Photonic computing can offer huge bandwidth to overcome the electronic bottleneck of electronic circuits, and it also plays important roles in all optical networks. Integrated photonic waveguide is becoming one of the most promising resources to implement photonic computing due to the compact size, light weight, and low power consumption. Photonic differentiator and integrator based on integrated photonic waveguide attract more and more attentions in recent years. In this paper, we review several schemes of ultrafast photonic differentiator and differential equation solver employing integrated silicon cascaded or doped microring an d MechZehnder interferometer to realize high-order photonic differentiator, tunable fractional-order differentiator, high-order all-optical differential equation solver and all-optical constant-coefficient tunable differential equation solver. And they are one of the most important development directions of the optical differential and integral.