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A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior
A robust adaptive grid method for singularly perturbed Burger-Huxley equations
1. | School of Mathematics and Statistics, Nanning Normal University, Nanning 530001, China |
2. | School of Big Data and Artificial Intelligence, Chizhou University, Chizhou, Anhui 247000, China |
In this paper, an adaptive grid method is proposed to solve one-dimensional unsteady singularly perturbed Burger-Huxley equation with appropriate initial and boundary conditions. Firstly, we use the classical backward-Euler scheme on a uniform mesh to approximate time derivative. The resulting nonlinear singularly perturbed semi-discrete problem is linearized by using Newton-Raphson-Kantorovich approximation method which is quadratically convergent. Then, an upwind finite difference scheme on an adaptive nonuniform grid is used for space derivative. The nonuniform grid is generated by equidistribution of a positive monitor function, which is similar to the arc-length function. It is shown that the presented adaptive grid method is first order uniform convergent in the time and spatial directions, respectively. Finally, numerical results are given to validate the theoretical results.
References:
[1] |
D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[2] |
B. Batiha, M. S. M. Noorani and I. Hashim,
Numerical simulation of the generalized Huxley equation by He's variational iteration method, Appl. Math. Comput., 186 (2007), 1322-1325.
doi: 10.1016/j.amc.2006.07.166. |
[3] |
R. E. Bellman and R. E. Kalaba, Quasilineaization and Nonlinear Boundary-Value Problems, Modern Analytic and Computional Methods in Science and Mathematics, Vol. 3 American Elsevier Publishing Co., Inc., New York 1965. |
[4] |
Y. Chen,
Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution, J. Comput. Appl. Math., 159 (2003), 25-34.
doi: 10.1016/S0377-0427(03)00563-6. |
[5] |
Y. Chen,
Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid, Adv. Comput. Math., 24 (2006), 197-212.
doi: 10.1007/s10444-004-7641-0. |
[6] |
Y. Chen and L.-B. Liu,
An adaptive grid method for singularly perturbed time-dependent convection-diffusion problems, Commum. Comput. Phys., 20 (2016), 1340-1358.
doi: 10.4208/cicp.240315.301215a. |
[7] |
C. Clavero, J. C. Jorge and F. Lisbona,
A uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems, J. Comput. Appl. Math., 154 (2003), 415-429.
doi: 10.1016/S0377-0427(02)00861-0. |
[8] |
M. T. Darvishi, S. Kheybari and F. Khani,
Spectral collocation method and Darvishi's preconditionings to solve the generalized Burgers-Huxley equation, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 2091-2103.
doi: 10.1016/j.cnsns.2007.05.023. |
[9] |
L. Duan and Q. Lu,
Bursting oscillations near codimension-two bifurcations in the Chay neuron model, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 59-63.
doi: 10.1515/IJNSNS.2006.7.1.59. |
[10] |
S. Gowrisankar and S. Natesan,
The parameter uniform numerical method for singularly perturbed parabolic reaction-diffusion problems on equidistributed grids, Appl. Math. Lett., 26 (2013), 1053-1060.
doi: 10.1016/j.aml.2013.05.017. |
[11] |
S. Gowrisankar and S. Natesan,
Uniformly convergent numerical method for singularly perturbed parabolic initial-boundary-value problems with equidistributed grids, Int. J. Comput. Math., 91 (2014), 553-577.
doi: 10.1080/00207160.2013.792925. |
[12] |
S. Gowrisankar and S. Natesan,
Robust numerical scheme for singularly perturbed convection-diffusion parabolic initial-boundary-value problems on equidistributed grids, Comput. Phys. Commun., 185 (2014), 2008-2019.
doi: 10.1016/j.cpc.2014.04.004. |
[13] |
V. Gupta and M. K. Kadalbajoo,
A singular perturbation approach to solve Burgers-Huxley equation via monotone finite difference scheme on layer-adaptive mesh, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1825-1844.
doi: 10.1016/j.cnsns.2010.07.020. |
[14] |
I. Hashim, M. S. M. Noorani and B. Batiha,
A note on the Adomian decomposition method for the generalized Huxley equation, Appl. Math. Comput., 181 (2006), 1439-1445.
doi: 10.1016/j.amc.2006.03.011. |
[15] |
I. Hashim, M. S. M. Noorani and M. R. Said Al-Hadidi,
Solving the generalized Burgers-Huxley equation using the adomian decomposition method, Math. Comput. Model., 43 (2006), 1404-1411.
doi: 10.1016/j.mcm.2005.08.017. |
[16] |
H. N. A. Ismail, K. Raslan and A. A. A. Rabboh,
Adomian decomposition method for Burgers-Huxley and Burgers-Fisher equations, Appl. Math. Comput., 159 (2004), 291-301.
doi: 10.1016/j.amc.2003.10.050. |
[17] |
M. Javidi,
A numerical solution of the generalized Burgers-Huxley equation by spectral collocation method, Appl. Math. Comput., 178 (2006), 338-344.
doi: 10.1016/j.amc.2005.11.051. |
[18] |
M. Javidi and A. Golbabai,
A new domain decomposition algorithm for generalized Burgers-Huxley equation based on Chebyshev polynomials and preconditioning, Chaos Solitons Fractals, 39 (2009), 849-857.
doi: 10.1016/j.chaos.2007.01.099. |
[19] |
A. Kaushik and M. D. Sharma,
A uniformly convergent numerical method on non-uniform mesh for singularly perturbed unsteady Burger-Huxley equation, Appl. Math. Comput., 195 (2008), 688-706.
doi: 10.1016/j.amc.2007.05.067. |
[20] |
A. J. Khattak,
A computational meshless method for the generalized Burger's-Huxley equation, Appl. Math. Model., 33 (2009), 3718-3729.
doi: 10.1016/j.apm.2008.12.010. |
[21] |
N. Kopteva,
Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (2001), 423-441.
doi: 10.1137/S0036142900368642. |
[22] |
N. Kopteva and M. Stynes,
A robust adaptive method for quasi-linear one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (2001), 1446-1467.
doi: 10.1137/S003614290138471X. |
[23] |
L.-B. Liu and Y. Chen,
A robust adaptive grid method for a system of two singularly perturbed convection-diffusion equations with weak coupling, J. Sci. Comput., 61 (2014), 1-16.
doi: 10.1007/s10915-013-9814-9. |
[24] |
S. Liu, T. Fan and Q. Lu,
The spike order of the winnerless competition (WLC) model and its application to the inhibition neural system, Int. J. Nonlin. Sci. Numer. Simul., 6 (2005), 133-138.
doi: 10.1515/IJNSNS.2005.6.2.133. |
[25] |
R. C. Mittal and A. Tripathi,
Numerical solutions of generalized Burgers-Fisher and generalized Burgers-Huxley equations using collocation of cubic $B$-splines, Int. J. Comput. Math., 92 (2015), 1053-1077.
doi: 10.1080/00207160.2014.920834. |
[26] |
R. Mohammadi,
B-spline collocation algorithm for numerical solution of the generalized Burger's-Huxley equation, Numer. Methods Partial Differential Equations, 29 (2013), 1173-1191.
doi: 10.1002/num.21750. |
[27] |
R. K. Mohanty, W. Dai and D. Liu,
Operator compact method of accuracy two in time and four in space for the solution of time dependent Burgers-Huxley equation, Numer. Algorithms, 70 (2015), 591-605.
doi: 10.1007/s11075-015-9963-z. |
[28] |
S. Murat, G. Görhan and Z. Asuman,
High-order finite difference schemes for numerical solutions of the generalized Burgers-Huxley equation, Numer. Methods Partial Differential Equations, 27 (2011), 1313-1326.
|
[29] |
H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Convection-diffusion and flow problems. Springer Series in Computational Mathematics, 24. Springer-Verlag, Berlin, 1996.
doi: 10.1007/978-3-662-03206-0. |
[30] |
J. Satsuma J, Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, World Scientific, Singapore, 1987. Google Scholar |
[31] |
X. Y. Wang, Z. S. Zhu and Y. K. Lu,
Solitary wave solutions of the generalised Burgers-Huxley equation, J. Phys. A, 23 (1990), 271-274.
doi: 10.1088/0305-4470/23/3/011. |
[32] |
A.-M. Wazwaz,
Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations, Appl. Math. Comput., 169 (2005), 639-656.
doi: 10.1016/j.amc.2004.09.081. |
[33] |
G.-J. Zhang, J.-X. Xu, H. Yao et al., Mechanism of bifurcation-dependent coherence resonance of an excitable neuron model, Int. J. Nonlin. Sci. Numer. Simul., 7 (2006), 447-450. Google Scholar |
show all references
References:
[1] |
D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[2] |
B. Batiha, M. S. M. Noorani and I. Hashim,
Numerical simulation of the generalized Huxley equation by He's variational iteration method, Appl. Math. Comput., 186 (2007), 1322-1325.
doi: 10.1016/j.amc.2006.07.166. |
[3] |
R. E. Bellman and R. E. Kalaba, Quasilineaization and Nonlinear Boundary-Value Problems, Modern Analytic and Computional Methods in Science and Mathematics, Vol. 3 American Elsevier Publishing Co., Inc., New York 1965. |
[4] |
Y. Chen,
Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution, J. Comput. Appl. Math., 159 (2003), 25-34.
doi: 10.1016/S0377-0427(03)00563-6. |
[5] |
Y. Chen,
Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid, Adv. Comput. Math., 24 (2006), 197-212.
doi: 10.1007/s10444-004-7641-0. |
[6] |
Y. Chen and L.-B. Liu,
An adaptive grid method for singularly perturbed time-dependent convection-diffusion problems, Commum. Comput. Phys., 20 (2016), 1340-1358.
doi: 10.4208/cicp.240315.301215a. |
[7] |
C. Clavero, J. C. Jorge and F. Lisbona,
A uniformly convergent scheme on a nonuniform mesh for convection-diffusion parabolic problems, J. Comput. Appl. Math., 154 (2003), 415-429.
doi: 10.1016/S0377-0427(02)00861-0. |
[8] |
M. T. Darvishi, S. Kheybari and F. Khani,
Spectral collocation method and Darvishi's preconditionings to solve the generalized Burgers-Huxley equation, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 2091-2103.
doi: 10.1016/j.cnsns.2007.05.023. |
[9] |
L. Duan and Q. Lu,
Bursting oscillations near codimension-two bifurcations in the Chay neuron model, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 59-63.
doi: 10.1515/IJNSNS.2006.7.1.59. |
[10] |
S. Gowrisankar and S. Natesan,
The parameter uniform numerical method for singularly perturbed parabolic reaction-diffusion problems on equidistributed grids, Appl. Math. Lett., 26 (2013), 1053-1060.
doi: 10.1016/j.aml.2013.05.017. |
[11] |
S. Gowrisankar and S. Natesan,
Uniformly convergent numerical method for singularly perturbed parabolic initial-boundary-value problems with equidistributed grids, Int. J. Comput. Math., 91 (2014), 553-577.
doi: 10.1080/00207160.2013.792925. |
[12] |
S. Gowrisankar and S. Natesan,
Robust numerical scheme for singularly perturbed convection-diffusion parabolic initial-boundary-value problems on equidistributed grids, Comput. Phys. Commun., 185 (2014), 2008-2019.
doi: 10.1016/j.cpc.2014.04.004. |
[13] |
V. Gupta and M. K. Kadalbajoo,
A singular perturbation approach to solve Burgers-Huxley equation via monotone finite difference scheme on layer-adaptive mesh, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1825-1844.
doi: 10.1016/j.cnsns.2010.07.020. |
[14] |
I. Hashim, M. S. M. Noorani and B. Batiha,
A note on the Adomian decomposition method for the generalized Huxley equation, Appl. Math. Comput., 181 (2006), 1439-1445.
doi: 10.1016/j.amc.2006.03.011. |
[15] |
I. Hashim, M. S. M. Noorani and M. R. Said Al-Hadidi,
Solving the generalized Burgers-Huxley equation using the adomian decomposition method, Math. Comput. Model., 43 (2006), 1404-1411.
doi: 10.1016/j.mcm.2005.08.017. |
[16] |
H. N. A. Ismail, K. Raslan and A. A. A. Rabboh,
Adomian decomposition method for Burgers-Huxley and Burgers-Fisher equations, Appl. Math. Comput., 159 (2004), 291-301.
doi: 10.1016/j.amc.2003.10.050. |
[17] |
M. Javidi,
A numerical solution of the generalized Burgers-Huxley equation by spectral collocation method, Appl. Math. Comput., 178 (2006), 338-344.
doi: 10.1016/j.amc.2005.11.051. |
[18] |
M. Javidi and A. Golbabai,
A new domain decomposition algorithm for generalized Burgers-Huxley equation based on Chebyshev polynomials and preconditioning, Chaos Solitons Fractals, 39 (2009), 849-857.
doi: 10.1016/j.chaos.2007.01.099. |
[19] |
A. Kaushik and M. D. Sharma,
A uniformly convergent numerical method on non-uniform mesh for singularly perturbed unsteady Burger-Huxley equation, Appl. Math. Comput., 195 (2008), 688-706.
doi: 10.1016/j.amc.2007.05.067. |
[20] |
A. J. Khattak,
A computational meshless method for the generalized Burger's-Huxley equation, Appl. Math. Model., 33 (2009), 3718-3729.
doi: 10.1016/j.apm.2008.12.010. |
[21] |
N. Kopteva,
Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (2001), 423-441.
doi: 10.1137/S0036142900368642. |
[22] |
N. Kopteva and M. Stynes,
A robust adaptive method for quasi-linear one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39 (2001), 1446-1467.
doi: 10.1137/S003614290138471X. |
[23] |
L.-B. Liu and Y. Chen,
A robust adaptive grid method for a system of two singularly perturbed convection-diffusion equations with weak coupling, J. Sci. Comput., 61 (2014), 1-16.
doi: 10.1007/s10915-013-9814-9. |
[24] |
S. Liu, T. Fan and Q. Lu,
The spike order of the winnerless competition (WLC) model and its application to the inhibition neural system, Int. J. Nonlin. Sci. Numer. Simul., 6 (2005), 133-138.
doi: 10.1515/IJNSNS.2005.6.2.133. |
[25] |
R. C. Mittal and A. Tripathi,
Numerical solutions of generalized Burgers-Fisher and generalized Burgers-Huxley equations using collocation of cubic $B$-splines, Int. J. Comput. Math., 92 (2015), 1053-1077.
doi: 10.1080/00207160.2014.920834. |
[26] |
R. Mohammadi,
B-spline collocation algorithm for numerical solution of the generalized Burger's-Huxley equation, Numer. Methods Partial Differential Equations, 29 (2013), 1173-1191.
doi: 10.1002/num.21750. |
[27] |
R. K. Mohanty, W. Dai and D. Liu,
Operator compact method of accuracy two in time and four in space for the solution of time dependent Burgers-Huxley equation, Numer. Algorithms, 70 (2015), 591-605.
doi: 10.1007/s11075-015-9963-z. |
[28] |
S. Murat, G. Görhan and Z. Asuman,
High-order finite difference schemes for numerical solutions of the generalized Burgers-Huxley equation, Numer. Methods Partial Differential Equations, 27 (2011), 1313-1326.
|
[29] |
H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Convection-diffusion and flow problems. Springer Series in Computational Mathematics, 24. Springer-Verlag, Berlin, 1996.
doi: 10.1007/978-3-662-03206-0. |
[30] |
J. Satsuma J, Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, World Scientific, Singapore, 1987. Google Scholar |
[31] |
X. Y. Wang, Z. S. Zhu and Y. K. Lu,
Solitary wave solutions of the generalised Burgers-Huxley equation, J. Phys. A, 23 (1990), 271-274.
doi: 10.1088/0305-4470/23/3/011. |
[32] |
A.-M. Wazwaz,
Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations, Appl. Math. Comput., 169 (2005), 639-656.
doi: 10.1016/j.amc.2004.09.081. |
[33] |
G.-J. Zhang, J.-X. Xu, H. Yao et al., Mechanism of bifurcation-dependent coherence resonance of an excitable neuron model, Int. J. Nonlin. Sci. Numer. Simul., 7 (2006), 447-450. Google Scholar |



Number of intervals |
||||||
32/ |
64/ |
128/ |
256/ |
512/ |
1024/ |
|
2.4400e-02 | 1.4033e-02 | 7.5889e-03 | 3.9547e-03 | 2.0227e-03 | 1.0233e-03 | |
1.8830e-02 | 1.1126e-02 | 6.5407e-03 | 3.6221e-03 | 1.9209e-03 | 1.0074e-03 | |
2.0658e-02 | 1.2614e-02 | 7.0567e-03 | 3.7952e-03 | 1.9563e-03 | 1.0231e-03 | |
2.5289e-02 | 1.7672e-02 | 9.0066e-03 | 4.8378e-03 | 2.5035e-03 | 1.2731e-03 | |
3.8607e-02 | 1.9497e-02 | 1.1221e-02 | 6.2852e-03 | 3.3405e-03 | 1.7233e-03 | |
9.3183e-02 | 7.0120e-02 | 4.4773e-02 | 2.0546e-02 | 1.0545e-02 | 5.1608e-03 | |
1.7017e-01 | 1.0083e-01 | 6.2216e-02 | 3.9526e-02 | 2.0493e-02 | 1.0027e-02 | |
2.0410e-01 | 1.6703e-01 | 9.2110e-02 | 5.2580e-02 | 2.8766e-02 | 1.6523e-02 | |
2.0450e-01 | 1.5975e-01 | 1.2612e-01 | 6.9772e-02 | 3.8531e-02 | 2.0526e-02 | |
2.5614e-01 | 2.1031e-01 | 1.3406e-01 | 8.5618e-02 | 4.8834e-02 | 2.6219e-02 |
Number of intervals |
||||||
32/ |
64/ |
128/ |
256/ |
512/ |
1024/ |
|
2.4400e-02 | 1.4033e-02 | 7.5889e-03 | 3.9547e-03 | 2.0227e-03 | 1.0233e-03 | |
1.8830e-02 | 1.1126e-02 | 6.5407e-03 | 3.6221e-03 | 1.9209e-03 | 1.0074e-03 | |
2.0658e-02 | 1.2614e-02 | 7.0567e-03 | 3.7952e-03 | 1.9563e-03 | 1.0231e-03 | |
2.5289e-02 | 1.7672e-02 | 9.0066e-03 | 4.8378e-03 | 2.5035e-03 | 1.2731e-03 | |
3.8607e-02 | 1.9497e-02 | 1.1221e-02 | 6.2852e-03 | 3.3405e-03 | 1.7233e-03 | |
9.3183e-02 | 7.0120e-02 | 4.4773e-02 | 2.0546e-02 | 1.0545e-02 | 5.1608e-03 | |
1.7017e-01 | 1.0083e-01 | 6.2216e-02 | 3.9526e-02 | 2.0493e-02 | 1.0027e-02 | |
2.0410e-01 | 1.6703e-01 | 9.2110e-02 | 5.2580e-02 | 2.8766e-02 | 1.6523e-02 | |
2.0450e-01 | 1.5975e-01 | 1.2612e-01 | 6.9772e-02 | 3.8531e-02 | 2.0526e-02 | |
2.5614e-01 | 2.1031e-01 | 1.3406e-01 | 8.5618e-02 | 4.8834e-02 | 2.6219e-02 |
Number of intervals |
|||||
32/ |
64/ |
128/ |
256/ |
512/ |
|
0.7981 | 0.8869 | 0.9403 | 0.9673 | 0.9831 | |
0.7591 | 0.7664 | 0.8526 | 0.9150 | 0.9311 | |
0.7117 | 0.8380 | 0.8948 | 0.9560 | 0.9352 | |
0.5170 | 0.9724 | 0.8966 | 0.9504 | 0.9756 | |
0.9856 | 0.7971 | 0.8362 | 0.9119 | 0.9549 | |
0.4102 | 0.6472 | 1.1238 | 0.9623 | 1.0309 | |
0.7550 | 0.6966 | 0.6544 | 0.9477 | 1.0312 | |
0.2892 | 0.8587 | 0.8088 | 0.8701 | 0.7999 | |
0.3563 | 0.3410 | 0.8540 | 0.8566 | 0.9086 | |
0.2844 | 0.6496 | 0.6469 | 0.8100 | 0.8973 |
Number of intervals |
|||||
32/ |
64/ |
128/ |
256/ |
512/ |
|
0.7981 | 0.8869 | 0.9403 | 0.9673 | 0.9831 | |
0.7591 | 0.7664 | 0.8526 | 0.9150 | 0.9311 | |
0.7117 | 0.8380 | 0.8948 | 0.9560 | 0.9352 | |
0.5170 | 0.9724 | 0.8966 | 0.9504 | 0.9756 | |
0.9856 | 0.7971 | 0.8362 | 0.9119 | 0.9549 | |
0.4102 | 0.6472 | 1.1238 | 0.9623 | 1.0309 | |
0.7550 | 0.6966 | 0.6544 | 0.9477 | 1.0312 | |
0.2892 | 0.8587 | 0.8088 | 0.8701 | 0.7999 | |
0.3563 | 0.3410 | 0.8540 | 0.8566 | 0.9086 | |
0.2844 | 0.6496 | 0.6469 | 0.8100 | 0.8973 |
Number of intervals |
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32/ |
64/ |
128/ |
256/ |
512/ |
1024/ |
|
2.5632e-02 | 1.6476e-02 | 9.9288e-03 | 5.6712e-03 | 3.1149e-03 | 1.7000e-03 | |
2.9611e-02 | 1.8504e-02 | 1.0796e-02 | 6.0104e-03 | 3.2417e-03 | 1.7105e-03 | |
3.1540e-02 | 2.0968e-02 | 1.1706e-02 | 6.3027e-03 | 3.3320e-03 | 1.7024e-03 | |
3.4637e-02 | 1.7587e-02 | 7.9159e-03 | 3.8469e-03 | 2.2367e-03 | 2.5185e-03 | |
8.3069e-02 | 6.3873e-02 | 3.7579e-02 | 1.9554e-02 | 9.7037e-03 | 3.0087e-03 | |
8.3319e-02 | 1.1804e-01 | 8.7025e-02 | 5.1153e-02 | 2.7435e-02 | 1.1428e-02 | |
5.9149e-02 | 1.2245e-01 | 1.4419e-01 | 9.5806e-02 | 5.5767e-02 | 2.7152e-02 | |
3.8012e-02 | 1.2070e-01 | 1.9830e-01 | 1.4945e-01 | 9.1763e-02 | 5.4821e-02 | |
3.7835e-02 | 8.8086e-02 | 1.9473e-01 | 2.0775e-01 | 1.3885e-01 | 9.6989e-02 | |
3.7788e-02 | 3.6581e-02 | 1.7951e-01 | 2.5413e-01 | 1.9134e-01 | 1.4338e-01 |
Number of intervals |
||||||
32/ |
64/ |
128/ |
256/ |
512/ |
1024/ |
|
2.5632e-02 | 1.6476e-02 | 9.9288e-03 | 5.6712e-03 | 3.1149e-03 | 1.7000e-03 | |
2.9611e-02 | 1.8504e-02 | 1.0796e-02 | 6.0104e-03 | 3.2417e-03 | 1.7105e-03 | |
3.1540e-02 | 2.0968e-02 | 1.1706e-02 | 6.3027e-03 | 3.3320e-03 | 1.7024e-03 | |
3.4637e-02 | 1.7587e-02 | 7.9159e-03 | 3.8469e-03 | 2.2367e-03 | 2.5185e-03 | |
8.3069e-02 | 6.3873e-02 | 3.7579e-02 | 1.9554e-02 | 9.7037e-03 | 3.0087e-03 | |
8.3319e-02 | 1.1804e-01 | 8.7025e-02 | 5.1153e-02 | 2.7435e-02 | 1.1428e-02 | |
5.9149e-02 | 1.2245e-01 | 1.4419e-01 | 9.5806e-02 | 5.5767e-02 | 2.7152e-02 | |
3.8012e-02 | 1.2070e-01 | 1.9830e-01 | 1.4945e-01 | 9.1763e-02 | 5.4821e-02 | |
3.7835e-02 | 8.8086e-02 | 1.9473e-01 | 2.0775e-01 | 1.3885e-01 | 9.6989e-02 | |
3.7788e-02 | 3.6581e-02 | 1.7951e-01 | 2.5413e-01 | 1.9134e-01 | 1.4338e-01 |
Number of intervals |
|||||
32/ |
64/ |
128/ |
256/ |
512/ |
|
0.6375 | 0.7307 | 0.8079 | 0.8644 | 0.8736 | |
0.6783 | 0.7773 | 0.8450 | 0.8907 | 0.9223 | |
0.5890 | 0.8409 | 0.8932 | 0.9196 | 0.9688 | |
0.9778 | 1.1517 | 1.0411 | 0.7823 | -0.1712 | |
0.3791 | 0.7653 | 0.9425 | 1.0108 | 1.6893 | |
-0.5026 | 0.4398 | 0.7666 | 0.8988 | 1.2634 | |
-1.0498 | -0.2357 | 0.5897 | 0.7807 | 1.0384 | |
-1.6666 | -0.7162 | 0.4080 | 0.7037 | 0.7432 | |
-1.2192 | -1.1445 | -0.0933 | 0.5814 | 0.5176 | |
0.0468 | -2.2948 | -0.5015 | 0.4093 | 0.4163 |
Number of intervals |
|||||
32/ |
64/ |
128/ |
256/ |
512/ |
|
0.6375 | 0.7307 | 0.8079 | 0.8644 | 0.8736 | |
0.6783 | 0.7773 | 0.8450 | 0.8907 | 0.9223 | |
0.5890 | 0.8409 | 0.8932 | 0.9196 | 0.9688 | |
0.9778 | 1.1517 | 1.0411 | 0.7823 | -0.1712 | |
0.3791 | 0.7653 | 0.9425 | 1.0108 | 1.6893 | |
-0.5026 | 0.4398 | 0.7666 | 0.8988 | 1.2634 | |
-1.0498 | -0.2357 | 0.5897 | 0.7807 | 1.0384 | |
-1.6666 | -0.7162 | 0.4080 | 0.7037 | 0.7432 | |
-1.2192 | -1.1445 | -0.0933 | 0.5814 | 0.5176 | |
0.0468 | -2.2948 | -0.5015 | 0.4093 | 0.4163 |
Number of intervals |
||||||
32/ |
64/ |
128/ |
256/ |
512/ |
1024/ |
|
2.4502e-02 | 1.4057e-02 | 7.5969e-03 | 3.9589e-03 | 2.0249e-03 | 1.0244e-03 | |
1.9156e-02 | 1.1404e-02 | 6.5847e-03 | 3.6356e-03 | 1.9226e-03 | 1.0085e-03 | |
2.2788e-02 | 1.3365e-02 | 7.2769e-03 | 3.8441e-03 | 1.9728e-03 | 1.0270e-03 | |
3.8767e-02 | 1.8983e-02 | 9.6122e-03 | 5.0867e-03 | 2.6211e-03 | 1.3306e-03 | |
4.4450e-02 | 2.0109e-02 | 1.0519e-02 | 5.9653e-02 | 3.1896e-03 | 1.6508e-03 | |
8.3339e-02 | 6.6120e-02 | 4.0769e-02 | 1.9284e-02 | 9.1775e-03 | 4.9998e-03 | |
1.8762e-01 | 8.4106e-02 | 5.7234e-02 | 3.8309e-02 | 1.9041e-02 | 9.2260e-03 | |
1.9755e-01 | 1.5347e-01 | 8.3864e-02 | 4.6945e-02 | 2.5508e-02 | 1.4930e-02 | |
2.1340e-01 | 1.5016e-01 | 1.1582e-01 | 6.1958e-02 | 3.3441e-02 | 1.7672e-02 | |
2.8299e-01 | 1.7036e-01 | 1.1805e-01 | 7.4251e-02 | 4.1879e-02 | 2.2413e-02 |
Number of intervals |
||||||
32/ |
64/ |
128/ |
256/ |
512/ |
1024/ |
|
2.4502e-02 | 1.4057e-02 | 7.5969e-03 | 3.9589e-03 | 2.0249e-03 | 1.0244e-03 | |
1.9156e-02 | 1.1404e-02 | 6.5847e-03 | 3.6356e-03 | 1.9226e-03 | 1.0085e-03 | |
2.2788e-02 | 1.3365e-02 | 7.2769e-03 | 3.8441e-03 | 1.9728e-03 | 1.0270e-03 | |
3.8767e-02 | 1.8983e-02 | 9.6122e-03 | 5.0867e-03 | 2.6211e-03 | 1.3306e-03 | |
4.4450e-02 | 2.0109e-02 | 1.0519e-02 | 5.9653e-02 | 3.1896e-03 | 1.6508e-03 | |
8.3339e-02 | 6.6120e-02 | 4.0769e-02 | 1.9284e-02 | 9.1775e-03 | 4.9998e-03 | |
1.8762e-01 | 8.4106e-02 | 5.7234e-02 | 3.8309e-02 | 1.9041e-02 | 9.2260e-03 | |
1.9755e-01 | 1.5347e-01 | 8.3864e-02 | 4.6945e-02 | 2.5508e-02 | 1.4930e-02 | |
2.1340e-01 | 1.5016e-01 | 1.1582e-01 | 6.1958e-02 | 3.3441e-02 | 1.7672e-02 | |
2.8299e-01 | 1.7036e-01 | 1.1805e-01 | 7.4251e-02 | 4.1879e-02 | 2.2413e-02 |
Number of intervals |
|||||||||
32/ |
64/ |
128/ |
256/ |
512/ |
|||||
0.8016 | 0.8878 | 0.9403 | 0.9672 | 1.1059 | |||||
0.7843 | 0.7924 | 0.8864 | 0.9191 | 0.9308 | |||||
0.7698 | 0.8771 | 0.9207 | 0.9624 | 0.9418 | |||||
1.0301 | 0.9818 | 0.9181 | 0.9566 | 0.9781 | |||||
1.1443 | 0.9348 | 0.8183 | 0.9032 | 0.9502 | |||||
0.3339 | 0.6976 | 1.0801 | 1.0712 | 0.8762 | |||||
1.1575 | 0.5553 | 0.5792 | 1.0086 | 1.0453 | |||||
0.3643 | 0.8718 | 0.8371 | 0.8800 | 0.7727 | |||||
0.5076 | 0.3746 | 0.9025 | 0.8897 | 0.9202 | |||||
0.7322 | 0.5292 | 0.6689 | 0.8262 | 0.9019 |
Number of intervals |
|||||||||
32/ |
64/ |
128/ |
256/ |
512/ |
|||||
0.8016 | 0.8878 | 0.9403 | 0.9672 | 1.1059 | |||||
0.7843 | 0.7924 | 0.8864 | 0.9191 | 0.9308 | |||||
0.7698 | 0.8771 | 0.9207 | 0.9624 | 0.9418 | |||||
1.0301 | 0.9818 | 0.9181 | 0.9566 | 0.9781 | |||||
1.1443 | 0.9348 | 0.8183 | 0.9032 | 0.9502 | |||||
0.3339 | 0.6976 | 1.0801 | 1.0712 | 0.8762 | |||||
1.1575 | 0.5553 | 0.5792 | 1.0086 | 1.0453 | |||||
0.3643 | 0.8718 | 0.8371 | 0.8800 | 0.7727 | |||||
0.5076 | 0.3746 | 0.9025 | 0.8897 | 0.9202 | |||||
0.7322 | 0.5292 | 0.6689 | 0.8262 | 0.9019 |
Number of intervals |
||||||
32/ |
64/ |
128/ |
256/ |
512/ |
1024/ |
|
2.4502e-02 | 1.4057e-02 | 7.5969e-03 | 3.9589e-03 | 2.0249e-03 | 1.6649e-03 | |
3.0271e-02 | 1.8725e-02 | 1.0868e-02 | 6.0332e-03 | 3.2486e-03 | 1.7125e-03 | |
3.7645e-02 | 2.2072e-02 | 1.2090e-02 | 6.4115e-03 | 3.3584e-03 | 1.7453e-03 | |
3.6325e-02 | 2.6153e-02 | 1.6509e-02 | 9.7972e-03 | 5.6201e-03 | 2.9364e-03 | |
9.9532e-02 | 5.4894e-02 | 2.7274e-02 | 1.2963e-02 | 6.0337e-03 | 3.6346e-03 | |
2.1833e-01 | 1.5551e-01 | 8.7820e-02 | 4.5033e-02 | 2.2291e-02 | 1.3620e-02 | |
2.2602e-01 | 2.6665e-01 | 1.8203e-01 | 1.0258e-01 | 5.2694e-02 | 2.9881e-02 | |
1.2823e-01 | 3.2914e-01 | 2.7178e-01 | 1.8208e-01 | 1.0356e-01 | 5.2147e-02 | |
4.4024e-02 | 3.2475e-01 | 3.2607e-01 | 2.5174e-01 | 1.6644e-01 | 8.0344e-02 | |
3.3007e-02 | 2.5627e-01 | 3.5442e-01 | 2.9823e-01 | 2.1963e-01 | 1.2277e-01 |
Number of intervals |
||||||
32/ |
64/ |
128/ |
256/ |
512/ |
1024/ |
|
2.4502e-02 | 1.4057e-02 | 7.5969e-03 | 3.9589e-03 | 2.0249e-03 | 1.6649e-03 | |
3.0271e-02 | 1.8725e-02 | 1.0868e-02 | 6.0332e-03 | 3.2486e-03 | 1.7125e-03 | |
3.7645e-02 | 2.2072e-02 | 1.2090e-02 | 6.4115e-03 | 3.3584e-03 | 1.7453e-03 | |
3.6325e-02 | 2.6153e-02 | 1.6509e-02 | 9.7972e-03 | 5.6201e-03 | 2.9364e-03 | |
9.9532e-02 | 5.4894e-02 | 2.7274e-02 | 1.2963e-02 | 6.0337e-03 | 3.6346e-03 | |
2.1833e-01 | 1.5551e-01 | 8.7820e-02 | 4.5033e-02 | 2.2291e-02 | 1.3620e-02 | |
2.2602e-01 | 2.6665e-01 | 1.8203e-01 | 1.0258e-01 | 5.2694e-02 | 2.9881e-02 | |
1.2823e-01 | 3.2914e-01 | 2.7178e-01 | 1.8208e-01 | 1.0356e-01 | 5.2147e-02 | |
4.4024e-02 | 3.2475e-01 | 3.2607e-01 | 2.5174e-01 | 1.6644e-01 | 8.0344e-02 | |
3.3007e-02 | 2.5627e-01 | 3.5442e-01 | 2.9823e-01 | 2.1963e-01 | 1.2277e-01 |
Number of intervals |
|||||
32/ |
64/ |
128/ |
256/ |
512/ |
|
0.8016 | 0.8878 | 0.9403 | 0.9672 | 0.2824 | |
0.6930 | 0.7849 | 0.8491 | 0.8931 | 0.9237 | |
0.7702 | 0.8684 | 0.9151 | 0.9329 | 0.9443 | |
0.4740 | 0.6637 | 0.7528 | 0.8018 | 0.9365 | |
0.8585 | 1.0091 | 1.0731 | 1.1033 | 0.7312 | |
0.4895 | 0.8244 | 0.9636 | 1.0145 | 0.7107 | |
-0.2385 | 0.5508 | 0.8274 | 0.9610 | 0.8184 | |
-1.3600 | 0.2763 | 0.5779 | 0.8141 | 0.9898 | |
-2.8830 | -0.0059 | 0.3732 | 0.5969 | 1.0507 | |
-2.9568 | -0.4678 | 0.2490 | 0.4388 | 0.8391 |
Number of intervals |
|||||
32/ |
64/ |
128/ |
256/ |
512/ |
|
0.8016 | 0.8878 | 0.9403 | 0.9672 | 0.2824 | |
0.6930 | 0.7849 | 0.8491 | 0.8931 | 0.9237 | |
0.7702 | 0.8684 | 0.9151 | 0.9329 | 0.9443 | |
0.4740 | 0.6637 | 0.7528 | 0.8018 | 0.9365 | |
0.8585 | 1.0091 | 1.0731 | 1.1033 | 0.7312 | |
0.4895 | 0.8244 | 0.9636 | 1.0145 | 0.7107 | |
-0.2385 | 0.5508 | 0.8274 | 0.9610 | 0.8184 | |
-1.3600 | 0.2763 | 0.5779 | 0.8141 | 0.9898 | |
-2.8830 | -0.0059 | 0.3732 | 0.5969 | 1.0507 | |
-2.9568 | -0.4678 | 0.2490 | 0.4388 | 0.8391 |
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