Replacing the finite difference methods for nonlinear two-point boundary value problems by successive application of the linear shooting method

Filipov, Stefan M.; Gospodinov, Ivan; Faragó, István

doi:10.1016/j.cam.2019.03.004

Cited by 17 publications

(3 citation statements)

References 20 publications

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“…Lodhi & Mishra, (2018) menyelesaikan masalah pertubasi singular menggunakan metode Septic B-sline untuk persamaan diferensial orde dua. Filipov et al, (2019) juga menggunakan metode beda hingga untuk menyelesaikan masalah kondisi batas nonlinear dimana skema beda hingganya diaplikasikan untuk metode shooting.…”

Section: Pendahuluanunclassified

Penyelesaian Numerik Masalah Syarat Batas Robin pada Persamaan Diferensial Cauchy-Euler

Habibah

Tuloli

Rimanada

et al. 2020

mjm

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This research studied how the numerical solution of the Cauchy-Euler differential equation with Robin boundary conditions. There were several numerical methods that can be used to get the numerical solution of a boundary value problem, namely the finite-difference method, the shooting method, the collocation method, and others. In this study, the numerical solution of Robin's boundary condition problem was obtained by the center finite-difference and the shooting methods. From the two methods, the numerical error was compared to the exact solution. The simulation results shown that the shooting method produces a better numerical solution for approximating the completion of the Cauchy-Euler differential equation than the finite-difference method since it produced smaller numerical errors.

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Section: Pendahuluanunclassified

Penyelesaian Numerik Masalah Syarat Batas Robin pada Persamaan Diferensial Cauchy-Euler

Habibah

Tuloli

Rimanada

et al. 2020

mjm

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“…Ahmad and Charan [15] make an attempt to compare the finite difference approach to the shooting method. In [16], the authors described the sequential implementation of the linear shooting approach. Lijun Zhang et al [17] also use the shooting method for solving their proposed governing equations.…”

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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“…Techniques to solve hybrid systems have existed since the 1960s utilizing multiple shooting methods [2], [3], [4], [5]. In these approaches, the interval is divided over multiple sub-intervals and the boundary-value problem (BVP) is converted to multiple initial-value problems (IVP).…”

Section: Introductionmentioning

confidence: 99%

Least-squares solutions of boundary-value problems in hybrid systems

Johnston,

Mortari

2019

Preprint

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This paper looks to apply the mathematical framework of the Theory of Functional Connections to the solution of boundaryvalue problems arising from hybrid systems. The Theory of Functional Connections is a technique to derive constrained expressions which are analytical expressions with embedded constraints. These expressions are particularly suitable to transform a large class of constrained optimization problems into unconstrained problems. The initial and most useful application of this technique is in the solution of differential equations where the problem can be posed as an unconstrained optimization problem and solved with simple numerical techniques (i.e. least-squares).A hybrid system is simply a sequence of different differential equations. The approach developed in this work derives an analytical constrained expression for the entire range of a hybrid system, enforcing both the boundary conditions as well as the continuity conditions across the sequence of differential equations. This reduces the searched solution space of the hybrid system to only admissible solutions. The transformation allows for a least-squares solution of the sequence for linear differential equations and a iterative least-squares solution for nonlinear differential equations.Lastly, the general formulation for "n" segments is developed and validation is provided through numerical tests for three differential equation sequences: linear/linear, linear/nonlinear, and nonlinear/nonlinear. The accuracy level obtained are all at machine-error, which is consistent with the accuracy experienced in past studies on the application of the Theory of Functional Connections to solve single ordinary differential equations.

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Replacing the finite difference methods for nonlinear two-point boundary value problems by successive application of the linear shooting method

Cited by 17 publications

References 20 publications

Penyelesaian Numerik Masalah Syarat Batas Robin pada Persamaan Diferensial Cauchy-Euler

Penyelesaian Numerik Masalah Syarat Batas Robin pada Persamaan Diferensial Cauchy-Euler

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

Least-squares solutions of boundary-value problems in hybrid systems

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