Transforming Generalized Conformable Linear PDEs into Classical Linear PDEs via Generalized Conformable Integral Operators

Abstract

This work investigates the reduction of Generalized Conformable Partial Differential Equations (GCPDEs) to classical Partial Differential Equations (PDEs) through a tailored change of variables based on integral conformable operators. We concentrate on second-order linear PDEs due to their central role in modeling physical and engineering systems. The proposed transformation yields an equivalent equation expressed in terms of conformable derivatives, preserving the original order and degree while introducing local kernel-dependent modulation. This equivalence establishes that solving a GCPDE can be systematically reduced to solving a standard PDE in transformed coordinates. We demonstrate the utility of this framework in modeling anomalous diffusion and anisotropic transport phenomena, where traditional formulations may fail to capture essential spatial or temporal variability.