Semi-analytical Approximation for Solving High-order Sturm-Liouville Problems

Abstract

In this paper, an algorithm for solving high-order non-singular Sturm-Liouville eigenvalue problems is proposed. A modified form of Adomian decomposition method is implemented to provide a semianalytical solution in the form of a rapidly convergent series. Convergent analysis and error estimate based on the Banach fixed-point is discussed. Five high-order Sturm-Liouville problems are solved numerically. Numerical results demonstrate reliability and efficiency of the proposed scheme.

Orthonormal Bernstein Galerkin technique for computations of higher order eigenvalue problems

“…Table 4 reveals that the absolute errors | | for all six eigenvalues are exceedingly small i.e., 1.0e+00. This shows that our proposed method agrees well with ADM [14] .…”

“…To assess the competency of the suggested method, four linear eigenvalue problems with two classes of boundary conditions have been worked out. The eigenvalues computed by the current technique are compared with the various numerical and analytical techniques [ 1 , 2 , 14 , 26 ] available in the literature. The Galerkin method is tested with finite intervals from eight to twelfth orders along with regular endpoint boundary conditions.…”

“…The first six eigenvalues utilizing Galerkin MWR exploiting for Eqs. (29a) and ( 29b ) using Bernstein polynomials along with their absolute error with ADM [14] are displayed in Table 4 . Table 4 reveals that the absolute errors | | for all six eigenvalues are exceedingly small i.e., 1.0e+00.…”

“…Several numerical methods, namely local adaptive differential quadrature have been implemented in various studies. A numerical method namely Adomian decomposition methods [ 13 , 14 ], energy inequalities method [15] , Galerkin based Septic B-Splines technique [16] , splines and non-polynomial spline techniques [17] , [18] , [19] , Local adaptive differential quadrature method [20] , reproducing kernel Hilbert space method [21] to solve the boundary value problems of higher order with multi-boundary conditions. The authors of [3] used both analytical and numerical methods to investigate the linear Electro-hydrodynamic stability problem of an eighth order differential equation.…”

Theory and numerical approaches of high order fractional Sturm–Liouville problems

“…Adomian decomposition method (ADM), variational iteration method (VIM), Chebyshev spectral collocation method (CSCM) and modified Adomian decomposition method are numerical or semi-analytic schemes available on this subject (see [6,33,34]).…”