Orthonormal Bernstein Galerkin technique for computations of higher order eigenvalue problems

Humaira Farzana; Samir Kumar Bhowmik; Md Shafiqul Islam

doi:10.1016/j.mex.2023.102006

Orthonormal Bernstein Galerkin technique for computations of higher order eigenvalue problems

MethodsX. 2023 Jan 5:10:102006. doi: 10.1016/j.mex.2023.102006. eCollection 2023.

Authors

Humaira Farzana¹, Samir Kumar Bhowmik², Md Shafiqul Islam³

Affiliations

¹ Department of Arts & Sciences, Ahsanullah University of Sciences & Technology, Dhaka 1215, Bangladesh.
² Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh.
³ Department of Applied Mathematics, University of Dhaka, Dhaka 1000, Bangladesh.

Abstract

The numerical approximation of eigenvalues of higher even order boundary value problems has sparked a lot of interest in recent years. However, it is always difficult to deal with higher-order BVPs because of the presence of boundary conditions. The objective of this work is to investigate a few higher order eigenvalue (Rayleigh numbers) problems utilizing the method of Galerkin weighted residual (MWR) and the effect of solution due to direct implementation of polynomial bases. The proposed method develops a precise matrix formulation for the eighth order eigenvalue and linear electro-hydrodynamic (EHD) stability problems.•The article explores the same for tenth and twelfth order eigenvalue problems.•This method involves computing numerical eigenvalues using Bernstein polynomials as the basis functions.•The novel weighted residual Galerkin technique's performance is numerically validated by comparing it to other numerical/analytical approaches in the literature.

Keywords: Eigenvalue/Rayleigh numbers; Galerkin Finite Element Method; Galerkin MWR; Orthonormal Bernstein polynomials.