"تحليل ودراسة المعادلات التكاملية باستخدام طريقة آدوميان: تطبيق على معادلات فوليترا" (Comprehensive Journal of Science, 2025)
Integral equations represent one of the most important mathematical tools used to describe various physical, engineering, and biological phenomena. They express the relationship between an unknown function and an integral containing the same function. Due to the difficulty of obtaining exact analytical solutions, several mathematical methods have been developed to approximate solutions. Among these, the Adomian Decomposition Method (ADM) has proven to be highly effective in solving both linear and nonlinear integral equations.This paper aims to analyze and study integral equations using the Adomian Decomposition Method, with a specific application to Volterra integral equations, in order to illustrate the mechanism of the method and its efficiency in obtaining accurate and rapidly convergent approximate solutions. The study also discusses the classification of integral equations into their main types, such as Fredholm and Volterra equations, and presents the iterative form of the Adomian method and its role in simplifying complex integral problems.The results show that the Adomian method provides an efficient and practical approach to solving integral equations without requiring complex initial assumptions or excessive mathematical simplifications, making it a valuable tool in mathematical modeling and modern engineering applications.
Keywords: Integral equations, Adomian decomposition method, Volterra equations, Fredholm equations, analytical methods, approximate solutions.