Representation formula for the Laplace equation

Question

I would like to know how to use the representation formula for the Laplace equation to actually compute values of a solution $u(\mathbf{x})$ in the interior of a domain $\Omega \subset \mathbb{R}^2$ (note here that $\mathbf{x} = (x_1,x_2) \in \mathbb{R}^2$). The representation formula is given by $$ u(\mathbf{x}) = \int_{\partial \Omega} \left( E(\mathbf{x} - \mathbf{y}) {\partial u \over \partial \mathbf{n}}(\mathbf{y}) - u(\mathbf{y}) {\partial E \over \partial \mathbf{n}} (\mathbf{x} - \mathbf{y})\right)\, ds(\mathbf{y}), $$ where $E(\mathbf{x} - \mathbf{y})$ is the fundamental solution $$ E(\mathbf{x} - \mathbf{y}) = \frac{1}{2 \pi} \, \ln(|\mathbf{x}-\mathbf{y}|). $$ Suppose that I am given a parametrization of the boundary $\partial \Omega$, say $(t,p(t))$ where $t \in [a,b]$. Let's suppose that I am given both $f(\mathbf{y}) = u(\mathbf{y})$ and $g(\mathbf{y}) = {\partial u \over \partial \mathbf{n}}(\mathbf{y})$ on the boundary $\partial \Omega$, i.e. I am given both the Dirichlet and the Neumann boundary values. Also assume that the boundary values $f$ and $g$ are both parametrized with respect to the same variable $t \in [a,b]$. Now how do I use this information to actually compute the solution $u(\mathbf{x})$ for $\mathbf{x}$ in the interior of $\Omega$?