Solving second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative method

Purpose The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method. Design/methodology/approach The authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived....

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Vydáno v:	Engineering computations Ročník 38; číslo 1; s. 107 - 130
Hlavní autoři:	Liu, Chein-Shan, Chang, Jiang-Ren
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Bradford Emerald Publishing Limited 27.01.2021 Emerald Group Publishing Limited
Témata:	Algorithms Approximation Boundary conditions Boundary value problems Closed form solutions Convergence Eigenvalues Eigenvectors Exact solutions Integral equations Iterative algorithms Iterative methods Numerical analysis Numerical methods Ordinary differential equations Orthogonality Thermal expansion Weighting functions Nonlinear boundary value problem Iterative algorithm Nonlinear boundary conditions Eigenfunctions as test functions Closed-form expansion coefficients
ISSN:	0264-4401, 1758-7077
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Abstract	Purpose The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method. Design/methodology/approach The authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived. By expanding the numerical solution in terms of the weighted eigenfunctions and using the orthogonality of eigenfunctions with respect to a weight function, and together with the non-separated/mixed boundary conditions, one can obtain the closed-form expansion coefficients with the aid of Drazin inversion formula. Findings When the authors develop the iterative algorithm, removing the time-varying terms as well as the nonlinear terms to the right-hand sides, to solve the nonlinear boundary value problem, it is convergent very fast and also provides very accurate numerical solutions. Research limitations/implications Basically, the authors’ strategy for the iterative numerical algorithm is putting the time-varying terms as well as the nonlinear terms on the right-hand sides. Practical implications Starting from an initial guess with zero value, the authors used the closed-form formula to quickly generate the new solution, until the convergence is satisfied. Originality/value Through the tests by six numerical experiments, the authors have demonstrated that the proposed iterative algorithm is applicable to the highly complex nonlinear boundary value problems with nonlinear boundary conditions. Because the coefficient matrix is set up outside the iterative loop, and due to the property of closed-form expansion coefficients, the presented iterative algorithm is very time saving and converges very fast.
AbstractList	PurposeThe purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method.Design/methodology/approachThe authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived. By expanding the numerical solution in terms of the weighted eigenfunctions and using the orthogonality of eigenfunctions with respect to a weight function, and together with the non-separated/mixed boundary conditions, one can obtain the closed-form expansion coefficients with the aid of Drazin inversion formula.FindingsWhen the authors develop the iterative algorithm, removing the time-varying terms as well as the nonlinear terms to the right-hand sides, to solve the nonlinear boundary value problem, it is convergent very fast and also provides very accurate numerical solutions.Research limitations/implicationsBasically, the authors’ strategy for the iterative numerical algorithm is putting the time-varying terms as well as the nonlinear terms on the right-hand sides.Practical implicationsStarting from an initial guess with zero value, the authors used the closed-form formula to quickly generate the new solution, until the convergence is satisfied.Originality/valueThrough the tests by six numerical experiments, the authors have demonstrated that the proposed iterative algorithm is applicable to the highly complex nonlinear boundary value problems with nonlinear boundary conditions. Because the coefficient matrix is set up outside the iterative loop, and due to the property of closed-form expansion coefficients, the presented iterative algorithm is very time saving and converges very fast. Purpose The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical method. Design/methodology/approach The authors introduce eigenfunctions as test functions, such that a weak-form integral equation is derived. By expanding the numerical solution in terms of the weighted eigenfunctions and using the orthogonality of eigenfunctions with respect to a weight function, and together with the non-separated/mixed boundary conditions, one can obtain the closed-form expansion coefficients with the aid of Drazin inversion formula. Findings When the authors develop the iterative algorithm, removing the time-varying terms as well as the nonlinear terms to the right-hand sides, to solve the nonlinear boundary value problem, it is convergent very fast and also provides very accurate numerical solutions. Research limitations/implications Basically, the authors’ strategy for the iterative numerical algorithm is putting the time-varying terms as well as the nonlinear terms on the right-hand sides. Practical implications Starting from an initial guess with zero value, the authors used the closed-form formula to quickly generate the new solution, until the convergence is satisfied. Originality/value Through the tests by six numerical experiments, the authors have demonstrated that the proposed iterative algorithm is applicable to the highly complex nonlinear boundary value problems with nonlinear boundary conditions. Because the coefficient matrix is set up outside the iterative loop, and due to the property of closed-form expansion coefficients, the presented iterative algorithm is very time saving and converges very fast.
Author	Chang, Jiang-Ren Liu, Chein-Shan
Author_xml	– sequence: 1 givenname: Chein-Shan surname: Liu fullname: Liu, Chein-Shan email: csliu58@163.com – sequence: 2 givenname: Jiang-Ren surname: Chang fullname: Chang, Jiang-Ren email: cjr@mail.ntou.edu.tw
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Keywords	Nonlinear boundary value problem Iterative algorithm Nonlinear boundary conditions Eigenfunctions as test functions Closed-form expansion coefficients
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Snippet	Purpose The purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical... PurposeThe purpose of this paper is to solve the second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative numerical...
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SubjectTerms	Algorithms Approximation Boundary conditions Boundary value problems Closed form solutions Convergence Eigenvalues Eigenvectors Exact solutions Integral equations Iterative algorithms Iterative methods Numerical analysis Numerical methods Ordinary differential equations Orthogonality Thermal expansion Weighting functions
Title	Solving second-order nonlinear boundary value problem with nonlinear boundary conditions by an iterative method
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