A novel technique to solve nonlinear higher-index Hessenberg differential-algebraic equations by Adomian...

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Applied Mathematics

Background:

• The Adomian decomposition method (ADM) is a powerful tool for nonlinear equations since 1980.
• Existing ADM applications for differential-algebraic equations (DAEs) require complex preprocessing like index reduction.
• These transformations can be computationally expensive and may yield non-physical solutions.

Purpose of the Study:

• To propose a novel technique applying ADM directly to nonlinear higher-index Hessenberg DAEs systems.
• To develop a simplified, efficient algorithm that avoids complex transformations.
• To reduce computational workload by solving linear algebraic systems iteratively.

Main Methods:

• Direct application of the Adomian decomposition method (ADM) to nonlinear higher-index Hessenberg DAEs.
• Avoidance of index reduction or other complex preprocessing steps.
• Iterative solution involving linear algebraic systems with constant coefficient matrices (except the first iteration).

Main Results:

• A straightforward, general algorithm for solving nonlinear higher-index Hessenberg DAEs.
• Significant reduction in computational complexity compared to transformation-based methods.
• Successful application to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints.

Conclusions:

• The proposed ADM technique offers an efficient and simplified approach for nonlinear higher-index DAEs.
• The method avoids drawbacks associated with traditional index reduction techniques.
• The approach is programmable in systems like Maple or Mathematica for practical simulations.