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In this paper, our aim is to determine the fundamental eigenfunction of a two nonlinear differential complex equation, which arises in microchip laser theory, using Adomian decompositon method.

In this paper our objective is to show how to approach the solution of any first order partial differential equation (p.d.e.), using Adomian decompositon method and α‐dense curves. Indicates that this method is perfectly adapted to the solution of such equations and especially in the nonlinear case. The results are of particular importance in biomedicine and biocybernetics.

Aim's to show how to approach the solution for a class of first order p.d.e. using Adomian decomposition method. Discusses the generalities of the method and α‐dense curves. Outlines the new approach and provides applications of its use.

Describes the Adomian decomposition method and provides a new proof for the evaluation of the Abbaoui‐Cherruault numbers. Demonstrates a new result for Bell polynominals and establishes and proves a new norm bound for Adomian polynominals. Demonstrates an application to the truncation error evaluation and applies the presented results to a microchip lasers problem.