A New Modification of the First Fredholm Fundamental Theorem for Analytical Solutions of Neutrosophic Fuzzy Fredholm Integral Equations With Separable Kernels

The fuzzy Fredholm integral equation is a fundamental equation that describes a wide range of phenomena in physics, chemistry, and fluid dynamics. Fuzzy neutrosophic numbers offer an effective way to handle the uncertainty in fuzzy Fredholm integral equations, helping identify the most suitable variables and factors for solving them. Therefore, in this article, the fuzzification and defuzzification of the Fredholm integral equation in a fuzzy neutrosophic environment are presented in detail, involving truth (T), indeterminacy (I), and false (F) components. Moreover, a new modification of the first Fredholm fundamental method is developed and implemented to solve the neutrosophic fuzzy second-kind Fredholm integral equation where the triangular neutrosophic number is used on the α, β, γ-cut level. Finally, numerical examples are carried out and solved using our proposed technique, and the results obtained indicate the effectiveness and feasibility of our presented approach.