A Computational Study of Radiative Transfer‐Type Equations Involving Fractional Derivatives and Integral Memory Terms

Source abstract

ABSTRACT In this work, we develop and analyze a numerical framework for a time‐fractional partial integro‐differential equation arising in radiative transfer phenomena. Radiative transfer models are essential for describing the propagation of radiation in complex media, where memory and nonlocal effects often play a significant role. To capture these effects, fractional order formulations have emerged as powerful alternatives to classical models. The existence and uniqueness of the solution are rigorously established, providing a solid theoretical foundation for the model. A stable and accurate numerical scheme is proposed for the considered problem. The scheme employs the L1 approximation on a graded temporal mesh to effectively handle initial time singularities. In addition, a central difference method is used for spatial discretization, while a composite trapezoidal rule is applied to approximate the integral term. Convergence analysis confirms the reliability of the method and provides explicit error bounds. Numerical experiments demonstrate the accuracy, efficiency, and robustness of the proposed approach, highlighting its effectiveness in capturing memory‐driven radiative transport processes. Numerical results confirm the physically consistent influence of the scattering coefficient on radiative attenuation. The results indicate that the method is well‐suited for practical simulations of fractional radiative transfer problems encountered in physics and engineering applications.