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
Journal of Communication Sciences and Information Technology
Volume 6, Issue 1, December 2024, Page 11-18 PDF (502.51 K)
Document Type: Original Article
DOI: 10.21608/jcsit.2024.332554.1012
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Authors
Reham Abd Elsadek Gomaa ¹; Ahmed Yehia Sayed ²; Emil Sobhi Shoukralla³; Alfaisal Abd Elhameed Hasan⁴; Nermin Abd Elsatar Saber⁵
¹Basic and applied science department, Faculty of Engineering, Arab academy for science, technology and maritime transport, Aswan, Egypt
²Physics and Engineering Mathematics Department, Faculty of Engineering at Mataria, Helwan University, Cairo, Egypt.
³Faculty of Electronic Engineering, Menofia University, Egypt
⁴Basic and Applied Sciences Department, College of Engineering and Technology, Arab Academy for Science, Technology and Maritime Transport (AASTMT), Aswan, Egypt
⁵Faculty of Energy and Environmental Engineering, British University in Egypt.
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
Volterra Integral equations; Fresnel diffraction integral; Diffraction theory; Legendre polynomials


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