Chebyshev differentiation matrices for efficient computation of the eigenvalues of fourth-order Sturm–Liouville problems

Taher, Anis Haytham Saleh; Malek, Alaeddin; Momeni-Masuleh, Sayed Hodjatollah

doi:10.1016/j.apm.2012.09.062

Cited by 23 publications

(5 citation statements)

References 18 publications

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“…Example 4.5. Consider the following fourth-order Sturm-Liouville problem related to mechanicals non-linear systems identification [7], [10], [14]    y (4) (x) − 2αx 2 y − 4αxy + (α 2 x 4 − 2α)y = λy(x), x ∈ (0, 5), y(0) = y (0) = 0, y(5) = y (5) = 0. x 13 λα 2 − 119 18532800…”

Section: Numerical Resultsmentioning

confidence: 99%

“…However many important phenomena occurring in various fields of science are described mathematically by highorder Sturm-Liouville problems. For example, the free vibration analysis of beam structures [6], [7], [8] is governed by a fourth-order Sturm-Liouville problem, and it is known that when a layer of fluid is heated from below and is subject to the action of rotation, instability may set as overstability, this instability my be modelled by a eighth-order Sturm-Liouville boundary value problem with appropriate boundary conditions specified. It may be noted that, when instability sets as ordinary convection, the marginal state will be characterized by sixth-order Sturm-Liouville boundary value problem [9], [10], [11], [12].…”

Section: Introductionmentioning

confidence: 99%

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

Taher

Malek

Thabet

2014

BJMCS

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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.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Taher

Malek

Thabet

2014

BJMCS

Self Cite

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“…Slightly fewer techniques are available for the direct eigenvalue problem of case m = 2 known as fourth-order Sturm-Liouville problem (FSLP). For example, Adomian decomposition method (ADM) [16], Chebyshev spectral collocation method (CSCM) [17], Homotopy Perturbation Method (HPM) [11], homotopy analysis method (HAM) [18], Differential quadrature method (DQM) and Boubaker polynomials expansion scheme (BPES) [19], Chebychev method (CM) [20], Spectral parameter power series (SPPS) [21], Chebyshev differentiation matrices (CDM) [22], variational iteration method (VIM) [23], Matrix methods (MM) [24], and Lie Group method [25] are the prominent techniques available.…”

Section: Introductionmentioning

confidence: 99%

Solutions of Direct and Inverse Even-Order Sturm-Liouville Problems Using Magnus Expansion

Perera

Böckmann

2019

Mathematics

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In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.

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“…On the other hand, Pseudospectral methods are a class of numerical methods which were introduced in 1970s [6,7,26]. Their application for solving engineering problems has become popular due to their computational feasibility and efficiency [12,27,31,32,34,35]. In the pseudospectral method, the unknown solution is expanded as a global polynomial interpolant based on some suitable collocation points.…”

Section: Introductionmentioning

confidence: 99%