Double Laplace–Sawi decomposition technique for solving nonlinear partial differential equations

Abstract

This study develops a double Laplace--Sawi decomposition technique for nonlinear partial differential equations by coupling the double Laplace--Sawi transform with Adomian polynomials.  In contrast with a purely formal transform calculation, explicit exponential-order hypotheses are imposed to guarantee existence of the single and double transforms and to justify interchange of integrations and differentiation.  Separate recursive formulations are derived for first- and second-order time operators, thereby covering problems with one or two initial conditions without ambiguity.  A sufficient contraction condition for convergence is established in a Banach space, together with an a posteriori truncation-error estimate.  Three nonlinear problems are examined.  The first corrects the transform and algebraic details of a nonlinear transport--reaction equation; the second demonstrates coefficient generation and finite-time blow-up for a cubic nonlinearity; and the third treats the inviscid Burgers equation, a genuinely nonlinear partial differential equation with coupled spatial dynamics and boundary data.  For the Burgers problem, the decomposition coefficients are proved by induction, the exact characteristic solution is recovered, and explicit truncation errors are reported.  The results show that the technique provides a systematic analytic construction on its stated convergence domain, while also identifying the limitations caused by local series convergence, singularities, and the increasing cost of high-order Adomian polynomials.