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Related Experiment Video

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The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

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Orthonormal Bernstein Galerkin technique for computations of higher order eigenvalue problems.

Humaira Farzana¹, Samir Kumar Bhowmik², Md Shafiqul Islam³

¹Department of Arts & Sciences, Ahsanullah University of Sciences & Technology, Dhaka 1215, Bangladesh.

|January 23, 2023

Summary

This study presents a novel Galerkin weighted residual method for approximating eigenvalues in higher-order boundary value problems (BVPs). The method accurately computes numerical eigenvalues for eighth, tenth, and twelfth-order problems using Bernstein polynomials.

Keywords:

Eigenvalue/Rayleigh numbers Galerkin Finite Element Method Galerkin MWR Orthonormal Bernstein polynomials

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Area of Science:

Numerical Analysis
Applied Mathematics
Computational Physics

Background:

Higher even order boundary value problems (BVPs) present significant numerical approximation challenges.
Existing methods struggle with the complexity introduced by boundary conditions in higher-order BVPs.

Purpose of the Study:

To investigate higher-order eigenvalue problems using the Galerkin weighted residual method (MWR).
To analyze the impact of polynomial bases on eigenvalue solutions.
To develop a precise matrix formulation for eighth, tenth, and twelfth-order eigenvalue problems, including linear electro-hydrodynamic (EHD) stability.

Main Methods:

Application of the Galerkin weighted residual method (MWR).
Direct implementation of polynomial bases, specifically Bernstein polynomials.
Development of a matrix formulation for eigenvalue computation.

Main Results:

Accurate numerical approximation of eigenvalues for eighth, tenth, and twelfth-order problems.
Demonstrated effectiveness of Bernstein polynomials as basis functions.
Successful application to linear electro-hydrodynamic (EHD) stability problems.

Conclusions:

The proposed MWR technique with Bernstein polynomials offers a precise and effective approach for solving higher-order eigenvalue problems.
The method provides a reliable alternative to existing numerical and analytical techniques.