GENERALIZATION OF THE ONE-DIMENSIONAL WAVE EQUATION VIA (p, q)− DEFORMATION

Abstract

. In this work, we present a comparative analysis of the q-deformed and (p, q)-deformed formulations of the one-dimensional wave equation within the framework of generalized calculus. The q-deformed wave equation, con- structed using the Jackson derivative, introduces a single deformation pa- rameter that encodes discrete-scale effects and leads to modified wave prop- agation governed by q-d’Alembert-type solutions. The (p, q)-deformed wave equation extends this approach by incorporating two independent deforma- tion parameters, allowing for asymmetric and multi-scale spatial dilations. We show that the q-deformed model is recovered as a special limiting case of the (p, q)-formalism, while both deformations reduce smoothly to the classi- cal wave equation in the undeformed limit. This comparison demonstrates that, although the q-deformation captures essential discretization features, the (p, q)-deformation provides a richer algebraic structure and greater flexi- bility for modeling wave phenomena in non-uniform and anisotropic media.