A hybrid collocation method for solving highly nonlinear boundary value problems.

Area of Science:

• Applied Mathematics
• Numerical Analysis
• Fluid Dynamics

Background:

• Nonlinear boundary value problems (BVPs) pose significant challenges in various scientific and engineering fields.
• Existing analytical and numerical methods often struggle with high nonlinearity and complex geometries.
• Accurate and efficient solution techniques are crucial for understanding phenomena like fluid flow and heat transfer.

Purpose of the Study:

• To introduce a novel hybrid collocation method for solving highly nonlinear two-point boundary value problems (BVPs).
• To demonstrate the method's efficiency and accuracy across a range of complex engineering applications.
• To provide a robust numerical tool for problems involving ordinary differential equations (ODEs).

Main Methods:

• A hybrid collocation approach integrating the Chebyshev collocation method with Laplace and differential transform methods.
• Application of the hybrid method to solve ODEs governing specific physical phenomena.
• Comparative analysis of the obtained results against established numerical techniques.

Main Results:

• The hybrid method provides reasonable and accurate approximate solutions for highly nonlinear BVPs.
• Demonstrated effectiveness in modeling Darcy-Brinkman-Forchheimer flow, magnetohydrodynamic flow in a channel, fin heat transfer, and squeezing flow.
• The proposed technique shows significant efficiency compared to existing methods.

Conclusions:

• The hybrid collocation method is a powerful and efficient tool for tackling challenging nonlinear BVPs.
• The method offers a reliable approach for problems in fluid mechanics, heat transfer, and other related disciplines.
• This technique advances the numerical solution of complex ODEs in engineering applications.