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In recent years, there has been a growing interest among researchers in the study of shallow water waves, driven by their wide applicability across various scientific disciplines. Within this field, the Camassa–Holm equation has garnered significant attention due to its ability to capture complex wave phenomena. However, researchers have extended their investigations to the Modified Camassa–Holm equation (MCH), which incorporates principles from fractional calculus, thereby offering a more comprehensive framework for understanding intricate dynamics. This article focuses on the analytical solution of the Fractional Modified Camassa–Holm (FMCH) equation, an extension of the MCH, achieved through a novel method known as the Residual Power Series Method (RPSM). By utilizing RPSM and initial conditions, the main aim is to unveil an analytical solution for the FMCH, contributing to an improved understanding and modeling of the complex real-world phenomena observed in waves.