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Numerical solutions of Volterra integro-differential equations using General Linear Method

The reviewing process of the paper is handled by Gafurjan Ibragimov, Siti Hasana Sapar and Siti Nur Iqmal Ibrahim

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  • In this paper, a third order General Linear Method for finding the numerical solution of Volterra integro-differential equation is considered. The order conditions of the proposed method are derived based on techniques of B-series and 'rooted trees'. The integral operator in Volterra integro-differential equation approximated using Simpson's rule and Lagrange interpolation is discussed. To illustrate the efficiency of third order General Linear Method, we compare the method with a third order Runge-Kutta method.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • Figure 1.  Log maximum error versus number of functions evaluations for Problem 1

    Figure 2.  Log maximum error versus number of functions evaluations for Problem 2

    Figure 3.  Log maximum error versus number of functions evaluations for Problem 3

    Figure 4.  Log maximum error versus number of functions evaluations for Problem 4

    Figure 5.  Log maximum error versus number of functions evaluations for Problem 5

    Table 1.  Matrix representation of coefficients of GLM.

    $ A_{s\times s} $ $ U_{s\times r} $
    $ B_{r\times s} $ $ V_{r\times r} $
     | Show Table
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    Table 2.  Matrix coefficients of GLM with $ s = 3 $, $ r = 2 $.

    $\left[ {\begin{array}{*{20}{l}} 0&0&0\\ {{a_{21}}}&0&0\\ {{a_{31}}}&{{a_{32}}}&0 \end{array}} \right] $ $\left[ {\begin{array}{*{20}{l}} {{u_{11}}}&{{u_{12}}}\\ {{u_{21}}}&{{u_{22}}}\\ {{u_{31}}}&{{u_{32}}} \end{array}} \right]$
    $\left[ {\begin{array}{*{20}{l}} {{b_{11}}}&{{b_{12}}}&{{b_{13}}}\\ {{b_{21}}}&{{b_{22}}}&{{b_{23}}} \end{array}} \right]$ $\left[ {\begin{array}{*{20}{c}} 1&{{v_{12}}}\\ {{v_{21}}}&0 \end{array}} \right]$
     | Show Table
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    Table 3.  Order conditions of GLM up to order three.

    No Order conditions
    1 $ b_{11}(u_{11}+u_{12})+b_{12}(u_{21}+u_{22})+b_{13}(u_{31}+u_{32})-v_{12}=1 $
    2 $ b_{21}(u_{11}+u_{12})+b_{22}(u_{21}+u_{22})+b_{23}(u_{31}+u_{32})=0 $
    3 $ -b_{11}u_{12}+b_{12}(a_{21}(u_{11} + u_{12})-u_{22})+b_{13}\big(a_{31}(u_{11} + u_{12})+a_{32}(u_{21} + u_{22})-u_{32}\big) $
    $ \qquad+v_{12}\xi_{22}=\frac{1}{2} $
    4 $ -b_{21}u_{12}+b_{22}(a_{21}(u_{11} + u_{12})-u_{22})+b_{23}\big(a_{31}(u_{11} + u_{12})+a_{32}(u_{21} + u_{22})-u_{32}\big)=0 $
    5 $ b_{11}u_{12}^{2}+b_{12}\big(a_{21}(u_{11}+u_{12})-u_{22}\big)^{2}+b_{13}\big(a_{31}(u_{11} + u_{12})+a_{32}(u_{21} + u_{22})-u_{32}\big)^{2} $
    $ \qquad+v_{12}\xi_{23}=\frac{1}{3} $
    6 $ b_{21}u_{12}^{2}+b_{22}\big(a_{21}(u_{11}+u_{12})-u_{22}\big)^{2}+b_{23}\big(a_{31}(u_{11} + u_{12})+a_{32}(u_{21} + u_{22})-u_{32}\big)^{2}=0 $
    7 $ b_{11}u_{12}\xi_{22}+b_{12}\big(\xi_{22}u_{22} - a_{21}u_{12}\big)+b_{13}\big(-a_{31}u_{12} + a_{32}(a_{21}(u_{11} + u_{12})-u_{22}) $
    $ \qquad +u_{32}\xi_{22}\big)+v_{12}\xi_{24}=\frac{1}{6} $
    8 $ b_{21}u_{12}\xi_{22}+b_{22}\big(\xi_{22}u_{22} - a_{21}u_{12}\big)+b_{23}\big(-a_{31}u_{12} + a_{32}(a_{21}(u_{11} + u_{12})-u_{22}) $
    $ \qquad+u_{32}\xi_{22}\big)=0 $
    9 $ -b_{11}u_{12}^{3}+b_{12}(a_{21}(u_{11} + u_{12})-u_{22})^{3}+b_{13}\big(a_{31}(u_{11} + u_{12}) +a_{32}(u_{21} + u_{22})-u_{32}\big)^{3} $
    $ \qquad+v_{12}\xi_{25}\frac{1}{4}=\frac{1}{4} $
    10 $ -b_{21}u_{12}^{3}+b_{22}(a_{21}(u_{11} + u_{12})-u_{22})^{3}+b_{23}\big(a_{31}(u_{11} + u_{12}) +a_{32}(u_{21} + u_{22}) $
    $ \qquad-u_{32}\big)^{3}=0 $
    11 $ -b_{11}u_{12}^{2}\xi_{22} + b_{12}(a_{21}(u_{11} + u_{12})-u_{22})(\xi_{22}u_{22}-a_{21}u_{12}) +b_{13}\big(a_{31}(u_{11} + u_{12})+a_{32} $
    $ \qquad (u_{21}+u_{22})-u_{32}\big)\big(-a_{31}u_{12}+a_{32}(a_{21}(u_{11}+u_{12})-u_{22})+u_{32}\xi_{22}\big)+v_{12}\xi_{26} =\frac{1}{8} $
    12 $ -b_{21}u_{12}^{2}\xi_{22} + b_{22}(a_{21}(u_{11} + u_{12})-u_{22})(\xi_{22}u_{22}-a_{21}u_{12}) +b_{23}\big(a_{31}(u_{11} + u_{12}) $
    $ \qquad+a_{32}(u_{21} + u_{22})-u_{32}\big)\big(-a_{31}u_{12}+a_{32}(a_{21}(u_{11}+u_{12})-u_{22})+u_{32}\xi_{22}\big)=0 $
    13 $ b_{11}u_{12}\xi_{23} + b_{12}(a_{21}u_{12}^{2}+\xi_{23}u_{22}) + b_{13}\big(a_{31}u_{12}^{2} + a_{32}(a_{21}(u_{11} + u_{12})-u_{22})^{2}+u_{32}\xi_{23}\big) $
    $ \qquad+v_{12}\xi_{27}=\frac{1}{12} $
    14 $ b_{21}u_{12}\xi_{23} + b_{22}(a_{21}u_{12}^{2}+\xi_{23}u_{22}) + b_{23}\big(a_{31}u_{12}^{2} + a_{32}(a_{21}(u_{11} + u_{12})-u_{22})^{2} $
    $ \qquad+u_{32}\xi_{23}\big)=0 $
    15 $ b_{11}u_{12}\xi_{24} + b_{12}(\xi_{22}a_{21}u_{12}+\xi_{24}u_{22}) + b_{13}\big(a_{31}u_{12}\xi_{22} + a_{32}(\xi_{22}u_{22} -a_{21}u_{12})+u_{32}\xi_{24}\big) $
    $ \qquad+v_{12}\xi_{28}=\frac{1}{24} $
    16 $ b_{21}u_{12}\xi_{24} + b_{22}(\xi_{22}a_{21}u_{12}+\xi_{24}u_{22}) + b_{23}\big(a_{31}u_{12}\xi_{22} + a_{32}(\xi_{22}u_{22} -a_{21}u_{12}) $
    $ \qquad+u_{32}\xi_{24}\big)=0 $
     | Show Table
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    Table 4.  Coefficients Set 1 of third order GLM

    $ u_{11}=1 $ $ u_{12}=0 $
    $ a_{21}=\frac{13}{18} $ $ u_{21}=\frac{7}{9} $ $ u_{22}=\frac{2}{9} $
    $ a_{31}=\frac{-17}{9} $ $ a_{32}=2 $ $ u_{31}=\frac{17}{9} $ $ u_{32}=\frac{-8}{9} $
    $ b_{11}=\frac{1}{6} $ $ b_{12}=\frac{2}{3} $ $ b_{13}=\frac{1}{6} $ $ v_{11}=1 $ $ v_{12}=0 $
    $ b_{21}=0 $ $ b_{22}=0 $ $ b_{23}=0 $ $ v_{21}=1 $ $ v_{22}=0 $
     | Show Table
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    Table 5.  Coefficients Set 2 of third order GLM

    $ u_{11}=1 $ $ u_{12}=0 $
    $ a_{21}=\frac{2}{3} $ $ u_{21}=\frac{5}{6} $ $ u_{22}=\frac{1}{6} $
    $ a_{31}=\frac{-5}{3} $ $ a_{32}=2 $ $ u_{31}=\frac{5}{3} $ $ u_{32}=\frac{-2}{3} $
    $ b_{11}=\frac{1}{6} $ $ b_{12}=\frac{2}{3} $ $ b_{13}=\frac{1}{6} $ $ v_{11}=1 $ $ v_{12}=0 $
    $ b_{21}=0 $ $ b_{22}=0 $ $ b_{23}=0 $ $ v_{21}=1 $ $ v_{22}=0 $
     | Show Table
    DownLoad: CSV

    Table 6.  Coefficients Set 3 of third order GLM

    $ u_{11}=1 $ $ u_{12}=0 $
    $ a_{21}=\frac{5}{6} $ $ u_{21}=\frac{2}{3} $ $ u_{22}=\frac{1}{3} $
    $ a_{31}=\frac{-7}{3} $ $ a_{32}=2 $ $ u_{31}=\frac{7}{3} $ $ u_{32}=\frac{-4}{3} $
    $ b_{11}=\frac{1}{6} $ $ b_{12}=\frac{2}{3} $ $ b_{13}=\frac{1}{6} $ $ v_{11}=1 $ $ v_{12}=0 $
    $ b_{21}=0 $ $ b_{22}=0 $ $ b_{23}=0 $ $ v_{21}=1 $ $ v_{22}=0 $
     | Show Table
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    Table 7.  Maximum global errors for Problem 1

    GLM, $ s=3 $ RK, $ s=3 $
    Step size MAXE
    $ h=0.1 $ $ 1.2347\times 10^{-6} $ $ 4.7137\times 10^{-6} $
    $ h=0.025 $ $ 9.9859\times 10^{-9} $ $ 7.1772\times 10^{-8} $
    $ h=0.01 $ $ 5.6041\times 10^{-10} $ $ 4.6094\times 10^{-9} $
    $ h=0.005 $ $ 6.7079\times 10^{-11} $ $ 5.7715\times 10^{-10} $
    $ h=0.001 $ $ 5.1845\times 10^{-13} $ $ 4.6243\times 10^{-12} $
     | Show Table
    DownLoad: CSV

    Table 8.  Maximum global errors for Problem 2

    GLM, $ s=3 $ RK, $ s=3 $
    Step size MAXE
    $ h=0.1 $ $ 2.4606\times 10^{-6} $ $ 6.9906\times 10^{-6} $
    $ h=0.025 $ $ 1.6319\times 10^{-8} $ $ 1.0137\times 10^{-7} $
    $ h=0.01 $ $ 8.3870\times 10^{-10} $ $ 6.4622\times 10^{-9} $
    $ h=0.005 $ $ 9.7077\times 10^{-11} $ $ 8.0749\times 10^{-10} $
    $ h=0.001 $ $ 7.2935\times 10^{-13} $ $ 6.4604\times 10^{-12} $
     | Show Table
    DownLoad: CSV

    Table 9.  Maximum global errors for Problem 3

    GLM, $ s=3 $ RK, $ s=3 $
    Step size MAXE
    $ h=0.1 $ $ 3.9332\times 10^{-6} $ $ 4.8141\times 10^{-5} $
    $ h=0.025 $ $ 1.4323\times 10^{-7} $ $ 1.4400\times 10^{-6} $
    $ h=0.01 $ $ 1.0939\times 10^{-8} $ $ 1.0134\times 10^{-7} $
    $ h=0.005 $ $ 1.4325\times 10^{-9} $ $ 1.3052\times 10^{-8} $
    $ h=0.001 $ $ 1.1851\times 10^{-11} $ $ 1.0688\times 10^{-10} $
     | Show Table
    DownLoad: CSV

    Table 10.  Maximum global errors for Problem 4

    GLM, $ s=3 $ RK, $ s=3 $
    Step size MAXE
    $ h=0.1 $ $ 4.5416\times 10^{-6} $ $ 1.4256\times 10^{-5} $
    $ h=0.025 $ $ 1.7061\times 10^{-8} $ $ 2.5908\times 10^{-7} $
    $ h=0.01 $ $ 1.4270\times 10^{-9} $ $ 1.7075\times 10^{-8} $
    $ h=0.005 $ $ 2.1002\times 10^{-10} $ $ 2.1553\times 10^{-9} $
    $ h=0.001 $ $ 1.8836\times 10^{-12} $ $ 1.7377\times 10^{-11} $
     | Show Table
    DownLoad: CSV

    Table 11.  Maximum global errors for Problem 5

    GLM, $ s=3 $ RK, $ s=3 $
    Step size MAXE
    $ h=0.1 $ $ 6.3429\times 10^{-6} $ $ 3.3432\times 10^{-5} $
    $ h=0.025 $ $ 4.4142\times 10^{-8} $ $ 6.3237\times 10^{-7} $
    $ h=0.01 $ $ 4.0164\times 10^{-9} $ $ 4.1259\times 10^{-8} $
    $ h=0.005 $ $ 5.4245\times 10^{-10} $ $ 5.1811\times 10^{-9} $
    $ h=0.001 $ $ 4.5689\times 10^{-12} $ $ 4.1572\times 10^{-11} $
     | Show Table
    DownLoad: CSV

    Table 12.  Total number of function evaluations Problems 1 - 5

    GLM, $ s=3 $ RK, $ s=3 $
    Step size TFE
    $ h=0.1 $ $ 34 $ $ 34 $
    $ h=0.025 $ $ 124 $ $ 124 $
    $ h=0.01 $ $ 304 $ $ 304 $
    $ h=0.005 $ $ 604 $ $ 604 $
    $ h=0.001 $ $ 3004 $ $ 3004 $
     | Show Table
    DownLoad: CSV
  • [1] J. C. Butcher, General linear methods, Acta Numerica, 15 (2006), 157-256.  doi: 10.1017/S0962492906220014.
    [2] J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley and Sons, Chichester, 2008. doi: 10.1002/9781119121534.
    [3] P. ChartierE. Hairer and G. Vilmart, Algebraic structures of B-series, Foundations of Computational Mathematics, 10 (2010), 407-427.  doi: 10.1007/s10208-010-9065-1.
    [4] J. R. DormandNumerical Methods for Differential Equations: A Computational Approach, CRC Press, Florida, 1992.  doi: 10.1201/9781351075107.
    [5] A. Filiz, A fourth-order robust numerical method for integro-differential equations, Asian Journal of Fuzzy and Applied Mathematics, 1 (2013), 28-33. 
    [6] A. Filiz, Numerical solution of linear volterra integro-differential equations using runge-kutta-felhberg method, Applied and Computational Mathematics, 1 (2014), 9-14. 
    [7] A. Filiz, General linear methods for ordinary differential equations, Mathematics and Computers in Simulation, 79 (2009), 1834-1845.  doi: 10.1016/j.matcom.2007.02.006.
    [8] P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, 1985.
    [9] F. RabieiF. A. HamidM. M. Rashidi and F. Ismail, Numerical simulation of fuzzy differential equations using general linear method and B-series, Advances in Mechanical Engineering, 9 (2010), 1-16. 
    [10] B. Raftari, Numerical solutions of the linear volterra integro-differential equations: Homotopy perturbation method and finite difference method, World Applied Sciences Journal, 9 (2010), 7-12. 
    [11] A. M. Wazwaz, Linear and Nonlinear Integral Equations, Springer, Beijing, 2011. doi: 10.1007/978-3-642-21449-3.
    [12] M. Zarebnia, Sinc numerical solution for the Volterra integro-differential equation, Nonlinear Sci. Numer. Simulat., 15 (2010), 700-706.  doi: 10.1016/j.cnsns.2009.04.021.
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