Tarig Projected Differential Transform Method for Solving Time-Fractional Gas Dynamics Equation

Abstract

A study on the investigation of numer ical solutions for two homogeneous nonlinear time fractional gas dynamics equations (TFGDEs) using an innovative hybrid approach, the Tarig Projected Differential Transform Method (TPDTM), have been analyzed. The new TPDTM simplifies computations easier. This method significantly advantages itself by efficiently handling nonlinear terms through the pro jected differential transform, thereby eliminating the need for Adomian’s and He’s polynomials. By com bining the Tarig transform with the Projected Differ ential Transform Method (PDTM), we establish key properties, including linearity, a convolution theorem, and the existence of solutions. Using TPDTM, we found the convergence and stability through absolute error analysis, demonstrating its high accuracy, effi ciency, and reliability. To authenticate these findings, a comprehensive comparative analysis is conducted against established numerical methods, namely the Laplace Adomian Decomposition Method (LADM), Finite Difference Method (FDM), Homotopy Pertur bation Method (HPM), and Finite Element Method (FEM). The results are presented in tabular and graphical (2D and 3D) formats, highlight TPDTM’s superior computational performance. Notably, the so lutions obtained from TPDTM and LADM exhibit strong consistency and uniformity, with TPDTM of fering a simplified approach to handling Adomian polynomials in LADM. Through two illustrative nu merical tests examples, TPDTM demonstrates en hanced computational efficiency and flexibility com pared to FDM, LADM,HPM,andFEM,emphasizing its potential as a robust tool for addressing nonlinear fractional differential equations with high accuracy and reliability. The performance of proposed method has been verified using polar and convergence plots, with numerical simulations in MATLAB software.