Detection of a Spatial Source Term Within a Multi-Dimensional, Multi-Term Time-Space Fractional Diffusion Equation

The main objective of this study was to identify the undetermined source term (ST) in a fractional space-time scattering equation with multiple terms, using data obtained from the most recent observations. To address this complex problem, we reformulated the equation by adopting a regularization-based optimization approach. This methodology not only makes it possible to determine the existence of a single minimum solution, but also to assess its stability. In the numerical context, we estimate and approach the function (ST) by applying the Levenberg–Marquardt regularization method, a powerful tool for solving inverse problems. In order to demonstrate the effectiveness of the proposed approach, we performed numerical simulations in one-dimensional and two-dimensional scenarios. These simulations illustrate our method’s ability to process complex data and provide accurate and stable solutions. Through this extended approach, we aimed to discover the single source term in a multi-term space-time fractional scattering equation, ensuring robust and reliable results, supported by the most recent observational data.