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**A class of non-uniform mesh three point arithmetic average discretization for \(y^{\prime\prime} = f(x, y, y^{\prime}\) and the estimates of \(y^{\prime}\).**
*(English)*
Zbl 1104.65315

Summary: We propose two new non-uniform mesh three point arithmetic average discretization strategy of order two and three, to solve non-linear ordinary differential equation \(y^{\prime\prime}= f(x, y, y^{\prime})\), \(a < x < b\) and the estimates of first-order derivative \(y^{\prime}\), where \(y = y(x)\), subject to the essential boundary conditions \(y(a) = A, y(b) = B\). Both methods are compact and directly applicable to singular problems. There is no need to discuss any special scheme for the singular problems. Error analysis of a proposed method is discussed. Numerical experiments are performed to study the convergence behaviors and efficiency of the proposed methods.

### MSC:

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

34B16 | Singular nonlinear boundary value problems for ordinary differential equations |

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

65L12 | Finite difference and finite volume methods for ordinary differential equations |

65L70 | Error bounds for numerical methods for ordinary differential equations |

### Keywords:

non-uniform mesh; non-linear equation; singular problem; diffusion-convection equation; Burgers’ equation; error analysis; numerical experiments; convergence
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\textit{R. K. Mohanty}, Appl. Math. Comput. 183, No. 1, 477--485 (2006; Zbl 1104.65315)

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### References:

[1] | Keller, H. B., Numerical Methods for Two Points Boundary Value Problems (1968), Waltham Mass: Blaisdell Pub. Co: Waltham Mass: Blaisdell Pub. Co New York · Zbl 0172.19503 |

[2] | Bogucz, F. A.; Walker, J. D.A., Fourth order finite difference methods for two point boundary value problems, IMA. J. Numer. Anal., 4, 69-82 (1984) · Zbl 0544.65052 |

[3] | Fox, L.; Mayers, D. F., Numerical Solution of Ordinary Differential Equations (1990), Chapman and Hall: Chapman and Hall London · Zbl 0643.34001 |

[4] | Jain, M. K.; Iyengar, S. R.K.; Subramanyam, G. S., Variable mesh methods for the numerical solution of two point singular perturbation problems, Comput. Methods Appl. Mech. Eng., 42, 273-286 (1984) · Zbl 0514.65065 |

[5] | Mohanty, R. K.; Evans, D. J.; Dey, Shivani, Three points discretization of order four and six for (d \(u\)/d \(x)\) of the solution of non-linear singular two point boundary value problems, Int. J. Comput. Math., 78, 123-139 (2001) · Zbl 0984.65075 |

[6] | Evans, D. J.; Mohanty, R. K., On the application of the SMAGE parallel algorithms on a non-uniform mesh for the solution of non-linear two-point boundary value problems with singularity, Int. J. Comput. Math., 82, 341-353 (2005) · Zbl 1064.65064 |

[7] | Mohanty, R. K., A family of variable mesh methods for the estimates of (d \(u\)/d \(r)\) and the solution of non-linear two-point boundary value problems with singularity, J. Comput. Appl. Math., 182, 173-187 (2005) · Zbl 1071.65113 |

[8] | Chawla, M. M.; Shivakumar, P. N., An efficient finite difference method for two-point boundary value problems, Neural Parallel Sci. Comput., 4, 387-396 (1996) · Zbl 1060.65626 |

[9] | Stephenson, J. W., Single cell discretization of order two and four for biharmonic problems, J. Comput. Phys., 55, 65-80 (1984) · Zbl 0542.65051 |

[10] | Hageman, L. A.; Young, D. M., Applied Iterative Methods (1981), Academic Press: Academic Press New York · Zbl 0459.65014 |

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