To infinity and some glimpses of beyond

Area of Science:

• Mathematical Modeling
• Computational Dynamics
• Numerical Analysis

Background:

• Dynamic models often encounter singularities, reaching infinity in finite time.
• Extending analysis and numerical computations beyond these singularities presents significant challenges.

Purpose of the Study:

• To propose a methodology for extending the analysis of dynamic models beyond finite-time singularities.
• To enable continued computation and well-behaved solutions past points of infinity.

Main Methods:

• Employing suitable transformations to alter the model's behavior near infinity.
• Utilizing complexification for regularization and exploring its dynamic impact.
• Applying compactification transformations to manage singularities.

Main Results:

• Demonstrated successful continuation of analysis and computation beyond singularities in Ordinary Differential Equations.
• Developed adaptive strategies for Partial Differential Equations, including buffer zones for transformation identification.
• Showcased the effectiveness of complexification and compactification in managing infinite behaviors.

Conclusions:

• The proposed transformation-based methodology offers a systematic way to bypass finite-time singularities in evolution equations.
• This approach enhances the analytical and numerical tractability of models with challenging dynamic behaviors.
• The techniques are potentially applicable to a wide range of mathematical and computational models facing infinities.