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Abstract
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The use of variable-coefficient time-fractional models is becoming more and more significant in the description of complex dynamical systems with heterogeneous structures and memory effects, including biological processes, anomalous diffusion, viscoelasticity, and signal processing. Standard numerical and analytical techniques, which frequently have low accuracy, instability, or high computational cost, find them especially difficult to handle due to their fractional and variable-coefficient nature. The influence of variable coefficients is difficult for traditional approximation techniques to accurately capture over extended periods of time. This highlights the need for strong semi-analytical methods that can efficiently handle both variable coefficients and fractional operators. By integrating residual corrections with the transform-based Elzaki approach, the Elzaki residual power series method (ERPSM) offers such a framework, systematically enhancing accuracy and convergence. By developing a quick, accurate, and adaptable approach to solving variable-coefficient time-fractional models, this research will help advance mathematical theory and support applications in the fields of physics, engineering, and applied sciences. v
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