The Tantawy technique for modeling fractional KdV and mKdV positron-acoustic solitary waves in an electron-positron-ion plasma with regularized $$\kappa -$$ distribution

This work investigates the dynamical behavior of both the integer and fractional-order positron-acoustic solitary waves (PASWs) in a collisionless, unmagnetized electron-positron-ion plasma composed of stationary positive ions, inertial cold positrons, inertialess hot Maxwellian positrons, and inertialess electrons obeying a regularized κ-distribution (RKD). Using the reductive perturbation technique, we derive two planar evolutionary wave equations, known as, a quadratic nonlinearity Korteweg-de Vries (KdV) equation, valid away from critical plasma compositions, and a cubic modified KdV (mKdV) equation, which governs the dynamics at the critical composition where the quadratic nonlinearity vanishes. By checking the sign of the quadratic nonlinearity coefficient in the KdV equation, it is found that the current model supports both compressive (for a positive sign) and rarefactive (for a negative sign) PASWs. At the same time, the sign of the cubic nonlinear coefficient to the mKdV equation describes the propagation of solitary (for positive sign) or shock (for negative sign) structures. Within the Caputo fractional-derivative framework, we further construct the corresponding time-fractional KdV (FKdV) and fractional mKdV (FmKdV) equations and solve them analytically using the Tantawy technique, yielding rapidly convergent series approximations of fractional PASWs. The fractional order serves as a quantitative measure of memory, and its deviation from the integer limit modifies the temporal evolution, amplitude, and width of PASWs, mimicking non-Markovian transport and anomalous dissipation in the plasma. A systematic parametric study reveals how the RKD cutoff and superthermality indices, the hot-positron and ion concentrations, and the fractional order shape the profiles of both planar KdV- and mKdV-type PASWs. The excellent agreement between the fractional approximations and the known integer solutions, together with small absolute errors at successive orders, highlights the accuracy and efficiency of the Tantawy technique for fractional evolutionary wave equations. The present results are relevant to space and astrophysical plasmas where superthermal electron populations and multicomponent electron-positron-ion mixtures are expected.