Novel exploration of topological degree method for noninstantaneous impulsive fractional integro-differential equation through the application of filtering system

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The primary focus of this paper is to explore the concept of an initial condition for a noninstantaneous impulsive fractional integro-differential equation of order 0<ϑ<1 in an n-dimensional Euclidean space. Using the Laplace transform method, we derive the solution representation of the given dynamical system and investigate the existence and uniqueness of the mild solution through the topological degree method and Gronwall’s inequality. Furthermore, we present a filter model featuring a finite impulsive response, which serves as a practical demonstration of the proposed system because it effectively captures the memory effects inherent in fractional-order systems and enhances system reliability with minimal input. Finally, the numerical computations with graphical illustrations provide concrete examples that validate the theoretical results, showcasing how the behavior of the system is impacted by changes in fractional order and emphasizing the versatility of our proposed method.