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Numerical solutions of viscoelastic bending wave equations with two term time kernels by Runge-Kutta convolution quadrature

The author is supported by NSFC grant 11671131, the Construct Program of the Key Discipline in Hunan Province, Performance Computing and Stochastic Information Processing (Ministry of Education of China).
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  • In this paper, we study the numerical solutions of viscoelastic bending wave equations

    $u_{t}(x,~t)-\int_{0}^{t}[\beta_{1}(t-s)\,u_{xx}(x,~s) - \beta_{2}(t-s)\,u_{xxxx}(x,~s)]ds = f(x,~t),$

    for $ 0<x<1,~ 0<t\leq T $, with self-adjoint boundary and initial value conditions, in which the functions $ \beta_{1}(t) $ and $ \beta_{2}(t) $ are completely monotonic on $ (0,~\infty) $ and locally integrable, but not constant. The equations are discretised in space by the finite difference method and in time by the Runge-Kutta convolution quadrature. The stability and convergence of the schemes are analyzed by the frequency domain and energy methods. Numerical experiments are provided to illustrate the accuracy and efficiency of the proposed schemes.

    Mathematics Subject Classification: Primary:45K05, 65J08;Secondary:65D32.

    Citation:

    \begin{equation} \\ \end{equation}
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  • Table 8.1.  The $ L_{2} $ errors and convergence rate of the 2-stage Radau IIA convolution quadrature of (8.1.1) with $ M=802 $, and $ t_{K}=1 $

    K $ e_{K} $ Rate
    2 $ 2.9416e-006 $ -
    4 $ 2.6658e-007 $ $ 3.4640 $
    8 $ 2.8401e-008 $ $ 3.2306 $
    2 $ 2.9416e-006 $ -
    8 $ 2.8401e-008 $ $ 3.3473 $
    Theory $ 3.0000 $
     | Show Table
    DownLoad: CSV

    Table 8.2.  The $ L_{2} $ errors and convergence rate of the 3-stage Radau IIA convolution quadrature of (8.1.1) with $ M=802 $ and $ t_{K}=1 $

    K $ e_{K} $ Rate
    2 $ 3.1196e-006 $ -
    4 $ 9.0652e-008 $ $ 5.1049 $
    8 $ 2.8981e-009 $ $ 4.9672 $
    2 $ 3.1196e-006 $ -
    8 $ 2.8981e-009 $ $ 5.0360 $
     | Show Table
    DownLoad: CSV

    Table 8.3.  The $ L_{2} $ errors and convergence rates of the 2-stage Radau IIA convolution quadrature of (8.1.1) with $ M=802 $, $ t_{K}=1 $, and $ f(x~t) = \frac{t^{5.5}}{\Gamma(6.5)} \sin(\pi x) e^{\pi x} (\pi x)^{2}(\pi-\pi x)^{2} $

    K $ e_{K} $ Rate
    2 $ 7.4612e-005 $ -
    4 $ 6.7979e-006 $ $ 3.4562 $
    8 $ 7.3481e-007 $ $ 3.2096 $
    2 $ 7.4612e-005 $ -
    8 $ 7.3481e-007 $ $ 3.3329 $
    Theory $ 3.0000 $
     | Show Table
    DownLoad: CSV

    Table 8.4.  The $ L_{2} $ errors and convergence rates of the 3-stage Radau IIA convolution quadrature of (8.1.1) with $ M=802 $, $ t_{K}=1 $, and $ f(x~t) = \frac{t^{5.5}}{\Gamma(6.5)} \sin(\pi x) e^{\pi x} (\pi x)^{2}(\pi-\pi x)^{2} $

    K $ e_{K} $ Rate
    2 $ 8.2319e-005 $ -
    4 $ 2.2979e-006 $ $ 5.1628 $
    8 $ 7.3507e-008 $ $ 4.9663 $
    2 $ 8.2319e-005 $ -
    8 $ 7.3507e-008 $ $ 5.0646 $
     | Show Table
    DownLoad: CSV

    Table 8.5.  The $ L_{2} $ errors and convergence rate of the 2-stage Radau IIA convolution quadrature of (5.1) with $ M=202 $, and $ t_{K}=1 $

    K $ e_{K} $ Rate
    2 $ 0.2789 $ -
    4 $ 0.0088 $ $ 4.9861 $
    8 $ 9.1479e-004 $ $ 3.2660 $
    16 $ 1.4166e-004 $ $ 2.6910 $
    2 $ 0.2789 $ -
    8 $ 9.1479e-004 $ $ 4.1260 $
    2 $ 0.2789 $ -
    16 $ 1.4166e-004 $ $ 3.6477 $
    4 $ 0.0088 $ -
    16 $ 1.4166e-004 $ $ 2.9785 $
     | Show Table
    DownLoad: CSV

    Table 8.6.  The $ L_{2} $ errors and convergence rate of the 3-stage Radau IIA convolution quadrature of (5.1) with $ M=202 $ and $ t_{K}=1 $

    K $ e_{K} $ Rate
    2 $ 0.0592 $ -
    4 $ 4.9980e-004 $ $ 6.8881 $
    8 $ 1.5134e-005 $ $ 5.0455 $
    2 $ 0.0592 $ -
    8 $ 1.5134e-005 $ $ 5.9668 $
     | Show Table
    DownLoad: CSV

    Table 8.7.  The $ L_{2} $ errors and convergence rates of the 2-stage Radau IIA convolution quadrature of (5.1) with $ M=202 $, $ t_{K}=1 $, and $ u_{0}(x)= \sin(\pi x) e^{\cos(\pi x)} $

    K $ e_{K} $ Rate
    4 $ 0.0593 $ -
    8 $ 0.0076 $ $ 2.9640 $
    16 $ 8.7669e-004 $ $ 3.1159 $
    4 $ 0.0593 $ -
    16 $ 8.7669e-004 $ $ 3.0399 $
     | Show Table
    DownLoad: CSV

    Table 8.8.  The $ L_{2} $ errors and convergence rates of the 3-stage Radau IIA convolution quadrature of (5.1) with $ M=202 $, $ t_{K}=1 $, and $ u_{0}(x)= (\sin(\pi x))^{3} $

    K $ e_{K} $ Rate
    2 $ 0.0664 $ -
    4 $ 0.0027 $ $ 4.6206 $
    8 $ 7.4457e-005 $ $ 5.1804 $
    2 $ 0.0664 $ -
    8 $ 7.4457e-005 $ $ 4.9003 $
     | Show Table
    DownLoad: CSV
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