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Abstract
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Integral equations of the third kind, particularly when nonlinear, represent some of the most challenging problems in applied mathematics. They naturally arise in mathematical models of physics, engineering, biology, and economics, where processes are governed by hereditary effects, feedback mechanisms, and nonlinear interactions. Unlike first- and second-kind integral equations, third-kind equations involve terms where the unknown function appears both inside the integral operator and outside it, often combined with nonlinearities that make analytical solutions unattainable. Developing robust and accurate numerical methods is therefore essential.
Nonlinear integral equations of the third kind rarely admit closed-form solutions. The inherent nonlinearity, combined with the ill-posed nature of third-kind operators, necessitates the development of numerical schemes capable of delivering stable and accurate approximations.
Such equations appear in potential theory, transport problems, population dynamics, viscoelasticity, and control theory, where solution accuracy directly affects predictions and interpretations. Without efficient numerical tools, the application of these models is limited.
Existing numerical techniques—such as iterative methods, quadrature schemes, or projection methods—often struggle with stability and convergence in the nonlinear and third-kind setting. Thus, new approaches tailored to these difficulties are required.
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