Solving Fractional Partial Differential Equations Using the Double Natural Transform Decomposition Method

Abstract

In this paper, we propose an analytical framework based on the double Natural transform decomposition method (DNTDM) for solving linear and nonlinear fractional partial differential equations (FPDEs) involving the Caputo fractional derivative. The proposed approach systematically combines the double Natural transform (DNT) with the Adomian decomposition method (ADM) to construct series solutions. The recursive structure of the method facilitates the treatment of nonlinear terms without linearization and incorporates the initial conditions through the transformation properties of the fractional derivatives. In addition, the use of the double Natural transform provides a unified treatment of the two independent variables, extending the standard Natural Transform Decomposition Method to a two-variable transform framework. Several illustrative examples are presented to demonstrate the accuracy and efficiency of the method. The obtained solutions are shown to agree with known exact solutions, confirming the reliability of the proposed approach. The DNTDM provides a systematic analytical and semi-analytical framework for a broad class of nonlinear FPDEs arising in applied sciences and engineering.