Analytical Solutions of Nonlinear Partial Differential Equations Using the Double Sawi–Adomian Decomposition Method

A double Sawi–Adomian decomposition framework is developed for nonlinear initial-value problems with one space and one time variable. The double transform converts the time derivative and the prescribed data into a Volterra integral equation, whereas Adomian polynomials organize the nonlinear term. In contrast with earlier transform–decomposition studies that mainly report formal series, the present formulation includes a contraction condition for existence and uniqueness, a convergence criterion for the decomposition series, and both a priori and residual-based truncation bounds. A second-order nonlinear equation and the inviscid Burgers equation are examined. Their decomposition components reproduce the Taylor expansions of tanh2ξτ and ξ/1+τ, respectively. Numerical error tables show the effect of the truncation order and also expose the finite radius of convergence in the Burgers example. Thus, the contribution is a certified double-Sawi formulation rather than a claim that changing the transform alone alters the underlying Adomian approximation.