This paper discusses iterative procedures for solving nonlinear integral equations, focusing on numerical methods for solving finite systems of nonlinear algebraic or transcendental equations. The author reviews conventional techniques for solving such systems and proposes a new approach to overcome their limitations. The problem involves solving a class of nonlinear integral equations that arise in kinetic theory of gases. The equations are discretized using Gaussian quadrature and Chebyshev polynomials, leading to a finite system of nonlinear equations. The solution requires an efficient iterative method due to the high computational cost of evaluating the kernel function. The paper surveys various numerical procedures for solving nonlinear equations, highlighting their deficiencies in this context. It proposes a class of iterative procedures based on heuristic considerations, which are illustrated with simple examples. These procedures aim to accelerate the convergence of iterative vector sequences. The paper also discusses the use of relaxation methods, Newton-Raphson-like methods, and sequence transformations such as the Aitken δ²-process and Wynn ε-algorithm. However, these methods are found to be insufficient for the class of problems under consideration due to the strong local coupling of variables and the need for high computational efficiency. The paper concludes that a dynamic, low-degree, coupled iterative process is needed to accelerate the convergence of the basic iteration. A class of such processes is suggested on a heuristic basis, with the extrapolation algorithm being particularly effective. The algorithm is shown to be useful in solving both linear and nonlinear equations arising from integral, ordinary, and partial differential equations. The paper also notes that while these methods are not guaranteed to converge globally, they can be effective in practice for the problems considered. The results of applying these methods to several examples are presented, demonstrating their effectiveness in accelerating convergence.This paper discusses iterative procedures for solving nonlinear integral equations, focusing on numerical methods for solving finite systems of nonlinear algebraic or transcendental equations. The author reviews conventional techniques for solving such systems and proposes a new approach to overcome their limitations. The problem involves solving a class of nonlinear integral equations that arise in kinetic theory of gases. The equations are discretized using Gaussian quadrature and Chebyshev polynomials, leading to a finite system of nonlinear equations. The solution requires an efficient iterative method due to the high computational cost of evaluating the kernel function. The paper surveys various numerical procedures for solving nonlinear equations, highlighting their deficiencies in this context. It proposes a class of iterative procedures based on heuristic considerations, which are illustrated with simple examples. These procedures aim to accelerate the convergence of iterative vector sequences. The paper also discusses the use of relaxation methods, Newton-Raphson-like methods, and sequence transformations such as the Aitken δ²-process and Wynn ε-algorithm. However, these methods are found to be insufficient for the class of problems under consideration due to the strong local coupling of variables and the need for high computational efficiency. The paper concludes that a dynamic, low-degree, coupled iterative process is needed to accelerate the convergence of the basic iteration. A class of such processes is suggested on a heuristic basis, with the extrapolation algorithm being particularly effective. The algorithm is shown to be useful in solving both linear and nonlinear equations arising from integral, ordinary, and partial differential equations. The paper also notes that while these methods are not guaranteed to converge globally, they can be effective in practice for the problems considered. The results of applying these methods to several examples are presented, demonstrating their effectiveness in accelerating convergence.