Abstract
The Galerkin method of weighted residual (MWR) for computing natural frequencies of some physical problems such as the Helmholtz equation and some second-order boundary value problems has been revised in this article. The use of one and two-dimensional characteristic Bernstein polynomials as the basis functions have been presented by the Galerkin MWR. The vibration of non-homogeneous membranes with Dirichlet type boundary conditions is also studied here. The useful properties of Bernstein polynomials, its derivatives and function approximations have also been illustrated. Besides, the efficiency and applicability of the proposed technique have been demonstrated through some numerical experiments.
Highlights
Mathematical model derived from multi-dimensional differential equations play a crucial role in modeling a variety of scientific and engineering application problems
The Galerkin method of weighted residual (MWR) for computing natural frequencies of some physical problems such as the Helmholtz equation and some second-order boundary value problems has been revised in this article
The use of one and two-dimensional characteristic Bernstein polynomials as the basis functions have been presented by the Galerkin MWR
Summary
Mathematical model derived from multi-dimensional differential equations play a crucial role in modeling a variety of scientific and engineering application problems. Numerical technique utilizing Bernstein polynomials basis to give the approximate solution of a parabolic partial differential equation illustrated in [3]. Bernstein Ritz-Galerkin method for solving an initial boundary value problem for two-dimensional (2-D) wave equation has been studied by [6]. Eigenvalue analysis for second order boundary value problems arises in several engineering application areas has been investigated using analytical or numerical techniques [22]. Bernstein basis has been exploited to solve ordinary and partial differential to solve eigenvalues as well as boundary value problems employing numerous techniques are in [23]-[28]. We have presented Bernstein polynomials based Galerkin method of weighted residual (MWR) technique that offers accurate solutions, are put up with in terms of truncated series of smooth polynomial functions.
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