Fractional Calculus in the Solution of the Klein–Gordon Equation

This paper investigates the Klein–Gordon equation within the framework of fractional calculus by incorporating non-integer time and spatial derivatives to model physical processes characterized by memory effects and nonlocal interactions. Fractional operators in the Riemann–Liouville and Caputo senses are employed, together with the Laplace transform and the Mittag–Leffler function, to reformulate and solve the associated initial value problem. The resulting solutions are examined both analytically and graphically in order to evaluate the influence of fractional orders on the temporal and spatial dynamics of the system. The analysis demonstrates that small variations in the fractional orders lead to significant qualitative changes in the behavior of the scalar field, indicating transitions between classical and fractional regimes. In particular, anomalous damping, power-law decay, and nonlocal propagation phenomena are observed, which are intrinsically linked to the properties of the Mittag–Leffler function. The principal contribution of this study is the systematic characterization of the role of fractional order in the Klein–Gordon equation. The proposed fractional model generalizes the classical formulation and provides a suitable mathematical framework for describing systems exhibiting anomalous dissipation, long-term memory, and non-Euclidean geometric effects. Keywords: Fractional calculus; Klein–Gordon equation; Caputo derivative; Riemann–Liouville derivative; Laplace transform; Mittag–Leffler function; nonlocal dynamics; memory effects.