Using Legendre polynomials Formulas at Fresnel Integral Diffraction

Gomaa, Reham Abd Elsadek; Sayed, Ahmed Yehia; Shoukralla, Emil Sobhi; Hasan, Alfaisal Abd Elhameed; Saber, Nermin Abd Elsatar

doi:10.21608/jcsit.2024.332554.1012

Using Legendre polynomials Formulas at Fresnel Integral Diffraction

Document Type : Original Article

Authors

¹ Basic and applied science department, Faculty of Engineering, Arab academy for science, technology and maritime transport, Aswan, Egypt

² Physics and Engineering Mathematics Department, Faculty of Engineering at Mataria, Helwan University, Cairo, Egypt.

³ Faculty of Electronic Engineering, Menofia University, Egypt

⁴ Basic and Applied Sciences Department, College of Engineering and Technology, Arab Academy for Science, Technology and Maritime Transport (AASTMT), Aswan, Egypt

⁵ Faculty of Energy and Environmental Engineering, British University in Egypt.

10.21608/jcsit.2024.332554.1012

Abstract

Many advanced scientific phenomena of optics, light, and physics can be represented mathematically in the form of Volterra integral equations. In this paper, we studied the diffraction phenomena of the light beam. Since the most important category of scalar diffraction theories is the Fresnel diffraction integral. These integrals have been used in numerous studies on the propagation effects of structured light beams. These scalar diffraction theories have been widely utilized in

studying the propagation of structured light. finally, we applied first-kind shifted Legendre

polynomials to find the interpolate solutions of weakly singular Volterra integral equations of the

second kind, where the Fresnel integral of diffraction will be involved. Numerical examples have

been included in order to show the efficiency of the presented method. The exact solution of the represented example is compared to the approximate solution and the absolute error is calculated to illustrate the efficiency of the proposed method.

Keywords