for various values of \(n\). For example, if \(n=2\) then show that \(x^{2}-y^{2}\) and \(x y\) are functions that satisfy the Laplace equation. In this way we can generate a whole set of basis functions.

For the square problem only even functions should be considered. Thus \(x^{2}-y^{2}\) is a good function, whereas \(x y\) is not. Similarly, so are some functions generated by \(z^{4}\), but not those generated by \(z^{3}\). Generate a set of, say, six such functions.

Use these functions to fit the boundary values of temperature in a least-square sense, which is the required solution by boundary collocation.