Fully discretized collocation methods for nonlinear singular Volterra integral equations

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Abstract

We consider a nonlinear weakly singular Volterra integral equation arising from a problem studied by Lighthill (1950) [1]. A series expansion for the solution is obtained and shown to be convergent in a neighbourhood of the origin. Owing to the singularity of the solution at the origin, the global convergence order of product integration and collocation methods is not optimal. However, the optimal orders can be recovered if we use the fully discretized collocation methods based on graded meshes. A theoretical proof is given and we present some numerical results which illustrate the performance of the methods.
Original languageUnknown
Pages (from-to)84-101
JournalJournal of Computational and Applied Mathematics
Volume247
Issue numberNA
DOIs
Publication statusPublished - 1 Jan 2013

Keywords

    Cite this

    @article{25c8789c1de74476bc77472b8ca37c58,
    title = "Fully discretized collocation methods for nonlinear singular Volterra integral equations",
    abstract = "We consider a nonlinear weakly singular Volterra integral equation arising from a problem studied by Lighthill (1950) [1]. A series expansion for the solution is obtained and shown to be convergent in a neighbourhood of the origin. Owing to the singularity of the solution at the origin, the global convergence order of product integration and collocation methods is not optimal. However, the optimal orders can be recovered if we use the fully discretized collocation methods based on graded meshes. A theoretical proof is given and we present some numerical results which illustrate the performance of the methods.",
    keywords = "Cordial equation, Nonlinear-Volterra integral equation, Fully discretized collocation methods, Graded meshes",
    author = "Rebelo, {Magda Stela de Jesus}",
    note = "Sem PDf conforme Despacho",
    year = "2013",
    month = "1",
    day = "1",
    doi = "10.1016/j.cam.2013.01.002",
    language = "Unknown",
    volume = "247",
    pages = "84--101",
    journal = "Journal of Computational and Applied Mathematics",
    issn = "0377-0427",
    publisher = "Elsevier Science B.V., Inc",
    number = "NA",

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    TY - JOUR

    T1 - Fully discretized collocation methods for nonlinear singular Volterra integral equations

    AU - Rebelo, Magda Stela de Jesus

    N1 - Sem PDf conforme Despacho

    PY - 2013/1/1

    Y1 - 2013/1/1

    N2 - We consider a nonlinear weakly singular Volterra integral equation arising from a problem studied by Lighthill (1950) [1]. A series expansion for the solution is obtained and shown to be convergent in a neighbourhood of the origin. Owing to the singularity of the solution at the origin, the global convergence order of product integration and collocation methods is not optimal. However, the optimal orders can be recovered if we use the fully discretized collocation methods based on graded meshes. A theoretical proof is given and we present some numerical results which illustrate the performance of the methods.

    AB - We consider a nonlinear weakly singular Volterra integral equation arising from a problem studied by Lighthill (1950) [1]. A series expansion for the solution is obtained and shown to be convergent in a neighbourhood of the origin. Owing to the singularity of the solution at the origin, the global convergence order of product integration and collocation methods is not optimal. However, the optimal orders can be recovered if we use the fully discretized collocation methods based on graded meshes. A theoretical proof is given and we present some numerical results which illustrate the performance of the methods.

    KW - Cordial equation

    KW - Nonlinear-Volterra integral equation

    KW - Fully discretized collocation methods

    KW - Graded meshes

    U2 - 10.1016/j.cam.2013.01.002

    DO - 10.1016/j.cam.2013.01.002

    M3 - Article

    VL - 247

    SP - 84

    EP - 101

    JO - Journal of Computational and Applied Mathematics

    JF - Journal of Computational and Applied Mathematics

    SN - 0377-0427

    IS - NA

    ER -