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Efficient Legendre dual-Petrov-Galerkin methods for odd-order differential equations

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    * Corresponding author

This work was supported in part by National Natural Science Foundation of China (Nos. 11601331 and 11571238)

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  • Efficient Legendre dual-Petrov-Galerkin methods for solving odd-order differential equations are proposed. Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier-like series. Numerical results indicate that the suggested methods are extremely accurate and efficient, and suitable for the odd-order equations.

    Mathematics Subject Classification: 65N35, 33C45, 35J58.

    Citation:

    \begin{equation} \\ \end{equation}
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  • Figure 1.  Numerical errors of scheme (20)

    Figure 2.  Numerical errors of scheme (20)

    Figure 3.  Numerical errors of scheme (41)

    Figure 4.  Numerical errors of scheme (41)

    Figure 5.  Numerical errors of scheme (59)

    Figure 6.  Numerical errors of scheme (59)

    Table 1.  Condition numbers of the usual Legendre dual-Petrov-Galerkin method for problem (19)

    $ \rm Matrices $ $ N=20 $ $ N=40 $ $ N=60 $ $ N=80 $ $ N=100 $
    $ \alpha(p_k, q_l) $ 3.1644e+05 2.3889e+07 3.3847e+08 2.3036e+09 1.0362e+10
    $ \beta(p_k, \partial_xq_l) $ 1.5106e+04 3.6487e+05 2.5120e+06 1.0070e+07 2.9818e+07
    $ \gamma(\partial_xp_k, \partial_xq_l) $ 2.4015e+03 3.9365e+04 2.0296e+05 6.4966e+05 1.6010e+06
    $ (\partial_xp_k, \partial^2_xq_l) $ 2.5926e+02 2.3471e+03 8.4079e+03 2.0652e+04 4.1318e+04
    $ \mathcal{A}(p_k, q_l) $ 2.5961e+02 2.3422e+03 8.3910e+03 2.0616e+04 4.1257e+04
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    Table 2.  Condition numbers of our new Legendre dual-Petrov-Galerkin method for problem (19)

    $ \rm Matrices $ $ N=20 $ $ N=40 $ $ N=60 $ $ N=80 $ $ N=100 $
    $ \alpha(\varphi_k, \psi_l) $ 8.2281e+02 1.0014e+04 4.6167e+04 1.3919e+05 3.3027e+05
    $ \beta(\varphi_k, \partial_x\psi_l) $ 4.2539e+01 1.6248e+02 3.6050e+02 6.3660e+02 9.9077e+02
    $ \gamma(\partial_x\varphi_k, \partial_x\psi_l) $ 1.3709e+00 1.3731e+00 1.3734e+00 1.3736e+00 1.3736e+00
    $ (\partial_x\varphi_k, \partial^2_x\psi_l) $ 1.3826e+00 1.3842e+00 1.3846e+00 1.3848e+00 1.3848e+00
    $ \mathcal{A}(\varphi_k, \psi_l) $ 1.0000e+00 1.0000e+00 1.0000e+00 1.0000e+00 1.0000e+00
     | Show Table
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    Table 3.  Condition numbers of the usual Legendre dual-Petrov-Galerkin method for problem (39)

    $ \rm Matrices $ $ N=20 $ $ N=40 $ $ N=60 $ $ N=80 $ $ N=100 $
    $ \alpha(r_k, s_l) $ 2.2919e+07 1.4122e+10 7.8699e+11 1.4829e+13 1.5013e+14
    $ \beta(\partial_xr_k, \partial^2_xs_l) $ 1.0527e+05 1.1264e+07 1.8445e+08 1.3612e+09 6.4521e+09
    $ (\partial^2_xr_k, \partial^3_xs_l) $ 3.7507e+03 1.3987e+05 1.1653e+06 5.2074e+06 1.6553e+07
    $ \mathcal{B}(r_k, s_l) $ 3.7593e+03 1.3996e+05 1.1656e+06 5.2083e+06 1.6555e+07
     | Show Table
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    Table 4.  Condition numbers of our new Legendre dual-Petrov-Galerkin method for problem (39)

    $ \rm Matrices $ $ N=20 $ $ N=40 $ $ N=60 $ $ N=80 $ $ N=100 $
    $ \alpha(\Phi_k, \Psi_l) $ 6.1180e+06 2.2260e+09 8.8437e+10 1.2957e+12 1.0730e+13
    $ \beta(\partial_x\Phi_k, \partial^2_x\Psi_l) $ 5.1022e+02 5.8057e+03 2.6067e+04 7.7483e+04 1.8226e+05
    $ (\partial^2_x\Phi_k, \partial^3_x\Psi_l) $ 1.0683e+00 1.0684e+00 1.0684e+00 1.0684e+00 1.0684e+00
    $ \mathcal{B}(\Phi_k, \Psi_l) $ 1.0000e+00 1.0000e+00 1.0000e+00 1.0000e+00 1.0000e+00
     | Show Table
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