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Nonlinear Volterra integral equations involve both nonlinear functions and integrals. The general form of a non-linear Volterra equation of the second kind has the form t y(t) = f(t) + 0 K(s,t)ϕ(y(s))ds, (1) where the used parameters are as follows: source f ∈ C1[0,T], kernel K ∈ C2([0,T]×[0,T]), ϕ ∈ C1[R] are known and function y is unknown. The equations of this type are used in various f ields such as physics, biology, economics, and engineering to model complex systems that involve memory effects or interactions between different components. This research is a smart combination of the two methods: the F-transform and Fixed Point Theorem to approach a numerical solution of the nonlinear Volterra integral equation [2]. We propose to compute an approximate solution to the equation (1) using the F-transform [37]. In details, we construct an operational matrix for the Volterra integral equation as a Hadamard product of two matrices, one of which refers to the Volterra operator, and the other to its kernel. By this, the entire equation can be reduced to a simpler form. This makes our method efficient and low computational. The supporting statements, including the convergence of the method, and estimate its computational complexity will be provided. Additionally, in order to guarantee that the nonlinear equation (1) has a unique solution, we show the applicability of the Fixed Point Theorem, which plays a key role in solving many problems in applied mathematics such as neutron transport, population biology, economics, applied mechanics, etc, see [8–10]. In parallel, our methodology is validated through the development of software that confirms the efficacy of our method on a range of examples. Our work demonstrates the joint progress of theoretical results and the development of practical software for solving complex mathematical problems.
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