# 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
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##### References:
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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