This can be done numerically in Mathematica. Here is an example.

    sol = NDSolve[{Laplacian[u[x, y], {x, y}] == 0, DirichletCondition[u[x, y] == 0, x <= 0.35], 
    DirichletCondition[u[x, y] == 1, x >= 0.35]}, u, {x, y} \[Element] Disk[]]
    ContourPlot[u[x, y] /. sol, {x, -1, 1}, {y, -1, 1}]
[![enter image description here][1]][1]

  
Edit. A typo in the code. The right picture instead of an incorrect one.

  [1]: https://i.sstatic.net/ReaKf.png