Solving the equation - uxx - ∈Uyy = f(x, y, u) by an O(h4) finite difference method

E. Lungu, T. Motsumi, Tadeusz Styś

    Research output: Contribution to journalArticle

    2 Citations (Scopus)

    Abstract

    The semi-linear equation - uxx - ∈uyy = f(x,y,u) with Dirichlet boundary conditions is solved by an O(h4) finite difference method, which has local truncation error O(h2) at the mesh points neighboring the boundary and O(h4) at most interior mesh points. It is proved that the finite difference method is O(h4) uniformly convergent as h → 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using the method. To illustrate the method, the equation of twisting a springy rod is solved.

    Original languageEnglish
    Pages (from-to)395-407
    Number of pages13
    JournalNumerical Methods for Partial Differential Equations
    Volume16
    Issue number4
    Publication statusPublished - Jul 2000

    Fingerprint

    Finite difference method
    Difference Method
    Finite Difference
    Algebraic Equation
    Iterative methods
    Linear equations
    Mesh
    Semilinear Equations
    Truncation Error
    Mathematica
    Implicit Method
    Diagonal matrix
    Boundary conditions
    Sparse matrix
    Dirichlet Boundary Conditions
    Gauss
    Elimination
    Testing
    Interior
    Iteration

    All Science Journal Classification (ASJC) codes

    • Analysis
    • Applied Mathematics
    • Computational Mathematics

    Cite this

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    abstract = "The semi-linear equation - uxx - ∈uyy = f(x,y,u) with Dirichlet boundary conditions is solved by an O(h4) finite difference method, which has local truncation error O(h2) at the mesh points neighboring the boundary and O(h4) at most interior mesh points. It is proved that the finite difference method is O(h4) uniformly convergent as h → 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using the method. To illustrate the method, the equation of twisting a springy rod is solved.",
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    Solving the equation - uxx - ∈Uyy = f(x, y, u) by an O(h4) finite difference method. / Lungu, E.; Motsumi, T.; Styś, Tadeusz.

    In: Numerical Methods for Partial Differential Equations, Vol. 16, No. 4, 07.2000, p. 395-407.

    Research output: Contribution to journalArticle

    TY - JOUR

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    AU - Lungu, E.

    AU - Motsumi, T.

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