# American Institute of Mathematical Sciences

January 2019, 24(1): 231-256. doi: 10.3934/dcdsb.2018110

## A comparative study on nonlocal diffusion operators related to the fractional Laplacian

 1 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA 2 Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

* Corresponding author: Yanzhi Zhang

Received  April 2017 Revised  December 2017 Published  March 2018

In this paper, we study four nonlocal diffusion operators, including the fractional Laplacian, spectral fractional Laplacian, regional fractional Laplacian, and peridynamic operator. These operators represent the infinitesimal generators of different stochastic processes, and especially their differences on a bounded domain are significant. We provide extensive numerical experiments to understand and compare their differences. We find that these four operators collapse to the classical Laplace operator as $\alpha \to2$. The eigenvalues and eigenfunctions of these four operators are different, and the $k$-th (for $k \in {\mathbb N}$) eigenvalue of the spectral fractional Laplacian is always larger than those of the fractional Laplacian and regional fractional Laplacian. For any $\alpha \in (0, 2)$, the peridynamic operator can provide a good approximation to the fractional Laplacian, if the horizon size $\delta$ is sufficiently large. We find that the solution of the peridynamic model converges to that of the fractional Laplacian model at a rate of ${\mathcal O}(\delta ^{-\alpha })$. In contrast, although the regional fractional Laplacian can be used to approximate the fractional Laplacian as $\alpha \to2$, it generally provides inconsistent result from that of the fractional Laplacian if $\alpha \ll 2$. Moreover, some conjectures are made from our numerical results, which could contribute to the mathematics analysis on these operators.

Citation: Siwei Duo, Hong Wang, Yanzhi Zhang. A comparative study on nonlocal diffusion operators related to the fractional Laplacian. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 231-256. doi: 10.3934/dcdsb.2018110
##### References:
 [1] N. Abatangelo and L. Dupaigne, Nonhomogeneous boundary conditions for the spectral fractional Laplacian, Ann I H Poincare C, 34 (2017), 439-467. doi: 10.1016/j.anihpc.2016.02.001. [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] R. Bañuelos and T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal., 211 (2004), 355-423. doi: 10.1016/j.jfa.2004.02.005. [4] K. Bogdan, K. Burdzy and Z. Chen, Censored stable processes, Probab. Theory Rel., 127 (2003), 89-152. doi: 10.1007/s00440-003-0275-1. [5] C. Burcur, Some observations on the Green function for the ball in the fractional Laplace framework, Commun. Pur. Appl. Anal., 15 (2016), 657-699. doi: 10.3934/cpaa.2016.15.657. [6] B. A. Carreras, V. E. Lynch and G. M. Zaslavsky, Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model, Phys. Plasmas, 8 (2001), 5096-5103. doi: 10.1063/1.1416180. [7] Z.-Q. Chen, P. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329. [8] Z.-Q. Chen, P. Kim and R. Song, Two-sided heat kernel estimates for censored stable-like processes, Probab. Theory Rel., 146 (2010), 361-399. doi: 10.1007/s00440-008-0193-3. [9] Z.-Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains, J. Funct. Anal., 226 (2005), 90-113. doi: 10.1016/j.jfa.2005.05.004. [10] Z.-Q. Chen, P. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329. [11] Z.-Q. Chen, P. Kim and R. Song, Dirichlet heat kernel estimates for rotationally symmetric Lévy processes, Proc. Lond. Math. Soc., 109 (2014), 90-120. doi: 10.1112/plms/pdt068. [12] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004. [13] M. D'Elia and M. Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Comput. Math. Appl., 66 (2013), 1245-1260. doi: 10.1016/j.camwa.2013.07.022. [14] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [15] 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. [16] S. Duo, L. Ju and Y. Zhang, A fast algorithm for solving the space-time fractional diffusion equation, Comput. Math. Appl., 2017. https://doi.org/10.1016/j.camwa.2017.04.008. doi: 10.1016/j.camwa.2017.04.008. [17] S. Duo, H.-W. van Wyk and Y. Zhang, A novel and accurate weighted trapezoidal finite difference method for the fractional laplacian, J. Comput. Phys., 355 (2018), 233-252. doi: 10.1016/j.jcp.2017.11.011. [18] 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. [19] B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal., 15 (2012), 536-555. [20] R. L. Frank, Eigenvalue bounds for the fractional Laplacian: A review, preprint, arXiv: 1603.09736. [21] Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329. doi: 10.1007/s00220-006-0054-9. [22] Q. Guan and M. Gunzburger, Analysis and approximation of a nonlocal obstacle problem, J. Comput. Appl. Math., 313 (2017), 102-118. doi: 10.1016/j.cam.2016.09.012. [23] Q. Guan and Z. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424. doi: 10.1142/S021949370500150X. [24] Q. Guan and Z. Ma, Reflected symmetric α-stable processes and regional fractional Laplacian, Probab. Theory Rel., 134 (2006), 649-694. doi: 10.1007/s00440-005-0438-3. [25] M. Gunzburger and R. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598. doi: 10.1137/090766607. [26] K. Kaleta, Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval, Studia Math., 209 (2012), 267-287. doi: 10.4064/sm209-3-5. [27] M. Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379-2402. doi: 10.1016/j.jfa.2011.12.004. [28] M. Kwasnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2015), 7-51. [29] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972. [30] T. Mengesha and Q. Du, Nonlocal constrained value problems for a linear peridynamic Navier equation, J. Elast., 116 (2014), 27-51. doi: 10.1007/s10659-013-9456-z. [31] C. Mou and Y. Yi, Interior regularity for regional fractional Laplacian, Comm. Math. Phys., 340 (2015), 233-251. doi: 10.1007/s00220-015-2445-2. [32] R. Musina and A. I. Nazarov, On fractional Laplacians, Comm. Part. Diff. Eq., 39 (2014), 1780-1790. doi: 10.1080/03605302.2013.864304. [33] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26. doi: 10.5565/PUBLMAT_60116_01. [34] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9), 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [35] X. Ros-Oton and J. Serra, Regularity theory for general stable operators, J. Differ. Equations, 260 (2016), 8675-8715. doi: 10.1016/j.jde.2016.02.033. [36] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. [37] J. Serra, Cσ+α regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels, Calc. Var. Partail Diff., 54 (2015), 3571-3601. doi: 10.1007/s00526-015-0914-2. [38] R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783. [39] M. F. Shlesinger, B. J. West and J. Klafter, Lévy dynamics of enhanced diffusion: Application to turbulence, Phys. Rev. Lett., 58 (1987), 1100-1103. doi: 10.1103/PhysRevLett.58.1100. [40] S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48 (2000), 175-209. doi: 10.1016/S0022-5096(99)00029-0. [41] R. Song and Z. Vondraček, Potential theory of subordinate killed Brownian motion in a domain, Probab. Theory Rel., 125 (2003), 578-592. doi: 10.1007/s00440-002-0251-1. [42] R. Song and Z. Vondraček, On the relationship between subordinate killed and killed subordinate processes, Electron. Commun. Probab., 13 (2008), 325-336. doi: 10.1214/ECP.v13-1388. [43] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N. J., 1970. [44] S. Y. Yolcu and T. Yolcu, Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain, Commun. Contemp. Math., 15 (2013), 1250048, 15pp. [45] S. Y. Yolcu and T. Yolcu, Refined eigenvalue bounds on the Dirichlet fractional Laplacian, Journal of Math. Phys. , 56 (2015), 073506, 12pp.

show all references

##### References:
 [1] N. Abatangelo and L. Dupaigne, Nonhomogeneous boundary conditions for the spectral fractional Laplacian, Ann I H Poincare C, 34 (2017), 439-467. doi: 10.1016/j.anihpc.2016.02.001. [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] R. Bañuelos and T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal., 211 (2004), 355-423. doi: 10.1016/j.jfa.2004.02.005. [4] K. Bogdan, K. Burdzy and Z. Chen, Censored stable processes, Probab. Theory Rel., 127 (2003), 89-152. doi: 10.1007/s00440-003-0275-1. [5] C. Burcur, Some observations on the Green function for the ball in the fractional Laplace framework, Commun. Pur. Appl. Anal., 15 (2016), 657-699. doi: 10.3934/cpaa.2016.15.657. [6] B. A. Carreras, V. E. Lynch and G. M. Zaslavsky, Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model, Phys. Plasmas, 8 (2001), 5096-5103. doi: 10.1063/1.1416180. [7] Z.-Q. Chen, P. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329. [8] Z.-Q. Chen, P. Kim and R. Song, Two-sided heat kernel estimates for censored stable-like processes, Probab. Theory Rel., 146 (2010), 361-399. doi: 10.1007/s00440-008-0193-3. [9] Z.-Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains, J. Funct. Anal., 226 (2005), 90-113. doi: 10.1016/j.jfa.2005.05.004. [10] Z.-Q. Chen, P. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329. [11] Z.-Q. Chen, P. Kim and R. Song, Dirichlet heat kernel estimates for rotationally symmetric Lévy processes, Proc. Lond. Math. Soc., 109 (2014), 90-120. doi: 10.1112/plms/pdt068. [12] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004. [13] M. D'Elia and M. Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Comput. Math. Appl., 66 (2013), 1245-1260. doi: 10.1016/j.camwa.2013.07.022. [14] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. [15] 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. [16] S. Duo, L. Ju and Y. Zhang, A fast algorithm for solving the space-time fractional diffusion equation, Comput. Math. Appl., 2017. https://doi.org/10.1016/j.camwa.2017.04.008. doi: 10.1016/j.camwa.2017.04.008. [17] S. Duo, H.-W. van Wyk and Y. Zhang, A novel and accurate weighted trapezoidal finite difference method for the fractional laplacian, J. Comput. Phys., 355 (2018), 233-252. doi: 10.1016/j.jcp.2017.11.011. [18] 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. [19] B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal., 15 (2012), 536-555. [20] R. L. Frank, Eigenvalue bounds for the fractional Laplacian: A review, preprint, arXiv: 1603.09736. [21] Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329. doi: 10.1007/s00220-006-0054-9. [22] Q. Guan and M. Gunzburger, Analysis and approximation of a nonlocal obstacle problem, J. Comput. Appl. Math., 313 (2017), 102-118. doi: 10.1016/j.cam.2016.09.012. [23] Q. Guan and Z. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424. doi: 10.1142/S021949370500150X. [24] Q. Guan and Z. Ma, Reflected symmetric α-stable processes and regional fractional Laplacian, Probab. Theory Rel., 134 (2006), 649-694. doi: 10.1007/s00440-005-0438-3. [25] M. Gunzburger and R. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598. doi: 10.1137/090766607. [26] K. Kaleta, Spectral gap lower bound for the one-dimensional fractional Schrödinger operator in the interval, Studia Math., 209 (2012), 267-287. doi: 10.4064/sm209-3-5. [27] M. Kwaśnicki, Eigenvalues of the fractional Laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379-2402. doi: 10.1016/j.jfa.2011.12.004. [28] M. Kwasnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2015), 7-51. [29] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972. [30] T. Mengesha and Q. Du, Nonlocal constrained value problems for a linear peridynamic Navier equation, J. Elast., 116 (2014), 27-51. doi: 10.1007/s10659-013-9456-z. [31] C. Mou and Y. Yi, Interior regularity for regional fractional Laplacian, Comm. Math. Phys., 340 (2015), 233-251. doi: 10.1007/s00220-015-2445-2. [32] R. Musina and A. I. Nazarov, On fractional Laplacians, Comm. Part. Diff. Eq., 39 (2014), 1780-1790. doi: 10.1080/03605302.2013.864304. [33] X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26. doi: 10.5565/PUBLMAT_60116_01. [34] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9), 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [35] X. Ros-Oton and J. Serra, Regularity theory for general stable operators, J. Differ. Equations, 260 (2016), 8675-8715. doi: 10.1016/j.jde.2016.02.033. [36] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. [37] J. Serra, Cσ+α regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels, Calc. Var. Partail Diff., 54 (2015), 3571-3601. doi: 10.1007/s00526-015-0914-2. [38] R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783. [39] M. F. Shlesinger, B. J. West and J. Klafter, Lévy dynamics of enhanced diffusion: Application to turbulence, Phys. Rev. Lett., 58 (1987), 1100-1103. doi: 10.1103/PhysRevLett.58.1100. [40] S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48 (2000), 175-209. doi: 10.1016/S0022-5096(99)00029-0. [41] R. Song and Z. Vondraček, Potential theory of subordinate killed Brownian motion in a domain, Probab. Theory Rel., 125 (2003), 578-592. doi: 10.1007/s00440-002-0251-1. [42] R. Song and Z. Vondraček, On the relationship between subordinate killed and killed subordinate processes, Electron. Commun. Probab., 13 (2008), 325-336. doi: 10.1214/ECP.v13-1388. [43] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N. J., 1970. [44] S. Y. Yolcu and T. Yolcu, Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain, Commun. Contemp. Math., 15 (2013), 1250048, 15pp. [45] S. Y. Yolcu and T. Yolcu, Refined eigenvalue bounds on the Dirichlet fractional Laplacian, Journal of Math. Phys. , 56 (2015), 073506, 12pp.
Comparison of the function ${\mathcal L}u$ with $u$ defined in (3.1), where the operator ${\mathcal L}$ represents ${\mathcal L}_s$ (solid line), ${\mathcal L}_h$ (dashed line), ${\mathcal L}_r$ (dash-dot line), or ${\mathcal L}_p$ with $\delta = 4$ (dotted line). For easy comparison, the result for ${\mathcal L} = -\partial_{xx}$ (line with symbols "*") is included in the plot of $\alpha = 1.95$. For $\alpha = 1$, $1.5$ or $1.95$, the plots in $y$-direction are partially presented
Difference between the peridynamic operator and the fractional Laplacian versus the parameter $\alpha$, where $u(x)$ is defined in (3.1)
Comparison of the function ${\mathcal L}u$ with $u$ defined in (3.2) with $q = 2$. The operator ${\mathcal L}$ represents ${\mathcal L}_s$ (solid line), ${\mathcal L}_h$ (dashed line), ${\mathcal L}_r$ (dash-dot line), or ${\mathcal L}_p$ with $\delta = 4$ (dotted line).For easy comparison, the result for ${\mathcal L} = -\partial_{xx}$ (line with symbols "*") is included in the plot of $\alpha = 1.95$
Comparison of the function ${\mathcal L}u$ with $u$ defined in (3.2) with $q = 2$. The operator ${\mathcal L}$ represents ${\mathcal L}_s$ (solid line), ${\mathcal L}_h$ (dashed line), ${\mathcal L}_r$ (dash-dot line), or ${\mathcal L}_p$ with $\delta = 4$ (dotted line)
Comparison of the function ${\mathcal L}u$ with $u$ defined in (3.2) with $q = 1$. The operator ${\mathcal L}$ represents ${\mathcal L}_s$ (solid line), ${\mathcal L}_h$ (dashed line), ${\mathcal L}_r$ (dash-dot line), or ${\mathcal L}_p$ with $\delta = 4$ (dotted line)
The absolute (left panel) and relative (right panel) differences in the eigenvalues of the fractional Laplacian and the spectral fractional Laplacian
The first (left panel) and second (right panel) eigenfunctions of the spectral fractional Laplacian ${\mathcal L}_s$ (solid line), fractional Laplacian ${\mathcal L}_h$ (dashed line), and regional fractional Laplacian ${\mathcal L}_r$ (dash-dot line). Note that the eigenfunctions of the spectral fractional Laplacian ${\mathcal L}_s$ are independent of $\alpha > 0$
Comparison of the solution to (3.4) with ${\mathcal L}_s$ (solid line), ${\mathcal L}_h$ (dashed line), ${\mathcal L}_r$ (dash-dot line), or ${\mathcal L}_p$ with $\delta = 4$ (dotted line). For easy comparison, the result for ${\mathcal L} = -\partial_{xx}$ (line with symbols "*") is included in the plot of $\alpha = 1.95$
Effects of the horizon size $\delta$ on the solution of the nonlocal problem (3.4) with the peridynamic operator ${\mathcal L}_p$, where $\delta = 2$ (solid line), $1$ (dash-dot line), or $0.5$ (dashed line)
Time evolution of the solution $u(x, t)$ to the nonlocal diffusion equation (3.6) with ${\mathcal L}_s$ (upper row), ${\mathcal L}_h$ (middle row), and ${\mathcal L}_r$ (lower row)
Time evolution of the solution $u(x, t)$ to the nonlocal diffusion equation (3.6) with the peridynamic operator ${\mathcal L}_p$, where the horizon size $\delta = 0.1$ (top) or $\delta = 1$ (bottom)
Solutions of the nonlocal diffusion equation (3.6) at time $t = 0.1, 0.5, 1$, where the operator is chosen as ${\mathcal L}_s$ (solid line), ${\mathcal L}_h$ (dashed line), ${\mathcal L}_p$ (dotted line), or ${\mathcal L}_r$ (dash-dot line). For easy comparison, we include the solution of the classical diffusion equation (i.e., ${\mathcal L}_i = -\partial_{xx}$ in (3.6)) in the last row
Time evolution of the solution $u(x, t)$ to the nonlocal diffusion-reaction equation (3.10) with ${\mathcal L}_s$ (row one), ${\mathcal L}_h$ (row two), and ${\mathcal L}_r$ (row three)
Time evolution of the solution $u(x, t)$ to the nonlocal diffusion-reaction equation (3.6) with the peridynamic operator ${\mathcal L}_p$, where the horizon size $\delta = 0.1$ (top) or $\delta = 0.5$ (bottom)
Solutions of the nonlocal diffusion-reaction equation (3.10) at time $t = 0.1, 0.5, 1$, where the operator is chosen as ${\mathcal L}_s$ (solid line), ${\mathcal L}_h$ (dashed line), ${\mathcal L}_p$ (dotted line), or ${\mathcal L}_r$ (dash-dot line). For easy comparison, we include the solution of the classical diffusion equation (i.e., ${\mathcal L}_i = -\partial_{xx}$ in (3.10)) in the last row
Comparison of the eigenvalues for different operators, where the eigenvalues of the standard Dirichlet Laplace operator $-\Delta$ are presented in most right column. For each $k$, upper row: $\lambda_k^s$; middle row: $\lambda_k^h$; lower row: $\lambda_k^r$
 0.2 0.5 0.7 0.9 1 1.2 1.5 1. 8 1.95 1.999 1 1.0945 1.2533 1.3718 1.5014 1.5708 1.7193 1.9687 2.2543 2.4123 2.4663 2.4674 0.9575 0.9702 1.0203 1.1032 1.1578 1.2971 1.5976 2.0488 2.3520 2.465 0.0003 0.0038 0.017 0.064 0.1135 0.2939 0.8088 1.6602 2.2444 2.4628 2 1.2573 1.7725 2.2285 2.8018 3.1416 3.9498 5.5683 7.8500 9.3206 9.8583 9.8696 1.1966 1.6016 1.9733 2.4583 2.7549 3.487 5.06 7.5033 9.2082 9.8559 0.1878 0.4593 0.6729 0.9799 1.2026 1.8719 3.6509 6.7378 8.9854 9.8512 3 1.3635 2.1708 2.9598 4.0357 4.7124 6.4252 10.23 16.287 20.55 22.172 22.207 1.3191 2.0289 2.7294 3.6987 4.3171 5.9121 9.5948 15.8 20.384 22.169 0.3085 0.8626 1.3646 2.0823 2.576 3.9902 7.75 14.701 20.049 22.161 4 1.4442 2.5066 3.6201 5.2283 6.2832 9.0744 15.75 27.335 36.012 39.406 29.478 1.4106 2.3873 3.4131 4.9055 5.8925 8.535 15.02 26.725 35.794 39.401 0.3981 1.2091 2.014 3.2054 4.0292 6.3902 12.811 25.313 35.349 39.391 5 1.5101 2.8025 4.2322 6.3912 7.854 11.861 22.011 40.847 55.645 61.558 61.685 1.4817 2.6949 4.0371 6.0733 7.4607 11.293 21.191 40.115 55.374 61.552 0.4700 1.5149 2.6231 4.323 5.5171 8.9817 18.67 38.408 54.820 61.54 6 1.5662 3.07 4.8083 7.5309 9.4248 14.761 28.934 56.714 79.402 88.627 88.826 1.5422 2.973 4.6253 7.2206 9.0334 14.175 28.037 55.868 79.080 88.62 0.5306 1.7911 3.1993 5.43 7.0245 11.722 25.235 53.876 78.418 88.605 8 1.659 3.5449 5.8809 9.7564 12.566 20.847 44.547 95.187 139.14 157.51 157.91 1.6400 3.4612 5.7133 9.455 12.175 20.225 43.509 94.122 138.72 157.5 0.6296 2.2799 4.2751 7.6101 10.072 17.552 40.218 91.591 137.85 157.49 10 1.7347 3.9633 6.8752 11.926 15.708 27.249 62.256 142.24 215 246.06 246.74 1.7189 3.8886 6.7186 11.632 15.317 26.598 61.096 140.96 214.48 246.05 0.7095 2.709 5.2735 9.749 13.145 23.749 57.377 137.92 213.39 246.02 20 1.9926 5.605 11.169 22.255 31.416 62.601 176.09 495.30 830.7 983.56 986.96 1.9836 5.5525 11.042 21.981 31.025 61.854 174.45 493.09 829.69 983.53 0.9779 4.381 9.585 19.998 28.657 58.439 169.09 487.74 827.58 983.49
 0.2 0.5 0.7 0.9 1 1.2 1.5 1. 8 1.95 1.999 1 1.0945 1.2533 1.3718 1.5014 1.5708 1.7193 1.9687 2.2543 2.4123 2.4663 2.4674 0.9575 0.9702 1.0203 1.1032 1.1578 1.2971 1.5976 2.0488 2.3520 2.465 0.0003 0.0038 0.017 0.064 0.1135 0.2939 0.8088 1.6602 2.2444 2.4628 2 1.2573 1.7725 2.2285 2.8018 3.1416 3.9498 5.5683 7.8500 9.3206 9.8583 9.8696 1.1966 1.6016 1.9733 2.4583 2.7549 3.487 5.06 7.5033 9.2082 9.8559 0.1878 0.4593 0.6729 0.9799 1.2026 1.8719 3.6509 6.7378 8.9854 9.8512 3 1.3635 2.1708 2.9598 4.0357 4.7124 6.4252 10.23 16.287 20.55 22.172 22.207 1.3191 2.0289 2.7294 3.6987 4.3171 5.9121 9.5948 15.8 20.384 22.169 0.3085 0.8626 1.3646 2.0823 2.576 3.9902 7.75 14.701 20.049 22.161 4 1.4442 2.5066 3.6201 5.2283 6.2832 9.0744 15.75 27.335 36.012 39.406 29.478 1.4106 2.3873 3.4131 4.9055 5.8925 8.535 15.02 26.725 35.794 39.401 0.3981 1.2091 2.014 3.2054 4.0292 6.3902 12.811 25.313 35.349 39.391 5 1.5101 2.8025 4.2322 6.3912 7.854 11.861 22.011 40.847 55.645 61.558 61.685 1.4817 2.6949 4.0371 6.0733 7.4607 11.293 21.191 40.115 55.374 61.552 0.4700 1.5149 2.6231 4.323 5.5171 8.9817 18.67 38.408 54.820 61.54 6 1.5662 3.07 4.8083 7.5309 9.4248 14.761 28.934 56.714 79.402 88.627 88.826 1.5422 2.973 4.6253 7.2206 9.0334 14.175 28.037 55.868 79.080 88.62 0.5306 1.7911 3.1993 5.43 7.0245 11.722 25.235 53.876 78.418 88.605 8 1.659 3.5449 5.8809 9.7564 12.566 20.847 44.547 95.187 139.14 157.51 157.91 1.6400 3.4612 5.7133 9.455 12.175 20.225 43.509 94.122 138.72 157.5 0.6296 2.2799 4.2751 7.6101 10.072 17.552 40.218 91.591 137.85 157.49 10 1.7347 3.9633 6.8752 11.926 15.708 27.249 62.256 142.24 215 246.06 246.74 1.7189 3.8886 6.7186 11.632 15.317 26.598 61.096 140.96 214.48 246.05 0.7095 2.709 5.2735 9.749 13.145 23.749 57.377 137.92 213.39 246.02 20 1.9926 5.605 11.169 22.255 31.416 62.601 176.09 495.30 830.7 983.56 986.96 1.9836 5.5525 11.042 21.981 31.025 61.854 174.45 493.09 829.69 983.53 0.9779 4.381 9.585 19.998 28.657 58.439 169.09 487.74 827.58 983.49
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