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    Wyświetlanie 1-3 z 3
    Tytuł:
    A local truncation error estimation for a SubIval solver
    Autorzy:
    Sowa, M.
    Powiązania:
    https://bibliotekanauki.pl/articles/202058.pdf
    Data publikacji:
    2018
    Wydawca:
    Polska Akademia Nauk. Czytelnia Czasopism PAN
    Tematy:
    fractional calculus
    numerical method
    adaptive step size
    local truncation error
    circuit analysis
    rachunek różniczkowy
    metoda numeryczna
    analiza obwodów
    Opis:
    The paper concerns an analysis for SubIval (the subinterval-based method for fractional derivative computations in initial value problems). A time step size adaptive solver is discussed, for which the formula of a local truncation error is derived. A general form for a system of linear equations is given for the considered class of problems (for which the analysis is performed in the paper). Two circuit examples are introduced to display the usefulness of the SubIval solver. For the examples that have been chosen it is possible to obtain referential solutions through completely different methods. The results obtained through the numerical solver are compared with evaluations of the referential solutions. The error estimation results obtained for the time steps of the SubIval solver are compared with the actual errors, being the differences between the numerical solutions and the referential solutions. The paper also contains a comparison of the accuracy of results obtained through the SubIval solver with the accuracies of other solvers.
    Źródło:
    Bulletin of the Polish Academy of Sciences. Technical Sciences; 2018, 66, 4; 475-484
    0239-7528
    Pojawia się w:
    Bulletin of the Polish Academy of Sciences. Technical Sciences
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Use of partial derivatives to derive a convergent numerical scheme with its error estimates
    Autorzy:
    Qureshi, Sania
    Adeyeye, Oluwaseun
    Shaikh, Asif Ali
    Powiązania:
    https://bibliotekanauki.pl/articles/122734.pdf
    Data publikacji:
    2019
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    multi-derivative
    local truncation error
    stability
    absolute relative errors
    consistency
    principal term
    wielopochodna
    pochodna cząstkowa
    schemat numeryczny
    błąd względny
    błąd bezwzględny
    Opis:
    Using the idea of the partial derivative with respect to the ordinate of a given mathematical function, a new numerical scheme having third order convergence has been devised for solving initial value problems in ordinary differential equations. Such problems are deemed to be indispensable in diverse fields of science, medical and engineering and are most often required to be solved by the numerical schemes. In view of this, the proposed numerical scheme is found to be efficient in solving both autonomous and non-autonomous type of problems as supported by some numerical experiments in the present study. Using the Taylor expansion for the slopes involved in the scheme, the leading term of the local truncation error is shown to have contained Ϭ(h4) which proves third order accuracy of the scheme. In addition to this, consistency and linear stability analysis of the proposed scheme has extensively been discussed. Numerical experiments show better performance of the proposed numerical scheme when compared with existing numerical schemes of the same order as that of the scheme proposed. CPU time (seconds), maximum absolute relative error and the absolute relative error, computed at the last grid point of the integration interval for the associated initial value problem, are the parameters to test the performance of the proposed numerical scheme. MATLAB Version: 9.4.0.813654 (R2018a) in double-precision on a personal computer equipped with a Processor Intel (R) Core(TM) i3-4500U CPU@ 1.70 GHz running under the Windows 10 operating system has been employed in order to carry out all the required numerical computations.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2019, 18, 4; 73-83
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Local accuracy and error bounds of the improved Runge-Kutta numerical methods
    Autorzy:
    Qureshi, S.
    Memon, Z.
    Shaikh, A. A.
    Powiązania:
    https://bibliotekanauki.pl/articles/122862.pdf
    Data publikacji:
    2018
    Wydawca:
    Politechnika Częstochowska. Wydawnictwo Politechniki Częstochowskiej
    Tematy:
    error estimate
    remainder term
    principal error function
    truncation error
    Lotkin bound
    granice błędu
    metoda numeryczna Runge-Kutty
    błąd obcięcia
    granica Lotkina
    funkcja błędu
    szacowanie błędu
    Opis:
    In this paper, explicit Improved Runge-Kutta (IRK) methods with two, three and four stages have been analyzed in detail to derive the error estimates inherent in them whereas their convergence, order of local accuracy, stability and arithmetic complexity have been proved in the relevant literature. Using single and multivariate Taylor series expansion for a mathematical function of one and two variables respectively, slopes involved in the IRK methods have been expanded in order to obtain the general expression for the leading or principal term in the local truncation error of the methods. In addition to this, principal error functions of the methods have also been derived using the idea of Lotkin bounds which consequently gave rise to the error estimates for the IRK methods. Later, these error estimates were compared with error estimates of the two, three, and four-stage standard explicit Runge-Kutta (RK) methods to show the better performance of the IRK methods in terms of the error bounds on the constant step-size h used for solving the initial value problems in ordinary differential equations. Finally, a couple of initial value problems have been tested to determine the maximum absolute global errors, absolute errors at the final nodal point of the integration interval and the CPU times (seconds) for all the methods under consideration to get a better idea of how the methods behave in a particular situation especially when it comes to analyzing the error terms.
    Źródło:
    Journal of Applied Mathematics and Computational Mechanics; 2018, 17, 4; 73-84
    2299-9965
    Pojawia się w:
    Journal of Applied Mathematics and Computational Mechanics
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-3 z 3

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