Self-adjoint singularly perturbed boundary value problems: an adaptive variational approach.

*(English)*Zbl 1290.65064A modified variational algorithm for the numerical solution of self-adjoint singularly perturbed boundary value problems is presented. The algorithm is based on a mixed piecewise domain decomposition and a certain manipulation of the variational iterative approach. Uniform convergence to the exact solution is proved and numerical results are given.

Reviewer: Thomas Sonar (Braunschweig)

##### MSC:

65L11 | Numerical solution of singularly perturbed problems involving ordinary differential equations |

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

34B05 | Linear boundary value problems for ordinary differential equations |

34E15 | Singular perturbations for ordinary differential equations |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

##### Keywords:

variational method; singular perturbation; algorithm; boundary value problem; piecewise domain decomposition; convergence; numerical results
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\textit{S. A. Khuri} and \textit{A. Sayfy}, Math. Methods Appl. Sci. 36, No. 9, 1070--1079 (2013; Zbl 1290.65064)

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##### References:

[1] | Potter, Maximum Principles in Differential Equations (1967) |

[2] | Boglaev, A variational difference scheme for a boundary value problem with a small parameter in the highest derivative, U.S.S.R., Computational Mathematics and Mathematical Physics 21 (4) pp 71– (1981) · Zbl 0508.65044 |

[3] | Schatz, On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimension, Mathematics of Computation 40 pp 47– (1983) · Zbl 0518.65080 |

[4] | Niijima, On a three-point difference scheme for a singular perturbation problem without a first derivative term II, Memoirs of Numerical Mathematics 7 pp 11– (1980) · Zbl 0484.65054 |

[5] | Niijima, On a three-point difference scheme for a singular perturbation problem without a first derivative term I, Memoirs of Numerical Mathematics 7 pp 1– (1980) · Zbl 0484.65054 |

[6] | Miller, Numerical Analysis of Singular Perturbation Problems pp 467– (1979) |

[7] | Liu, The Lie-group shooting method for singularly perturbed two-point boundary value problems, CMES: Computer Modeling in Engineering & Sciences 15 pp 179– (2006) · Zbl 1152.65452 |

[8] | Liu, The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems, Communications in Nonlinear Science and Numerical Simulation 17 pp 1506– (2012) · Zbl 1244.65113 |

[9] | Kadalbajoo, Fitted mesh B-spline collocation method for solving self-adjoint singularly perturbed boundary value problems, Applied Mathematics and Computation 161 pp 973– (2005) · Zbl 1073.65062 |

[10] | Surla, Numerical Methods and Approximation Theory II pp 19– (1985) |

[11] | Sakai, On exponential splines, Journal of Approximation Theory 47 pp 122– (1986) · Zbl 0603.41007 |

[12] | Kadalbajoo, Geometric mesh FDM for self-adjoint singular perturbation boundary value problems, Applied Mathematics and Computation 190 pp 1646– (2007) · Zbl 1124.65064 |

[13] | Kumar, Wavelet optimized finite difference method using interpolating wavelets for self-adjoint singularly perturbed problems, Journal of Computational and Applied Mathematics 230 pp 803– (2009) · Zbl 1167.65407 |

[14] | Hao, Search for variational principles in electrodynamics by Lagrange method, International Journal of Nonlinear Sciences and Numerical Simulation 6 (2) pp 209– (2005) · Zbl 1401.78004 |

[15] | He, Variational iteration method for delay differential equations, Communications in Nonlinear Science and Numerical Simulation 2 (4) pp 235– (1997) · Zbl 0924.34063 |

[16] | He, Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation 114 pp 115– (2000) · Zbl 1027.34009 |

[17] | Inokuti, Variational Method in the Mechanics of Solids pp 156– (1978) |

[18] | Liu, Variational approach to nonlinear electrochemical system, International Journal of Nonlinear Sciences and Numerical Simulation 5 (1) pp 95– (2004) · Zbl 06942051 |

[19] | Liu, Generalized variational principles for ion acoustic plasma waves by He’s semi-inverse method, Chaos Solitons & Fractals 23 (2) pp 573– (2005) · Zbl 1135.76597 |

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