# American Institute of Mathematical Sciences

April  2020, 25(4): 1543-1563. doi: 10.3934/dcdsb.2019239

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

 School of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

* Corresponding author

Received  January 2019 Revised  June 2019 Published  November 2019

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

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.

Citation: Shan Li, Shi-Mi Yan, Zhong-Qing Wang. Efficient Legendre dual-Petrov-Galerkin methods for odd-order differential equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1543-1563. doi: 10.3934/dcdsb.2019239
##### References:

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##### References:
Numerical errors of scheme (20)
Numerical errors of scheme (20)
Numerical errors of scheme (41)
Numerical errors of scheme (41)
Numerical errors of scheme (59)
Numerical errors of scheme (59)
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
 $\rm Matrices$ $N=20$ $N=40$ $N=60$ $N=80$ $N=100$ $\alpha(p_k, q_l)$ 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
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
 $\rm Matrices$ $N=20$ $N=40$ $N=60$ $N=80$ $N=100$ $\alpha(\varphi_k, \psi_l)$ 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
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
 $\rm Matrices$ $N=20$ $N=40$ $N=60$ $N=80$ $N=100$ $\alpha(r_k, s_l)$ 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
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
 $\rm Matrices$ $N=20$ $N=40$ $N=60$ $N=80$ $N=100$ $\alpha(\Phi_k, \Psi_l)$ 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
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