Solution of nonlinear higher-index Hessenberg DAEs by Adomian polynomials and differential transform method

Area of Science:

• Numerical Analysis
• Applied Mathematics
• Computational Science

Background:

• Higher-index Hessenberg differential-algebraic equations (DAEs) are crucial in various applications.
• Solving these higher-index DAEs presents significant numerical and analytical challenges.

Purpose of the Study:

• To introduce a novel analytical method for solving two classes of higher-index Hessenberg DAEs.
• To address the difficulties associated with the numerical and analytical treatment of these equations.

Main Methods:

• The study employs Adomian polynomials combined with the differential transform method (DTM).
• Nonlinear terms are handled using Adomian polynomials within the DTM framework.
• A nonsingular linear algebraic system is derived from the recursion system for solving power series coefficients.

Main Results:

• The proposed method effectively solves higher-index Hessenberg DAEs without requiring index reduction or linearization.
• Two test problems demonstrate the method's efficacy.
• A Laplace-Padé resummation technique is utilized to enhance the convergence domain of the series solutions.

Conclusions:

• The integration of Adomian polynomials and DTM offers an efficient analytical solution for higher-index Hessenberg DAEs.
• The method's ability to bypass traditional complexities like index reduction makes it a valuable tool.
• The post-treatment with Laplace-Padé resummation improves the practical applicability of the obtained series solutions.