Atomic Solutions of Non-Homogeneous Parabolic Partial Differential Equations via Tensor Product Theory

In this paper, we develop and apply the atomic solution method to obtain exact solutions of a class of non-homogeneous parabolic partial differential equations (PPDEs).The approach is based on the tensor product theory in Banach spaces together with key properties of atomic operators.By exploiting these structures, the PDE is decomposed into simpler components, allowing us to construct exact solutions in terms of atomic functions.Several cases are examined, leading to nine distinct atomic solutions under different assumptions on the source terms.The method highlights the effectiveness of the atomic framework in handling non-homogeneous PDEs and demonstrates its potential as a systematic tool for solving problems that arise in heat flow, diffusion, and related physical processes.