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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

K. BogdanK. Burdzy and Z. Chen, Censored stable processes, Probab. Theory Rel., 127 (2003), 89-152.  doi: 10.1007/s00440-003-0275-1.  Google Scholar

[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.  Google Scholar

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B. A. CarrerasV. 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.  Google Scholar

[7]

Z.-Q. ChenP. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329.   Google Scholar

[8]

Z.-Q. ChenP. 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.  Google Scholar

[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.  Google Scholar

[10]

Z.-Q. ChenP. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329.   Google Scholar

[11]

Z.-Q. ChenP. 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.  Google Scholar

[12]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004.  Google Scholar

[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.  Google Scholar

[14]

E. Di NezzaG. 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.  Google Scholar

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Q. DuM. GunzburgerR. 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.  Google Scholar

[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.  Google Scholar

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S. DuoH.-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.  Google Scholar

[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.  Google Scholar

[19]

B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal., 15 (2012), 536-555.   Google Scholar

[20]

R. L. Frank, Eigenvalue bounds for the fractional Laplacian: A review, preprint, arXiv: 1603.09736. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

Q. Guan and Z. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424.  doi: 10.1142/S021949370500150X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

M. Kwasnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2015), 7-51.   Google Scholar

[29]

N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

R. Musina and A. I. Nazarov, On fractional Laplacians, Comm. Part. Diff. Eq., 39 (2014), 1780-1790.  doi: 10.1080/03605302.2013.864304.  Google Scholar

[33]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.  doi: 10.5565/PUBLMAT_60116_01.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

M. F. ShlesingerB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N. J., 1970.  Google Scholar

[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.  Google Scholar

[45]

S. Y. Yolcu and T. Yolcu, Refined eigenvalue bounds on the Dirichlet fractional Laplacian, Journal of Math. Phys. , 56 (2015), 073506, 12pp.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

K. BogdanK. Burdzy and Z. Chen, Censored stable processes, Probab. Theory Rel., 127 (2003), 89-152.  doi: 10.1007/s00440-003-0275-1.  Google Scholar

[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.  Google Scholar

[6]

B. A. CarrerasV. 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.  Google Scholar

[7]

Z.-Q. ChenP. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329.   Google Scholar

[8]

Z.-Q. ChenP. 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.  Google Scholar

[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.  Google Scholar

[10]

Z.-Q. ChenP. Kim and R. Song, Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1307-1329.   Google Scholar

[11]

Z.-Q. ChenP. 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.  Google Scholar

[12]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004.  Google Scholar

[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.  Google Scholar

[14]

E. Di NezzaG. 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.  Google Scholar

[15]

Q. DuM. GunzburgerR. 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.  Google Scholar

[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.  Google Scholar

[17]

S. DuoH.-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.  Google Scholar

[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.  Google Scholar

[19]

B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal., 15 (2012), 536-555.   Google Scholar

[20]

R. L. Frank, Eigenvalue bounds for the fractional Laplacian: A review, preprint, arXiv: 1603.09736. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

Q. Guan and Z. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424.  doi: 10.1142/S021949370500150X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

M. Kwasnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2015), 7-51.   Google Scholar

[29]

N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

R. Musina and A. I. Nazarov, On fractional Laplacians, Comm. Part. Diff. Eq., 39 (2014), 1780-1790.  doi: 10.1080/03605302.2013.864304.  Google Scholar

[33]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26.  doi: 10.5565/PUBLMAT_60116_01.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

M. F. ShlesingerB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N. J., 1970.  Google Scholar

[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.  Google Scholar

[45]

S. Y. Yolcu and T. Yolcu, Refined eigenvalue bounds on the Dirichlet fractional Laplacian, Journal of Math. Phys. , 56 (2015), 073506, 12pp.  Google Scholar

Figure 1.  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
Figure 2.  Difference between the peridynamic operator and the fractional Laplacian versus the parameter $\alpha $, where $u(x)$ is defined in (3.1)
Figure 3.  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 $\alpha = 1.95$
Figure 4.  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)
Figure 5.  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)
Figure 6.  The absolute (left panel) and relative (right panel) differences in the eigenvalues of the fractional Laplacian and the spectral fractional Laplacian
Figure 7.  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 $\alpha > 0$
Figure 8.  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 $\alpha = 1.95$
Figure 9.  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)
Figure 10.  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)
Figure 11.  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)
Figure 12.  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
Figure 13.  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)
Figure 14.  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)
Figure 15.  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
Table 1.  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: $\lambda_k^r$
0.20.50.70.911.21.51. 81.951.999
11.09451.25331.37181.50141.57081.71931.96872.25432.41232.46632.4674
0.95750.97021.02031.10321.15781.29711.59762.04882.35202.4650
0.00030.00380.01700.06400.11350.29390.80881.66022.24442.4628
21.25731.77252.22852.80183.14163.94985.56837.85009.32069.85839.8696
1.19661.60161.97332.45832.75493.48705.06007.50339.20829.8559
0.18780.45930.67290.97991.20261.87193.65096.73788.98549.8512
31.36352.17082.95984.03574.71246.425210.23016.28720.55022.17222.207
1.31912.02892.72943.69874.31715.91219.594815.80020.38422.169
0.30850.86261.36462.08232.57603.99027.750014.70120.04922.161
41.44422.50663.62015.22836.28329.074415.75027.33536.01239.40629.478
1.41062.38733.41314.90555.89258.535015.02026.72535.79439.401
0.39811.20912.01403.20544.02926.390212.81125.31335.34939.391
51.51012.80254.23226.39127.854011.86122.01140.84755.64561.558 61.685
1.48172.69494.03716.07337.460711.29321.19140.11555.37461.552
0.47001.51492.62314.32305.51718.981718.67038.40854.82061.540
61.56623.07004.80837.53099.424814.76128.93456.71479.40288.62788.826
1.54222.97304.62537.22069.033414.17528.03755.86879.08088.620
0.53061.79113.19935.43007.024511.72225.23553.87678.41888.605
81.65903.54495.88099.756412.56620.84744.54795.187139.14157.51157.91
1.64003.46125.71339.455012.17520.22543.50994.122138.72157.50
0.62962.27994.27517.610110.07217.55240.21891.591137.85157.49
101.73473.96336.875211.92615.70827.24962.256142.24215.00246.06246.74
1.71893.88866.718611.63215.31726.59861.096140.96214.48246.05
0.70952.70905.27359.749013.14523.74957.377137.92213.39246.02
201.99265.605011.16922.25531.41662.601176.09495.30830.70983.56986.96
1.98365.552511.04221.98131.02561.854174.45493.09829.69983.53
0.97794.38109.585019.99828.65758.439169.09487.74827.58983.49
0.20.50.70.911.21.51. 81.951.999
11.09451.25331.37181.50141.57081.71931.96872.25432.41232.46632.4674
0.95750.97021.02031.10321.15781.29711.59762.04882.35202.4650
0.00030.00380.01700.06400.11350.29390.80881.66022.24442.4628
21.25731.77252.22852.80183.14163.94985.56837.85009.32069.85839.8696
1.19661.60161.97332.45832.75493.48705.06007.50339.20829.8559
0.18780.45930.67290.97991.20261.87193.65096.73788.98549.8512
31.36352.17082.95984.03574.71246.425210.23016.28720.55022.17222.207
1.31912.02892.72943.69874.31715.91219.594815.80020.38422.169
0.30850.86261.36462.08232.57603.99027.750014.70120.04922.161
41.44422.50663.62015.22836.28329.074415.75027.33536.01239.40629.478
1.41062.38733.41314.90555.89258.535015.02026.72535.79439.401
0.39811.20912.01403.20544.02926.390212.81125.31335.34939.391
51.51012.80254.23226.39127.854011.86122.01140.84755.64561.558 61.685
1.48172.69494.03716.07337.460711.29321.19140.11555.37461.552
0.47001.51492.62314.32305.51718.981718.67038.40854.82061.540
61.56623.07004.80837.53099.424814.76128.93456.71479.40288.62788.826
1.54222.97304.62537.22069.033414.17528.03755.86879.08088.620
0.53061.79113.19935.43007.024511.72225.23553.87678.41888.605
81.65903.54495.88099.756412.56620.84744.54795.187139.14157.51157.91
1.64003.46125.71339.455012.17520.22543.50994.122138.72157.50
0.62962.27994.27517.610110.07217.55240.21891.591137.85157.49
101.73473.96336.875211.92615.70827.24962.256142.24215.00246.06246.74
1.71893.88866.718611.63215.31726.59861.096140.96214.48246.05
0.70952.70905.27359.749013.14523.74957.377137.92213.39246.02
201.99265.605011.16922.25531.41662.601176.09495.30830.70983.56986.96
1.98365.552511.04221.98131.02561.854174.45493.09829.69983.53
0.97794.38109.585019.99828.65758.439169.09487.74827.58983.49
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