On the localized and periodic solutions to the time-fractional Klein-Gordan equations: Optimal additive function method and new iterative method

This investigation explores two numerical approaches: the optimal auxiliary function method (OAFM) and the new iterative method (NIM). These techniques address the physical fractional-order Klein-Gordon equations (FOKGEs), a class of partial differential equations (PDEs) that model various physical phenomena in engineering and diverse plasma models. The OAFM is a recently introduced method capable of efficiently solving several nonlinear differential equations (DEs), whereas the NIM is a well-established method specifically designed for solving fractional DEs. Both approaches are utilized to analyze different variations in FOKGE. By conducting numerous numerical experiments on the FOKGE, we compare the accuracy, efficiency, and convergence of these two proposed methods. This study is expected to yield significant findings that will help researchers study various nonlinear phenomena in fluids and plasma physics.