Merdan, Mehmet; Şişman, Şeyma

This work derives analytical solutions to fractional partial differential equations utilizing the Hilfer fractional derivative through the use of the Laplace transform method. The suggested method offers a cohesive solution framework that concurrently retrieves the Liouville-Caputo and Riemann-Liouville formulations as particular instances for designated selections of the Hilfer parameters. Exact novel solution representations are derived for the fractional KdV, modified KdV, K(2,2), and Klein-Gordon equations under appropriate initial conditions, emphasizing the impact of fractional order and type parameters on wave propagation characteristics. The approach circumvents linearization or perturbation assumptions and demonstrates a rapid convergence rate. Numerical simulations and graphical representations produced with the Maple program corroborate the analytical findings and illustrate the efficacy and universality of the proposed method.