Integral Equations And Moment Theory In The Study Of Complex Systems: A Real And Complex Analysis

Abstract

Complex systems are characterized by nonlinearity, feedback loops, and emergent behavior, which pose significant challenges to conventional modeling techniques. The analytical representation and solution of such systems are discussed in this paper using integral equations and moment theory as the main mathematical tools. The primary aim is to develop a systematic and formal framework in which one can model dynamic systems in integral formulations and reconstructions in terms of moments, with the aid of tools of real and complex analysis. This methodology is driven by the necessity to develop methods of solutions that are not based on empirical observations or computer-based simulations, but still allow for gaining profound insights into the underlying structure and dynamics of complex systems. The paper derives Fredholm and Volterra integral equations and investigates their solvability, uniqueness, and stability in specified spaces of functions. It also connects classical moment problems with such integral structures to recover unknown functions given sequences. The theoretical findings are illustrated with symbolic examples, as well as to explain analytical modeling techniques. The results indicate a logical framework of how to treat inverse problems and analytically model cumulative system behavior. This helps to develop mathematical modeling in areas like control systems, dynamical networks, and physical processes. The framework provides a foundation for extensions in the future with operator theory, numerical methods, and practical applications.