<p>The interaction of photons within a medium involves a range of complex processes, including scattering, absorption, and re-emission, all of which are effectively described by the radiative transfer equation (RTE). This study aims to provide a comprehensive solution to the RTE by examining four distinct types of scattering: isotropic, linear anisotropic, pure quadratic, and Rayleigh. To achieve this aim, we utilize two advanced and well-established computational techniques: the Legendre polynomial (P_N) method and the first-type Chebyshev polynomial (T_N) method. These methods were chosen for their ability to accurately and efficiently approximate solutions to the RTE across various scattering scenarios. The discrete eigenvalues of the RTE are computed using Wolfram Mathematica, ensuring high precision in the numerical results. The computational framework is specifically designed to address the challenges posed by various scattering types, with a focus on the convergence of solutions as the number of iterations increases. In this study, solutions are computed up to the 13th iteration, enabling a thorough analysis of the convergence behaviour and accuracy of the methods. Additionally, the results are validated through comparison with established benchmark datasets from existing literature, further reinforcing the credibility and reliability of the computational methods employed in this research. Ultimately, this work contributes to the advancement of computational methodologies in the study of photon transport, establishing a solid foundation for future innovations and applications across various scientific domains. The findings of this research represent a significant step toward enhancing the accuracy and efficiency of radiative transfer models, which are crucial for understanding complex physical systems and phenomena.</p>