# American Institute of Mathematical Sciences

April  2022, 15(4): 851-876. doi: 10.3934/dcdss.2022016

## Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA

* Corresponding author: Yanzhi Zhang

Received  March 2021 Revised  December 2021 Published  April 2022 Early access  February 2022

Fund Project: This work was partially supported by the US National Science Foundation under Grant Number DMS-1913293 and DMS-1953177

In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian $(- \Delta)^\frac{{ \alpha}}{{2}}$ for $\alpha \in (0, 2)$. One main advantage is that our method can easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of ${\mathcal O}(h^2)$, while ${\mathcal O}(h^4)$ for quadratic basis functions with $h$ a small mesh size. This accuracy can be achieved for any $\alpha \in (0, 2)$ and can be further increased if higher-degree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies $u \in C^{m, l}(\bar{ \Omega})$ for $m \in {\mathbb N}$ and $0 < l < 1$, our method has an accuracy of ${\mathcal O}(h^{\min\{m+l, \, 2\}})$ for constant and linear basis functions, while ${\mathcal O}(h^{\min\{m+l, \, 4\}})$ for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.

Citation: Yixuan Wu, Yanzhi Zhang. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 851-876. doi: 10.3934/dcdss.2022016
##### References:
 [1] G. Acosta, F. M. Bersetche and J. P. Borthagaray, A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian, Comput. Math. Appl., 74 (2017), 784-816.  doi: 10.1016/j.camwa.2017.05.026. [2] G. Acosta and J. P. Borthagaray, A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM J. Numer. Anal., 55 (2017), 472-495.  doi: 10.1137/15M1033952. [3] G. Acosta, J. P. Borthagaray and N. Heuer, Finite element approximations of the nonhomogeneous fractional Dirichlet problem, IMA J. Numer. Anal., 39 (2019), 1471-1501.  doi: 10.1093/imanum/dry023. [4] M. Ainsworth and C. Glusa, Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains, Contemporary Computational Mathematics–-a Celebration of the 80th Birthday of Ian Sloan, Vol. 1, 2, 17–57, Springer, Cham, 2018. [5] U. Biccari, M. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 387-409.  doi: 10.1515/ans-2017-0014. [6] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.  doi: 10.4064/sm-123-1-43-80. [7] A. Bonito and J. E. Pasciak, Numerical approximation of fractional powers of regularly accretive operators, IMA J. Numer. Anal., 37 (2017), 1245-1273.  doi: 10.1093/imanum/drw042. [8] J. Burkardt, Y. Wu and Y. Zhang, A unified meshfree pseudospectral method for solving both classical and fractional PDEs, SIAM J. Sci. Comput., 43 (2021), A1389–A1411. doi: 10.1137/20M1335959. [9] Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.  doi: 10.1137/110833294. [10] S. Duo, H. W. van Wyk and Y. Zhang, A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem, J. Comput. Phys., 355 (2018), 233-252.  doi: 10.1016/j.jcp.2017.11.011. [11] S. Duo, H. Wang and Y. Zhang, A comparative study on nonlocal diffusion operators related to the fractional Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 231-256.  doi: 10.3934/dcdsb.2018110. [12] S. Duo and Y. Zhang, Computing the ground and first excited states of the fractional Schrödinger equation in an infinite potential well, Commun. Comput. Phys., 18 (2015), 321-350.  doi: 10.4208/cicp.300414.120215a. [13] S. Duo and Y. Zhang, Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications, Comput. Methods. Appl. Mech. Eng., 355 (2019), 639-662.  doi: 10.1016/j.cma.2019.06.016. [14] S. Duo and Y. Zhang, Numerical approximations for the tempered fractional Laplacian: Error analysis and applications, J. Sci. Comput., 81 (2019), 569-593.  doi: 10.1007/s10915-019-01029-7. [15] Y. Huang and A. Oberman, Numerical methods for the fractional Laplacian: A finite difference–quadrature approach, SIAM J. Numer. Anal., 52 (2014), 3056-3084.  doi: 10.1137/140954040. [16] F. Izsák and B. J. Szekeres, Models of space-fractional diffusion: A critical review, Appl. Math. Lett., 71 (2017), 38-43.  doi: 10.1016/j.aml.2017.03.006. [17] K. Kirkpatrick and Y. Zhang, Fractional Schrödinger dynamics and decoherence, Phys. D, 332 (2016), 41-54.  doi: 10.1016/j.physd.2016.05.015. [18] M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002. [19] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972. [20] A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. M. Meerschaert, M. Ainsworth and G. E. Karniadakis, What is the fractional Laplacian? A comparative review with new results, J. Comput. Phys., 404 (2020), 109009.  doi: 10.1016/j.jcp.2019.109009. [21] X. Ros-Oton and J. Serra, Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris, 350 (2012), 505-508.  doi: 10.1016/j.crma.2012.05.011. [22] J. A. Rosenfeld, S. A. Rosenfeld and W. E. Dixon, A mesh-free pseudospectral approach to estimating the fractional Laplacian via radial basis functions, J. Comput. Phys., 390 (2019), 306-322.  doi: 10.1016/j.jcp.2019.02.015. [23] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. [24] C. Sheng, J. Shen, T. Tang, L.-L. Wang and H. Yuan, Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains, SIAM J. Numer. Anal., 58 (2020), 2435-2464.  doi: 10.1137/19M128377X. [25] T. Tang, L.-L. Wang, H. Yuan and T. Zhou, Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains, SIAM J. Sci. Comput., 42 (2020), A585–A611. doi: 10.1137/19M1244299. [26] Y. Wu and Y. Zhang, A universal solution scheme for fractional and classical PDEs, arXiv: 2102.00113, 2021. [27] Y. Wu and Y. Zhang, Variable-order Laplacian and its computations with meshfree methods, preprint, 2021.

show all references

##### References:
 [1] G. Acosta, F. M. Bersetche and J. P. Borthagaray, A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian, Comput. Math. Appl., 74 (2017), 784-816.  doi: 10.1016/j.camwa.2017.05.026. [2] G. Acosta and J. P. Borthagaray, A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM J. Numer. Anal., 55 (2017), 472-495.  doi: 10.1137/15M1033952. [3] G. Acosta, J. P. Borthagaray and N. Heuer, Finite element approximations of the nonhomogeneous fractional Dirichlet problem, IMA J. Numer. Anal., 39 (2019), 1471-1501.  doi: 10.1093/imanum/dry023. [4] M. Ainsworth and C. Glusa, Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains, Contemporary Computational Mathematics–-a Celebration of the 80th Birthday of Ian Sloan, Vol. 1, 2, 17–57, Springer, Cham, 2018. [5] U. Biccari, M. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 387-409.  doi: 10.1515/ans-2017-0014. [6] K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.  doi: 10.4064/sm-123-1-43-80. [7] A. Bonito and J. E. Pasciak, Numerical approximation of fractional powers of regularly accretive operators, IMA J. Numer. Anal., 37 (2017), 1245-1273.  doi: 10.1093/imanum/drw042. [8] J. Burkardt, Y. Wu and Y. Zhang, A unified meshfree pseudospectral method for solving both classical and fractional PDEs, SIAM J. Sci. Comput., 43 (2021), A1389–A1411. doi: 10.1137/20M1335959. [9] Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.  doi: 10.1137/110833294. [10] S. Duo, H. W. van Wyk and Y. Zhang, A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem, J. Comput. Phys., 355 (2018), 233-252.  doi: 10.1016/j.jcp.2017.11.011. [11] S. Duo, H. Wang and Y. Zhang, A comparative study on nonlocal diffusion operators related to the fractional Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 231-256.  doi: 10.3934/dcdsb.2018110. [12] S. Duo and Y. Zhang, Computing the ground and first excited states of the fractional Schrödinger equation in an infinite potential well, Commun. Comput. Phys., 18 (2015), 321-350.  doi: 10.4208/cicp.300414.120215a. [13] S. Duo and Y. Zhang, Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications, Comput. Methods. Appl. Mech. Eng., 355 (2019), 639-662.  doi: 10.1016/j.cma.2019.06.016. [14] S. Duo and Y. Zhang, Numerical approximations for the tempered fractional Laplacian: Error analysis and applications, J. Sci. Comput., 81 (2019), 569-593.  doi: 10.1007/s10915-019-01029-7. [15] Y. Huang and A. Oberman, Numerical methods for the fractional Laplacian: A finite difference–quadrature approach, SIAM J. Numer. Anal., 52 (2014), 3056-3084.  doi: 10.1137/140954040. [16] F. Izsák and B. J. Szekeres, Models of space-fractional diffusion: A critical review, Appl. Math. Lett., 71 (2017), 38-43.  doi: 10.1016/j.aml.2017.03.006. [17] K. Kirkpatrick and Y. Zhang, Fractional Schrödinger dynamics and decoherence, Phys. D, 332 (2016), 41-54.  doi: 10.1016/j.physd.2016.05.015. [18] M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002. [19] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972. [20] A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. M. Meerschaert, M. Ainsworth and G. E. Karniadakis, What is the fractional Laplacian? A comparative review with new results, J. Comput. Phys., 404 (2020), 109009.  doi: 10.1016/j.jcp.2019.109009. [21] X. Ros-Oton and J. Serra, Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris, 350 (2012), 505-508.  doi: 10.1016/j.crma.2012.05.011. [22] J. A. Rosenfeld, S. A. Rosenfeld and W. E. Dixon, A mesh-free pseudospectral approach to estimating the fractional Laplacian via radial basis functions, J. Comput. Phys., 390 (2019), 306-322.  doi: 10.1016/j.jcp.2019.02.015. [23] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. [24] C. Sheng, J. Shen, T. Tang, L.-L. Wang and H. Yuan, Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains, SIAM J. Numer. Anal., 58 (2020), 2435-2464.  doi: 10.1137/19M128377X. [25] T. Tang, L.-L. Wang, H. Yuan and T. Zhou, Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains, SIAM J. Sci. Comput., 42 (2020), A585–A611. doi: 10.1137/19M1244299. [26] Y. Wu and Y. Zhang, A universal solution scheme for fractional and classical PDEs, arXiv: 2102.00113, 2021. [27] Y. Wu and Y. Zhang, Variable-order Laplacian and its computations with meshfree methods, preprint, 2021.
Comparison of our method with linear basis (i.e., $\varphi^1$) or quadratic basis (i.e., $\varphi^2$) and the method in [15] with linear basis (i.e., Huang2014) in approximating function $(- \Delta)^\frac{{ \alpha}}{{2}}u(x)$, where $u(x) = (1-x^2)^6_+$ and the error function is defined in (28). (a) $\alpha = 0.6$; (b) $\alpha = 1.5$
Numerical error $|e_ \Delta(x)|$ at point $x = 0$ (left) and $x = 0.5$ (right) in approximating function $(- \Delta)^\frac{{ \alpha}}{{2}}u$ with $u(x) = (1-x^2)_+^{1+\lfloor \alpha\rfloor}$, where $\alpha = 0.5$ (blue), $1$ (red), and $1.7$ (green)
Numerical errors $\|e_ \Delta\|_\infty$ in approximating function $(- \Delta)^\frac{{ \alpha}}{{2}}u$, where $\alpha = 0.5$ (blue), $1$ (red), and $1.7$ (green). (a) $u = (1-x^2)_+^{3.1+ \alpha}$; (b) $u = (1-x^2)_+^{4.1+ \alpha}$
Comparison of our method with linear basis (i.e., $\varphi^1$) or quadratic basis (i.e., $\varphi^2$) and the finite difference method in [10] (i.e., FDM) in approximating function $(- \Delta)^\frac{{ \alpha}}{{2}}u(x)$, where $u(x) = (1-x^2)^{4.1+ \alpha}_+$, and $\alpha = 0.5$ (a) or $1.7$ (b)
Effects of splitting parameter $\gamma$ in approximating function $(- \Delta)^\frac{{ \alpha}}{{2}}u$ on $(-1, 1)$ with $u(x) = 1/(1+x^2)$, where $\alpha = 0.5$ (blue), $1$ (red), and $1.7$ (green)
Numerical errors $|e_u(x)|$ at point $x = 0$ in solving the 1D fractional Poisson problem (1)–(2) with $f(x) = 1$ and $g(x) = 0$, where $\alpha = 0.5$ (blue), $1$ (red), or $1.7$ (green)
Numerical errors $\|e_u\|_\infty$ in solving the 1D Poisson problem (1)–(2) with $g(x) = 0$ and $f(x)$ in (29), where the exact solution is $u(x) = (1-x^2)_+^s$. From (a) to (d): $s = \alpha, \, 2, \, 3, \, 4$, where $\alpha = 0.5$ (blue), $1$ (red), or $1.7$ (green)
Comparison of our method with linear basis (i.e., $\varphi^1$) or quadratic basis (i.e., $\varphi^2$) and the finite difference method in [10] (i.e., FDM) in solving the fractional Poisson equation with exact solution $u(x) = (1-x^2)^{4}_+$, where $\alpha = 0.5$ (a) or $1.7$ (b)
Numerical errors $\|e_u\|_\infty$ in solving the 1D tempered fractional Poisson problem with exact solution $u(x) = (1-x^2)_+^2$, where $\alpha = 0.6$ (blue), $1$ (red), or $1.5$ (green)
Numerical errors in the solution of the 2D fractional Poisson problems with basis function $\varphi^1$ and mesh size $h = 1/64$
Numerical errors $\|e_ \Delta\|_\infty$ and convergence rates (c.r.) in approximating function $(- \Delta)^\frac{{ \alpha}}{{2}}u$ with $u = (1-x^2)_+^{1+\lfloor \alpha \rfloor}$ and basis function $\varphi^p$ (for $p = 0, 1, 2$)
 $h$ 1/16 1/32 1/64 1/128 1/256 1/512 $\alpha = 0.5$ $\varphi^0$ 7.5879e-3 5.3220e-3 3.7499e-3 2.6475e-3 1.8707e-3 1.3224e-3 c.r. 0.5117 0.5051 0.5023 0.5010 0.5005 $\varphi^1$ 7.8596e-3 5.4986e-3 3.8696e-3 2.7304e-3 1.9287e-3 1.3632e-3 c.r. 0.5154 0.5069 0.5031 0.5014 0.5007 $\varphi^2$ 1.3338e-2 9.3477e-3 6.5851e-3 4.6488e-3 3.2848e-3 2.3219e-3 c.r. 0.5129 0.5054 0.5024 0.5011 0.5005 $\alpha = 1$ $\varphi^0$ 8.1722e-4 3.8342e-4 1.8911e-4 9.4360e-5 4.7189e-5 2.3604e-5 c.r. 1.0918 1.0197 1.0029 0.9997 0.9994 $\varphi^1$ 8.1722e-4 3.8342e-4 1.8911e-4 9.4360e-5 4.7189e-5 2.3604e-5 c.r. 1.0918 1.0197 1.0029 0.9997 0.9994 $\varphi^2$ 4.7698e-3 2.3572e-3 1.1717e-3 5.8413e-4 2.9164e-4 1.4571e-4 c.r. 1.0169 1.0085 1.0042 1.0021 1.0011 $\alpha = 1.7$ $\varphi^0$ 3.6356e-3 1.8041e-3 1.2777e-3 1.0195e-3 8.3276e-4 6.8083e-4 c.r. 1.0109 0.4977 0.3257 0.2919 0.2906 $\varphi^1$ 2.5288e-3 5.4873e-4 5.8948e-4 5.0041e-4 4.0137e-4 3.2097e-4 c.r. 2.2043 -0.1033 0.2363 0.3182 0.3225 $\varphi^2$ 9.9878e-2 7.8950e-2 6.3253e-2 5.1024e-2 4.1302e-2 3.3489e-2 c.r. 0.3392 0.3198 0.3010 0.3050 0.3025
 $h$ 1/16 1/32 1/64 1/128 1/256 1/512 $\alpha = 0.5$ $\varphi^0$ 7.5879e-3 5.3220e-3 3.7499e-3 2.6475e-3 1.8707e-3 1.3224e-3 c.r. 0.5117 0.5051 0.5023 0.5010 0.5005 $\varphi^1$ 7.8596e-3 5.4986e-3 3.8696e-3 2.7304e-3 1.9287e-3 1.3632e-3 c.r. 0.5154 0.5069 0.5031 0.5014 0.5007 $\varphi^2$ 1.3338e-2 9.3477e-3 6.5851e-3 4.6488e-3 3.2848e-3 2.3219e-3 c.r. 0.5129 0.5054 0.5024 0.5011 0.5005 $\alpha = 1$ $\varphi^0$ 8.1722e-4 3.8342e-4 1.8911e-4 9.4360e-5 4.7189e-5 2.3604e-5 c.r. 1.0918 1.0197 1.0029 0.9997 0.9994 $\varphi^1$ 8.1722e-4 3.8342e-4 1.8911e-4 9.4360e-5 4.7189e-5 2.3604e-5 c.r. 1.0918 1.0197 1.0029 0.9997 0.9994 $\varphi^2$ 4.7698e-3 2.3572e-3 1.1717e-3 5.8413e-4 2.9164e-4 1.4571e-4 c.r. 1.0169 1.0085 1.0042 1.0021 1.0011 $\alpha = 1.7$ $\varphi^0$ 3.6356e-3 1.8041e-3 1.2777e-3 1.0195e-3 8.3276e-4 6.8083e-4 c.r. 1.0109 0.4977 0.3257 0.2919 0.2906 $\varphi^1$ 2.5288e-3 5.4873e-4 5.8948e-4 5.0041e-4 4.0137e-4 3.2097e-4 c.r. 2.2043 -0.1033 0.2363 0.3182 0.3225 $\varphi^2$ 9.9878e-2 7.8950e-2 6.3253e-2 5.1024e-2 4.1302e-2 3.3489e-2 c.r. 0.3392 0.3198 0.3010 0.3050 0.3025
Numerical errors $\|e_ \Delta\|_\infty$ and convergence rate (c.r.) in approximating function $(- \Delta)^\frac{{ \alpha}}{{2}}u$ with $u = (1-x^2)_+^{2.1+ \alpha}$ and basis function $\varphi^p$ (for $p = 0, 1, 2$).
 $h$ 1/16 1/32 1/64 1/128 1/256 1/512 $\alpha = 0.5$ $\varphi^0$ 9.7624e-5 2.8827e-5 7.6872e-6 1.9687e-6 4.9667e-7 1.2457e-7 c.r. 1.7598 1.9069 1.9653 1.9868 1.9953 $\varphi^1$ 2.1391e-4 5.9663e-5 1.5540e-5 3.9426e-6 9.9077e-7 2.4817e-7 c.r. 1.8421 1.9409 1.9787 1.9925 1.9972 $\varphi^2$ 1.0716e-4 2.4168e-5 5.5431e-6 1.2823e-6 2.9789e-7 6.9347e-8 c.r. 2.1486 2.1243 2.1120 2.1059 2.1029 $\alpha = 1$ $\varphi^0$ 5.9137e-4 7.5126e-5 9.4842e-6 2.0487e-6 6.4898e-7 1.7163e-7 c.r. 2.9767 2.9857 2.2109 1.6584 1.9189 $\varphi^1$ 5.9137e-4 7.5126e-5 9.4842e-6 2.0487e-6 6.4898e-7 1.7163e-7 c.r. 2.9767 2.9857 2.2109 1.6584 1.9189 $\varphi^2$ 2.4438e-4 5.4962e-5 1.2583e-5 2.9079e-6 6.7516e-7 1.5713e-7 c.r. 2.1526 2.1270 2.1134 2.1067 2.1033 $\alpha = 1.7$ $\varphi^0$ 1.4385e-2 5.3211e-3 1.5476e-3 4.0205e-4 9.9477e-5 2.4066e-5 c.r. 1.4348 1.7817 1.9446 2.0150 2.0474 $\varphi^1$ 1.5206e-2 5.5501e-3 1.6318e-3 4.2867e-4 1.0729e-4 2.6267e-5 c.r. 1.4540 1.7660 1.9285 1.9983 2.0302 $\varphi^2$ 8.0506e-4 1.0070e-4 1.6011e-5 3.1025e-6 6.6674e-7 1.4996e-7 c.r. 2.9991 2.6529 2.3675 2.2182 2.1526
 $h$ 1/16 1/32 1/64 1/128 1/256 1/512 $\alpha = 0.5$ $\varphi^0$ 9.7624e-5 2.8827e-5 7.6872e-6 1.9687e-6 4.9667e-7 1.2457e-7 c.r. 1.7598 1.9069 1.9653 1.9868 1.9953 $\varphi^1$ 2.1391e-4 5.9663e-5 1.5540e-5 3.9426e-6 9.9077e-7 2.4817e-7 c.r. 1.8421 1.9409 1.9787 1.9925 1.9972 $\varphi^2$ 1.0716e-4 2.4168e-5 5.5431e-6 1.2823e-6 2.9789e-7 6.9347e-8 c.r. 2.1486 2.1243 2.1120 2.1059 2.1029 $\alpha = 1$ $\varphi^0$ 5.9137e-4 7.5126e-5 9.4842e-6 2.0487e-6 6.4898e-7 1.7163e-7 c.r. 2.9767 2.9857 2.2109 1.6584 1.9189 $\varphi^1$ 5.9137e-4 7.5126e-5 9.4842e-6 2.0487e-6 6.4898e-7 1.7163e-7 c.r. 2.9767 2.9857 2.2109 1.6584 1.9189 $\varphi^2$ 2.4438e-4 5.4962e-5 1.2583e-5 2.9079e-6 6.7516e-7 1.5713e-7 c.r. 2.1526 2.1270 2.1134 2.1067 2.1033 $\alpha = 1.7$ $\varphi^0$ 1.4385e-2 5.3211e-3 1.5476e-3 4.0205e-4 9.9477e-5 2.4066e-5 c.r. 1.4348 1.7817 1.9446 2.0150 2.0474 $\varphi^1$ 1.5206e-2 5.5501e-3 1.6318e-3 4.2867e-4 1.0729e-4 2.6267e-5 c.r. 1.4540 1.7660 1.9285 1.9983 2.0302 $\varphi^2$ 8.0506e-4 1.0070e-4 1.6011e-5 3.1025e-6 6.6674e-7 1.4996e-7 c.r. 2.9991 2.6529 2.3675 2.2182 2.1526
Numerical errors $\|e_ \Delta\|_\infty$ and convergence rate (c.r.) in approximating function $(- \Delta)^\frac{{ \alpha}}{{2}}u$ on (-1, 1) with $u(x) = 1/(1+x^2)$ and basis function $\varphi^p$ (for $p = 0, 1, 2$)
 $h$ 1/16 1/32 1/64 1/128 1/256 $\alpha = 0.5$ $\varphi^0$ 2.0023e-5 5.4222e-6 1.3919e-6 3.5121e-7 8.8084e-8 c.r. 1.8847 1.9618 1.9866 1.9954 $\varphi^1$ 3.0000e-5 8.3090e-6 2.1718e-6 5.5132e-7 1.3857e-7 c.r. 1.8522 1.9357 1.9780 1.9923 $\varphi^2$ 1.7384e-7 1.2148e-8 7.8796e-10 4.9569e-11 2.6986e-12 c.r. 3.8391 3.9464 3.9906 4.1991 $\alpha = 1$ $\varphi^0$ 1.1056e-4 1.7994e-5 3.2863e-6 6.6985e-7 1.4849e-7 c.r. 2.6193 2.4530 2.2945 2.1735 $\varphi^1$ 1.1056e-4 1.7994e-5 3.2863e-6 6.6985e-7 1.4849e-7 c.r. 2.6193 2.4530 2.2945 2.1735 $\varphi^2$ 8.0637e-7 2.5951e-8 8.3813e-10 2.7735e-11 9.3070e-13 c.r. 4.9576 4.9525 4.9174 4.8972 $\alpha = 1.7$ $\varphi^0$ 2.2284e-3 4.6838e-4 9.8834e-5 2.0988e-5 4.4908e-6 c.r. 2.2503 2.2446 2.2355 2.2245 $\varphi^1$ 2.3784e-3 5.1283e-4 1.1135e-4 2.4403e-5 5.4026e-6 c.r. 2.2135 2.2033 2.1900 2.1753 $\varphi^2$ 3.1706e-5 1.6865e-6 8.9337e-8 4.7546e-9 2.6046e-10 c.r. 4.2326 4.2386 4.2319 4.1902
 $h$ 1/16 1/32 1/64 1/128 1/256 $\alpha = 0.5$ $\varphi^0$ 2.0023e-5 5.4222e-6 1.3919e-6 3.5121e-7 8.8084e-8 c.r. 1.8847 1.9618 1.9866 1.9954 $\varphi^1$ 3.0000e-5 8.3090e-6 2.1718e-6 5.5132e-7 1.3857e-7 c.r. 1.8522 1.9357 1.9780 1.9923 $\varphi^2$ 1.7384e-7 1.2148e-8 7.8796e-10 4.9569e-11 2.6986e-12 c.r. 3.8391 3.9464 3.9906 4.1991 $\alpha = 1$ $\varphi^0$ 1.1056e-4 1.7994e-5 3.2863e-6 6.6985e-7 1.4849e-7 c.r. 2.6193 2.4530 2.2945 2.1735 $\varphi^1$ 1.1056e-4 1.7994e-5 3.2863e-6 6.6985e-7 1.4849e-7 c.r. 2.6193 2.4530 2.2945 2.1735 $\varphi^2$ 8.0637e-7 2.5951e-8 8.3813e-10 2.7735e-11 9.3070e-13 c.r. 4.9576 4.9525 4.9174 4.8972 $\alpha = 1.7$ $\varphi^0$ 2.2284e-3 4.6838e-4 9.8834e-5 2.0988e-5 4.4908e-6 c.r. 2.2503 2.2446 2.2355 2.2245 $\varphi^1$ 2.3784e-3 5.1283e-4 1.1135e-4 2.4403e-5 5.4026e-6 c.r. 2.2135 2.2033 2.1900 2.1753 $\varphi^2$ 3.1706e-5 1.6865e-6 8.9337e-8 4.7546e-9 2.6046e-10 c.r. 4.2326 4.2386 4.2319 4.1902
Numerical errors $\|e_u\|_\infty$ and convergence rate (c.r.) in solving the 1D Poisson problem on $\Omega = (-1, 1)$, where $f(x) = 1$ in (1) and $g(x) = 0$ in (2)
 $h$ 1/16 1/32 1/64 1/128 1/256 1/512 $\alpha = 0.6$ $\varphi^0$ 7.4494e-2 5.9980e-2 4.8507e-2 3.9314e-2 3.1898e-2 2.5895e-2 c.r. 0.3126 0.3063 0.3031 0.3016 0.3008 $\varphi^1$ 7.5493e-2 6.0790e-2 4.9164e-2 3.9847e-2 3.2331e-2 2.6247e-2 c.r. 0.3125 0.3062 0.3031 0.3016 0.3008 $\varphi^2$ 8.4532e-2 6.8106e-2 5.5102e-2 4.4671e-2 3.6249e-2 2.9429e-2 c.r. 0.3117 0.3057 0.3028 0.3014 0.3007 $\alpha = 1$ $\varphi^0$ 4.9166e-2 3.4508e-2 2.4310e-2 1.7158e-2 1.2121e-2 8.5671e-3 c.r. 0.5107 0.5054 0.5027 0.5013 0.5007 $\varphi^1$ 4.9166e-2 3.4508e-2 2.4310e-2 1.7158e-2 1.2121e-2 8.5671e-3 c.r. 0.5107 0.5054 0.5027 0.5013 0.5007 $\varphi^2$ 5.7935e-2 4.0695e-2 2.8682e-2 2.0248e-2 1.4306e-2 1.0112e-2 c.r. 0.5096 0.5047 0.5023 0.5012 0.5006 $\alpha = 1.5$ $\varphi^0$ 1.6161e-2 9.5429e-3 5.6545e-3 3.3563e-3 1.9939e-3 1.1851e-3 c.r. 0.7600 0.7550 0.7525 0.7513 0.7506 $\varphi^1$ 1.5976e-2 9.4344e-3 5.5905e-3 3.3184e-3 1.9714e-3 1.1717e-3 c.r. 0.7599 0.7550 0.7525 0.7513 0.7506 $\varphi^2$ 2.2627e-2 1.3365e-2 7.9205e-3 4.7018e-3 2.7934e-3 1.6603e-3 c.r. 0.7596 0.7548 0.7524 0.7512 0.7506
 $h$ 1/16 1/32 1/64 1/128 1/256 1/512 $\alpha = 0.6$ $\varphi^0$ 7.4494e-2 5.9980e-2 4.8507e-2 3.9314e-2 3.1898e-2 2.5895e-2 c.r. 0.3126 0.3063 0.3031 0.3016 0.3008 $\varphi^1$ 7.5493e-2 6.0790e-2 4.9164e-2 3.9847e-2 3.2331e-2 2.6247e-2 c.r. 0.3125 0.3062 0.3031 0.3016 0.3008 $\varphi^2$ 8.4532e-2 6.8106e-2 5.5102e-2 4.4671e-2 3.6249e-2 2.9429e-2 c.r. 0.3117 0.3057 0.3028 0.3014 0.3007 $\alpha = 1$ $\varphi^0$ 4.9166e-2 3.4508e-2 2.4310e-2 1.7158e-2 1.2121e-2 8.5671e-3 c.r. 0.5107 0.5054 0.5027 0.5013 0.5007 $\varphi^1$ 4.9166e-2 3.4508e-2 2.4310e-2 1.7158e-2 1.2121e-2 8.5671e-3 c.r. 0.5107 0.5054 0.5027 0.5013 0.5007 $\varphi^2$ 5.7935e-2 4.0695e-2 2.8682e-2 2.0248e-2 1.4306e-2 1.0112e-2 c.r. 0.5096 0.5047 0.5023 0.5012 0.5006 $\alpha = 1.5$ $\varphi^0$ 1.6161e-2 9.5429e-3 5.6545e-3 3.3563e-3 1.9939e-3 1.1851e-3 c.r. 0.7600 0.7550 0.7525 0.7513 0.7506 $\varphi^1$ 1.5976e-2 9.4344e-3 5.5905e-3 3.3184e-3 1.9714e-3 1.1717e-3 c.r. 0.7599 0.7550 0.7525 0.7513 0.7506 $\varphi^2$ 2.2627e-2 1.3365e-2 7.9205e-3 4.7018e-3 2.7934e-3 1.6603e-3 c.r. 0.7596 0.7548 0.7524 0.7512 0.7506
Numerical errors $\|e_u\|_\infty$ and convergence rate (c.r.) in solving the 2D Poisson problem on $\Omega = (-1, 1)^2$ with $f$ and $g$ defined in (29)–(30), where linear basis $\varphi^1$ is used
 $h$ 1/4 1/8 1/16 1/32 1/64 1/128 $\alpha = 0.2$ 5.3181e-4 1.1946e-4 2.8883e-5 7.1505e-6 1.7827e-6 4.4531e-7 c.r. 2.1544 2.0482 2.0141 2.0040 2.0011 $\alpha = 0.7$ 2.3855e-3 5.1805e-4 1.2151e-4 2.9565e-5 7.3092e-6 1.8190e-6 c.r. 2.2031 2.0921 2.0391 2.0161 2.0065 $\alpha = 1$ 3.9406e-3 8.3747e-4 1.9083e-4 4.5384e-5 1.1056e-5 2.7276e-6 c.r. 2.2343 2.1338 2.0720 2.0374 2.0191 $\alpha = 1.4$ 6.9983e-3 1.4880e-3 3.2910e-4 7.5175e-5 1.7618e-5 4.2102e-6 c.r. 2.2337 2.1767 2.1302 2.0932 2.0651 $\alpha = 1.9$ 1.4264e-2 3.3824e-3 8.0943e-4 1.9424e-4 4.6676e-5 1.1230e-5 c.r. 2.0762 2.0631 2.0591 2.0571 2.0554
 $h$ 1/4 1/8 1/16 1/32 1/64 1/128 $\alpha = 0.2$ 5.3181e-4 1.1946e-4 2.8883e-5 7.1505e-6 1.7827e-6 4.4531e-7 c.r. 2.1544 2.0482 2.0141 2.0040 2.0011 $\alpha = 0.7$ 2.3855e-3 5.1805e-4 1.2151e-4 2.9565e-5 7.3092e-6 1.8190e-6 c.r. 2.2031 2.0921 2.0391 2.0161 2.0065 $\alpha = 1$ 3.9406e-3 8.3747e-4 1.9083e-4 4.5384e-5 1.1056e-5 2.7276e-6 c.r. 2.2343 2.1338 2.0720 2.0374 2.0191 $\alpha = 1.4$ 6.9983e-3 1.4880e-3 3.2910e-4 7.5175e-5 1.7618e-5 4.2102e-6 c.r. 2.2337 2.1767 2.1302 2.0932 2.0651 $\alpha = 1.9$ 1.4264e-2 3.3824e-3 8.0943e-4 1.9424e-4 4.6676e-5 1.1230e-5 c.r. 2.0762 2.0631 2.0591 2.0571 2.0554
 [1] Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure and Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008 [2] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445 [3] Maoding Zhen, Jinchun He, Haoyun Xu. Critical system involving fractional Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (1) : 237-253. doi: 10.3934/cpaa.2019013 [4] Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905 [5] Vladimir Georgiev, Koichi Taniguchi. On fractional Leibniz rule for Dirichlet Laplacian in exterior domain. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1101-1115. doi: 10.3934/dcds.2019046 [6] De Tang, Yanqin Fang. Regularity and nonexistence of solutions for a system involving the fractional Laplacian. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431 [7] Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393 [8] Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236 [9] Selma Yildirim Yolcu, Türkay Yolcu. Sharper estimates on the eigenvalues of Dirichlet fractional Laplacian. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2209-2225. doi: 10.3934/dcds.2015.35.2209 [10] Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154 [11] Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1813-1845. doi: 10.3934/dcds.2016.36.1813 [12] Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268 [13] Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068 [14] Hua Chen, Hong-Ge Chen. Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 301-317. doi: 10.3934/dcds.2021117 [15] Nicola Abatangelo. Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5555-5607. doi: 10.3934/dcds.2015.35.5555 [16] Jen-Yen Lin, Hui-Ju Chen, Ruey-Lin Sheu. Augmented Lagrange primal-dual approach for generalized fractional programming problems. Journal of Industrial and Management Optimization, 2013, 9 (4) : 723-741. doi: 10.3934/jimo.2013.9.723 [17] Mingqi Xiang, Binlin Zhang, Die Hu. Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping. Electronic Research Archive, 2020, 28 (2) : 651-669. doi: 10.3934/era.2020034 [18] Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1837-1855. doi: 10.3934/dcdss.2021007 [19] Simone Creo, Maria Rosaria Lancia, Paola Vernole. Transmission problems for the fractional $p$-Laplacian across fractal interfaces. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022047 [20] Jinguo Zhang, Dengyun Yang. Fractional $p$-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups. Electronic Research Archive, 2021, 29 (5) : 3243-3260. doi: 10.3934/era.2021036

2021 Impact Factor: 1.865