This paper introduces the concept of on-chip temporal optical computing, based on dispersive Fourier transform and suitably designed modulation module, to perform mathematical operations of interest, such as differentiation, integration, or convolution in time domain. The desired mathematical operation is performed as signal propagates through a fully reconfigurable on-chip photonic signal processor. Although a few number of photonic temporal signal processors have been introduced recently, they are usually bulky or they suffer from limited reconfigurability which is of great importance to implement large-scale general-purpose photonic signal processors. To address these limitations, this paper demonstrates a fully reconfigurable photonic integrated signal processing system. As the key point, the reconfigurability is achieved by taking advantages of dispersive Fourier transformation, linearly chirp modulation using four wave mixing, and applying the desired arbitrary transfer function through a cascaded Mach-Zehnder modulator and phase modulator. Our demonstration reveals an operation time of $200~ps$ with high resolution of $300~fs$. To have an on-chip photonic signal processor, a broadband photonic crystal waveguide with an extremely large group-velocity dispersion of $2.81 \times {10^{6}}~\frac{ps^2}{km}$ is utilized. Numerical simulations of the proposed structure reveal a great potential for chip-scale fully reconfigurable all-optical signal processing through a bandwidth of $400~GHz$.