Application of the Fourier transform Adomian decomposition method for initial value problems

Purpose The purpose of this study is to apply the Fourier Transform Adomian Decomposition Method (FTADM) to solve initial value problems for types of partial differential equations (PDEs). Through careful calculations, the article constructs and analyzes approximate solutions to these equations and provides the corresponding error tables. The findings indicate that both FTADM and the Laplace Adomian Decomposition Method (LADM) exhibit consistent convergence rates for solving linear and nonlinear equation systems. And under certain equations, FTADM exhibits a faster convergence rate than ODM. Design/methodology/approach The study employs FTADM as the primary solution method, combining Fourier transforms with Adomian decomposition techniques to enhance the computational efficiency of nonlinear partial differential equations. The article first introduces the theoretical framework of FTADM and then applies it to six different partial differential equations, including the KdV equation, Fokker-Planck equation, Burgers equation, mKdV equation, viscous Burgers equation and Boussinesq-Burgers equation. Finally, numerical comparisons of the convergence rates of the approximate solutions obtained by FTADM, LADM and ODM are presented to validate the method's effectiveness. Findings FTADM effectively solves nonlinear partial differential equations and provides high-accuracy approximate solutions. The convergence rates of the approximate solutions obtained from FTADM and LADM are the same, indicating that FTADM is stable and reliable for nonlinear problems. In all tested equations and systems, the solutions generated by FTADM closely match known analytical solutions, validating the method's effectiveness. Error analysis reveals that as the number of iterations increases, the precision of the FTADM results improves, further confirming its convergence. Originality/value This article verifies the effectiveness of this method by using the FTADM to solve complex nonlinear partial differential equations. The comparison of the convergence of FTADM and LADM solutions, providing numerical evidence to support FTADM's convergence. The results of the study provide experimental data that support future rigorous mathematical proofs of FTADM's convergence, laying the foundation for further research.