Digital inpainting methods provide an important tool in the restoration of images in a wide range of applications. We present mathematical methods with certain higher order partial differential equations for the inpainting of ancient frescoes. In particular we discuss the Cahn-Hilliard equation for the inpainting of binary structure and a higher order total variation approach. As an example for the preformance of our algorithms we consider the recently discovered Neidhart frescoes in Vienna. Keywords: Image interpolation, Cahn-Hilliard equation, total variation minimization, fresco restoration