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A new numerical approach for solving high-order non-linear ordinary differential equations

Journal Article


Abstract


  • There have been many numerical solution approaches to ordinary dierential equations in the literature.

    However, very few are eective in solving non-linear ordinary dierential equations (ODEs), particularly

    when they are of order higher than one. With modern symbolic calculation packages, such as Maple and

    Mathematica, being readily available to researchers, we shall present a new numerical method in this

    paper. Based on the repeated use of a symbolic calculation package and a second-order nite-dierence

    scheme, our method is particularly suitable for solving high-order non-linear dierential equations arising

    from initial-value problems. One important feature of our approach is that if the highest-order derivative

    in an ODE can be written explicitly in terms of all the other terms of lower orders, our method requires

    no iterations at all. On the other hand, if the highest-order derivative in an ODE cannot be written

    explicitly in terms of all the other lower-order terms, iterations are only required before the actual time

    marching begins.Copyright ? 2003 John Wiley & Sons, Ltd.

Publication Date


  • 2003

Citation


  • Zhu, S. & Phan, H. (2003). A new numerical approach for solving high-order non-linear ordinary differential equations. Communications in Numerical Methods in Engineering, 19 (8), 601-614.

Scopus Eid


  • 2-s2.0-0042326141

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/767

Has Global Citation Frequency


Number Of Pages


  • 13

Start Page


  • 601

End Page


  • 614

Volume


  • 19

Issue


  • 8

Place Of Publication


  • England

Abstract


  • There have been many numerical solution approaches to ordinary dierential equations in the literature.

    However, very few are eective in solving non-linear ordinary dierential equations (ODEs), particularly

    when they are of order higher than one. With modern symbolic calculation packages, such as Maple and

    Mathematica, being readily available to researchers, we shall present a new numerical method in this

    paper. Based on the repeated use of a symbolic calculation package and a second-order nite-dierence

    scheme, our method is particularly suitable for solving high-order non-linear dierential equations arising

    from initial-value problems. One important feature of our approach is that if the highest-order derivative

    in an ODE can be written explicitly in terms of all the other terms of lower orders, our method requires

    no iterations at all. On the other hand, if the highest-order derivative in an ODE cannot be written

    explicitly in terms of all the other lower-order terms, iterations are only required before the actual time

    marching begins.Copyright ? 2003 John Wiley & Sons, Ltd.

Publication Date


  • 2003

Citation


  • Zhu, S. & Phan, H. (2003). A new numerical approach for solving high-order non-linear ordinary differential equations. Communications in Numerical Methods in Engineering, 19 (8), 601-614.

Scopus Eid


  • 2-s2.0-0042326141

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/767

Has Global Citation Frequency


Number Of Pages


  • 13

Start Page


  • 601

End Page


  • 614

Volume


  • 19

Issue


  • 8

Place Of Publication


  • England