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

Stefan M. Filipov, Ivan D. Gospodinov, I. Faragó

Research output: Contribution to journalArticle

Abstract

This paper studies numerical solution of nonlinear two-point boundary value problems for second order ordinary differential equations. First, it establishes a connection between the finite difference method and the quasi-linearization method. We prove that using finite differences to discretize the sequence of linear differential equations arising from quasi-linearization (Newton method on operator level) leads to the usual iteration formula of the Newton finite difference method. From the provided derivation, it can easily be inferred that such a relation holds also for the Picard and the constant-slope methods. Based on this result, we propose a way of replacing the Newton, Picard, and constant-slope finite difference methods by respective successive application of the linear shooting method. This approach has a number of advantages. It removes the necessity of solving systems of algebraic equations, hence working with matrices, altogether. Compared to the usual finite difference method with general solver, it reduces the number of computational operations from O(N 3 ), where N is the number of mesh-points, to only O(N).

Original languageEnglish
Pages (from-to)46-60
Number of pages15
JournalJournal of Computational and Applied Mathematics
Volume358
DOIs
Publication statusPublished - Oct 1 2019

Fingerprint

Shooting Method
Nonlinear Boundary Value Problems
Two-point Boundary Value Problem
Finite difference method
Boundary value problems
Difference Method
Finite Difference
Quasilinearization Method
Linearization
Slope
Newton-Raphson method
Ordinary differential equations
Second-order Ordinary Differential Equations
Mathematical operators
Differential equations
Linear differential equation
Newton Methods
Algebraic Equation
Numerical Solution
Mesh

Keywords

  • Constant-slope
  • Finite difference method
  • Linear shooting
  • Newton
  • Picard
  • Quasi-linearization

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

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title = "Replacing the finite difference methods for nonlinear two-point boundary value problems by successive application of the linear shooting method",
abstract = "This paper studies numerical solution of nonlinear two-point boundary value problems for second order ordinary differential equations. First, it establishes a connection between the finite difference method and the quasi-linearization method. We prove that using finite differences to discretize the sequence of linear differential equations arising from quasi-linearization (Newton method on operator level) leads to the usual iteration formula of the Newton finite difference method. From the provided derivation, it can easily be inferred that such a relation holds also for the Picard and the constant-slope methods. Based on this result, we propose a way of replacing the Newton, Picard, and constant-slope finite difference methods by respective successive application of the linear shooting method. This approach has a number of advantages. It removes the necessity of solving systems of algebraic equations, hence working with matrices, altogether. Compared to the usual finite difference method with general solver, it reduces the number of computational operations from O(N 3 ), where N is the number of mesh-points, to only O(N).",
keywords = "Constant-slope, Finite difference method, Linear shooting, Newton, Picard, Quasi-linearization",
author = "Filipov, {Stefan M.} and Gospodinov, {Ivan D.} and I. Farag{\'o}",
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AU - Filipov, Stefan M.

AU - Gospodinov, Ivan D.

AU - Faragó, I.

PY - 2019/10/1

Y1 - 2019/10/1

N2 - This paper studies numerical solution of nonlinear two-point boundary value problems for second order ordinary differential equations. First, it establishes a connection between the finite difference method and the quasi-linearization method. We prove that using finite differences to discretize the sequence of linear differential equations arising from quasi-linearization (Newton method on operator level) leads to the usual iteration formula of the Newton finite difference method. From the provided derivation, it can easily be inferred that such a relation holds also for the Picard and the constant-slope methods. Based on this result, we propose a way of replacing the Newton, Picard, and constant-slope finite difference methods by respective successive application of the linear shooting method. This approach has a number of advantages. It removes the necessity of solving systems of algebraic equations, hence working with matrices, altogether. Compared to the usual finite difference method with general solver, it reduces the number of computational operations from O(N 3 ), where N is the number of mesh-points, to only O(N).

AB - This paper studies numerical solution of nonlinear two-point boundary value problems for second order ordinary differential equations. First, it establishes a connection between the finite difference method and the quasi-linearization method. We prove that using finite differences to discretize the sequence of linear differential equations arising from quasi-linearization (Newton method on operator level) leads to the usual iteration formula of the Newton finite difference method. From the provided derivation, it can easily be inferred that such a relation holds also for the Picard and the constant-slope methods. Based on this result, we propose a way of replacing the Newton, Picard, and constant-slope finite difference methods by respective successive application of the linear shooting method. This approach has a number of advantages. It removes the necessity of solving systems of algebraic equations, hence working with matrices, altogether. Compared to the usual finite difference method with general solver, it reduces the number of computational operations from O(N 3 ), where N is the number of mesh-points, to only O(N).

KW - Constant-slope

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KW - Linear shooting

KW - Newton

KW - Picard

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