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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.
Methodsx
|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.
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.
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