Use of a shooting method to compute eigenvalues of fourth-order two-point boundary value problems

Jones, D. J.

doi:10.1016/0377-0427(93)90065-j

Cited by 9 publications

(6 citation statements)

References 5 publications

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“…Consider an axially loaded slender rod as an Euler Column of length "L" with flexural rigidity of "D=EI" under the compressive force "P". Differential boundary equations and analytical expression of eigenvalue for the buckling of Euler Columns under various boundary conditions are given by [1]. X and Y represent axial coordinate and deflection of rod respectively.…”

Section: Governing Equationsmentioning

confidence: 99%

“…In the case of eigenvalue BVP, Presence of an eigenvalue as an additional variable restricts the direct use of shooting method. D.J.Jone's [1] used an additional step to find eigenvalues by shooting method. Firstly use of the Least Square Method by incrementing the guessed λ from 0 to 100,000 in steps of 20 is employed to get the traces of first few eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

“…From equation(1), equation(8) & equation(18)(19)(20) 0 1, 0 1 0, a 1 (1) a 2 (1) 1 } , ={ c 1 , c 2 , c 3 } , = {0,0, δ}4. NUMERICAL RESULTS & DISCUSSIONIn order to establish a neutral medium for comparison; Secant method is applied as a root finding algorithmwhile Runge-kutta of 4 th order is used as numerical integrator for both methods.…”

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confidence: 99%

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Numerical Methods in Calculating Eigenvalues: Case Studies in Stability of Euler Columns

Jamil

Tütüncü

2019

ALKÜ Fen Bilimleri Dergisi

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A comparative analysis of well renowned "Shooting Method" with another numerical method "Complementary Functions Method" (CFM) is presented for calculating eigenvalue (λ). Contrary to the shooting method hit and trial approach, CFM exploits the properties of linear ordinary differential equation (LODE). In the case of linear eigenvalue Boundary value Problem (BVP), CFM generates an algebraic equation system with one unknown "λ" and, alone root finding method is sufficient to give required eigenvalue. However, the Shooting Method create a system of algebraic equations containing two unknowns "λ" and "missing initial conditions", that demands an additional numerical technique along with root finding method. These radical differences between two approaches, sets the basis for this comparative investigation. As a case study in Linear Elastic Stability, different cases of Euler columns are investigated by finding eigenvalues for each case numerically, under both methods. Comparison is performed on the basis of results accuracy and cost effectiveness for both numerical techniques while solving linear stability problems.

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Section: Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical Methods in Calculating Eigenvalues: Case Studies in Stability of Euler Columns

Jamil

Tütüncü

2019

ALKÜ Fen Bilimleri Dergisi

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“…Shooting method for a class of two-point singular nonlinear boundary value problems was discussed by Elgindi and Langer [6]. D. J. Jones [7] explored the shooting method for calculating eigenvalues of fourth-order two-point BVPs. In [8][9][10], authors discussed the shooting technique for linear and nonlinear BVPs.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Reliable Solutions to the Boundary Value Problems by Using Shooting Method

Arefin¹,

Nishu²,

Dhali³

et al. 2022

Mathematical Problems in Engineering

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This research aims to use the shooting method (SM) to find numerical solutions to the boundary value problems of ordinary differential equations (ODEs). Applied mathematics, theoretical physics, engineering, control, and optimization theory all have two-point boundary value problems. If the two-point boundary value problem cannot be solved analytically, numerical approaches must be used. The scenario in the two-point boundary value issue for a single second-order differential equation with prescribed initial and final values of the solution gives rise to shooting method. Firstly, the method is discussed, and some boundary value problems of ODEs are solved by using the proposed method. Obtained results are compared with the exact solution for the validation of the proposed method and represented both in graphical and tabular form. It has been found that the convergence rate of the shooting method to the exact solution is so high. As a finding of this research, it has been determined that the shooting method produces the best-fit numerical results of boundary value problems.

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“…Many authors have attempted to obtain higher accuracy rapidly by using a numerous methods. The shooting method to compute eigen-values of fourth-order two-point boundary value problems studied by D. J. Jones [1].Wang et al [2] investigated application of the shooting method to second order multi point integral boundary value problems. Kwong and Wong [3] have studied the shooting method and non-homogeneous multipoint BVPs of second-order ODE.…”

Section: Introductionmentioning

confidence: 99%

Numerical Study on the Boundary Value Problem by Using a Shooting Method

Rahman¹

2015

PAMJ

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Abstract:In the present paper, a shooting method for the numerical solution of nonlinear two-point boundary value problems is analyzed. Dirichlet, Neumann, and Sturm-Liouville boundary conditions are considered and numerical results are obtained. Numerical examples to illustrate the method are presented to verify the effectiveness of the proposed derivations. The solutions are obtained by the proposed method have been compared with the analytical solution available in the literature and the numerical simulation is guarantee the desired accuracy. Finally the results have been shown in graphically.

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Use of a shooting method to compute eigenvalues of fourth-order two-point boundary value problems

Cited by 9 publications

References 5 publications

Numerical Methods in Calculating Eigenvalues: Case Studies in Stability of Euler Columns

Numerical Methods in Calculating Eigenvalues: Case Studies in Stability of Euler Columns

Analysis of Reliable Solutions to the Boundary Value Problems by Using Shooting Method

Numerical Study on the Boundary Value Problem by Using a Shooting Method

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