Theoretical analysis of -transform decomposition method with applications of nonlinear ordinary differentialequations

One of the most noteworthy differential equations of the first order is the Riccati differential equation. It is applied in various branches of mathematics, including algebraic geometry, physics, and conformal mapping theory. The $$-transform Adomian decomposition method is employed in the current study to find exact solutions for different kinds of nonlinear differential equations. We give thorough detailed proofs for new theorems related to the $$-transform technique. The Adomian decomposition method and the $$-transform method serve as the foundation for this technique. For certain differential equations, the theoretical analysis of the $$-transform Adomian decomposition method is examined and is computed using readily computed terms. Our findings are contrasted with exact solutions found in the literature that were produced using alternative techniques. The significant features of the $$-transform Adomian decomposition method are described in the article. It has been shown that the $$-transform Adomian decomposition method is very efficient, useful, and adaptable to a broad variety of linear and nonlinear differential equations. Most of the symbolic and numerical calculations were performed using Mathematica.