Modeling Luminance Distribution in Optical Systems Using Fractional Boundary Value Problems via Green’s Function

This paper investigates the mathematical modeling of luminance distribution in optical systems through the framework of fractional boundary value problems (FBVPs). Starting from geometric optics principles, we formulate the luminance at an image point as an integral operator involving the object’s luminance and the optical device characteristics. We translate this physical problem into a fractional differential equation with boundary conditions, capturing intricate interactions in light propagation. Employing fractional calculus tools and integral equation theory, we establish the existence and uniqueness results for solutions using fixed-point methods, particularly the Banach contraction principle. Furthermore, the study explores Ulam–Hyers stability, demonstrating robustness of luminance distributions under small perturbations in system parameters. An illustrative example highlights applicability to realistic scenarios. Our results enrich the understanding of luminance proportionality between objects and their images and provide a solid analytical foundation for optical system design and analysis.