Publications

MURI Publications

1. M. Samiee, E. Kharazmi, M. Zayernouri and M. M. Meerschaert, “Petrov-Galerkin Method for Fully Distributed-Order Fractional Partial Differential Equations”, submitted, https://arxiv.org/pdf/1805.08242.pdf.
2. Y. Jiao, G. Malara and P. D. Spanos, “Boundary Element Method Based Approach for Estimating the Response of a System Governed by a Stochastic Nonlinear Fractional Diffusion Equation”, submitted.
3. G. Malara, Y. Jiao and P. D. Spanos, “Statistical Linearization Approach for Calculating the Response Statistics of a System Governed by a Nonlinear Fractional Diffusion Equation”, submitted.
4. P. Varghaei, E. Kharazmi, J. Suzuki, and M. Zayernouri, “Vibration analysis of geometrically nonlinear and fractional viscoelastic cantilever beams”, submitted, https://arxiv.org/pdf/1909.02142.pdf.
5. Z. Hao, G. Lin, Z. Zhang, “Regularity and spectral methods for two-sided fractional diffusion equations with a low-order term”, submitted, https://arxiv.org/pdf/1705.07209.pdf.
6. M. Gulian, and G. Pang. “Stochastic Solution of Elliptic and Parabolic Boundary Value Problems for the Spectral Fractional Laplacian”, submitted, https://arxiv.org/pdf/1812.01206.pdf.
7. H. Wang and X.C. Zheng, “Nonlinear variable-order fractional differential equations and their numerical approximations: Well-posedness, regularity and error estimate”, submitted.
8. X. Zheng, V.J. Ervin and H. Wang, “Well-posedness of the two-sided variable coefficient Caputo flux fractional diffusion equation and error estimate of its spectral approximation”, submitted, https://arxiv.org/pdf/1811.00582.pdf.
9. J. Jia, X. Zheng, H. Fu, P. Dai, H. Wang, “A fast method for variable-order space-fractional diffusion equations”, submitted, https://arxiv.org/pdf/1907.02697.pdf.  
10. X. Zheng, H. Wang, “Well-posedness and regularity of a variable-order space-time fractional diffusion equation, analysis and applications”, submitted.
11. Z. Fang, H. Sun, H. Wang, “A fast method for a variable-order time-fractional PDE”, submitted.
12. B. Li, H. Wang, J. Wang, “Well-posedness and numerical approximation of a fractional diffusion equation with a nonlinear variable order”, submitted.
13. X. Zheng, J. Cheng and H. Wang, “Uniqueness of determining the variable fractional order in the boundary-value problem of variable-order linear space-fractional diffusion equations with variable diffusivity coefficients”, submitted.
14. X. Zheng and H. Wang, “Finite element approximations to variable-order time-fractional diffusion equations and their analysis without regularity assumptions of the solutions”, submitted.
15. X. Li, Z. Mao, F. Song, H. Wang, and G. E. Karniadakis, “A fast solver for spectral element approximation applied to fractional differential equations using hierarchical matrix approximation”, in revision, https://arxiv.org/pdf/1808.02937.pdf.
16. M. Ainsworth and Z. Mao, “Fractional Phase Field Crystal Modelling: Analysis, Approximation and Pattern formation”, to appear IMA Journal of Applied Mathematics, 2020.
17. E. de Moraes, M. Zayernouri, M. Meerschaert, “An Integrated Sensitivity-Uncertainty Quantification Framework for Stochastic Phase-Field Modeling of Material Damage”, submitted.
18. J. Suzuki and M. Zayernouri, “A Data-Infused Self-Singularity-Capturing Scheme for Fractional Differential Equations”, submitted, https://arxiv.org/pdf/1810.12219.pdf.
19. E. de Moraes, H. Salehi, and M. Zayernouri, “Data-Driven Failure Prediction in Brittle Materials: A Phase-Field based Machine Learning Framework”, submitted.
20. E. Kharazmi, Z. Mao, G. Pang, M. Zayernouri, and G. E. Karniadakis, “Fractional Calculus and Numerical Methods for Fractional PDEs”, Paper Review (De Gruyter), 2020.
21. Q. Du, X. Li, and L. Yuan. “Analysis of coarse-grained lattice models and connections to nonlocal interactions”, preprint, 2020.
22. M. Samiee, A. A Safaei, M. Zayernouri, “A Fractional Subgrid-scale Model for Turbulent Flows: Theoretical Formulation and a Priori Study”, Phys. Fluids, (2020), in Press.
23. E. Kharazmi and M. Zayernouri, “Operator-Based Uncertainty Quantification of Stochastic Fractional PDEs, ASME Journal on Verification, Validation, and Uncertainty Quantification”, (2020), in Press.
24. M. Samiee, E. Kharazmi, M. Meerschaert, M. Zayernouri, “A Unified Petrov-Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations”, Commun. Appl. Math. Comput, (2020), in Press.
25. Y. Zhou, J. Suzuki, C. Zhang, and M. Zayernouri, “Fast Implicit-Explicit Time-Integration of Nonlinear Fractional Differential Equations”, to appear Journal of Applied Numerical Mathematics, 2020.
26. P. Varghaei, E. Kharazmi, J. Suzuki, and M. Zayernouri, “Nonlinear Vibration of Fractional Viscoelastic Cantilever Beam: Application to Structural Health Monitoring”, ASME Journal of Vibration and Acoustics, (2020), in Press.
27. 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., Vol. 404 (2020), pp. 109009.
28. H. Zhang, X. Jiang, F. Zeng and G. E. Karniadakis. “A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations.” J. Comput. Phys., Vol. 405 (2020), pp. 109141.
29. T. Wang, X.-F. Li, and H. Wang, “An error estimate of a Eulerian-Lagrangian localized adjoint method for a space-fractional advection diffusion equation”, Int’l J. Num. Anal. Modeling, Vol. 17 (2020), pp. 151-171.
30. P. Dai, Q. Wu, X. Zheng, H. Wang, “An efficient matrix splitting preconditioning technique for two-dimensional unsteady space-fractional diffusion equations”, J. Comput. Appl. Math., Vol. 371 (2020), pp. 112673.
31. N. Du, X. Guo, H. Wang, “A fast state-based peridynamic numerical model”, Commun. Comput. Phys., Vol. 27 (2020), pp. 274-291.
32. N. Du, X. Guo, H. Wang, “Fast upwind and Eulerian-Lagrangian control volume schemes for time-dependent directional space-fractional advection-dispersion equations”, J. Comput. Phys., Vol. 405 (2020), pp. 109127.
33. X. Zheng, V.J. Ervin, H. Wang, “An indirect finite element method for variable-coefficient space-fractional diffusion equations and its optimal order error estimates”, Commun. Appl. Math. Comp., Vol. 2 (2020), pp. 147-162.
34. X. Zheng, H. Wang, “An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes”, SIAM J. Numer. Anal., Vol. 58 (2020), pp. 330-352.
35. X. Zheng, H. Wang, “Well-posedness and smoothing properties of history-state based variable-order time-fractional diffusion equations”, J. Appl. Math. Phys. (ZAMP), Vol. 71 (2020), pp. 34.
36. Q. Du, B. Engquist, and X.C. Tian. “Multiscale modeling, homogenization and nonlocal effects: mathematical and computational issues”, Contemporary Mathematics, Celebrating 75th Anniversary of Mathematics of Computation, 2020.
37. H. Lee and Q Du. “Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications”, ESAIM-Math. Model. Num., Vol. 54 (2020), pp. 105–128.
38. Q. Du, L. Toniazzi, and Z. Zhou. “Stochastic representation of solution to nonlocal‐in‐time diffusion”, Stoch. Process. Their Appl, available on line 25 June 2019.
39. B. Toaldo and M. M. Meerschaert, “Relaxation patterns and semi-Markov dynamics”, Stochastic Processes and their Applications, Vol. 129 (2019), No. 8, pp. 2850-2879.
40. M. Samiee, M. Zayernouri, M. M. Meerschaert, “A Unified Spectral Method for FPDEs with Two-sided Derivatives; Part I: A Fast Solver”, J. Comput. Phys., Vol. 385 (2019), pp. 225-243.
41. M. Samiee, M. Zayernouri, M. M. Meerschaert, “A unified spectral method for FPDEs with two-sided derivatives; Part II: stability, and error analysis”, J. Comput. Phys., Vol. 385 (2019), pp. 244-261.
42. F. Sabzikar, A. I. McLeod, and M. M. Meerschaert, “Parameter estimation for ARTFIMA time series”, J. Stat. Plan. Inference, Vol. 200, (2019), pp. 129-145.
43. J. F. Kelly and M. M. Meerschaert, “The fractional advection-dispersion equation for contaminant transport”, Handbook of Fractional Calculus with Applications, 2019.
44. M. M. Meerschaert, E. Nane and P. Vellaisamy, “Inverse subordinators and time fractional equations”, Handbook of Fractional Calculus with Applications, 2019.
45. M. M. Meerschaert and H. P. Scheffler, “Continuous time random walks and space-time fractional differential equations”, Handbook of Fractional Calculus with Applications, 2019.
46. Y. Zhang and M. M. Meerschaert, “Lagrangian approximation of vector fractional differential equations”, Handbook of Fractional Calculus with Applications, 2019.
47. P. Kern, S. Lage and M. M. Meerschaert, “Semi-fractional diffusion equations”, Fract. Cal. Appl. Anal., Vol. 22 (2019), No. 2, pp. 326-357.
48. C. Li, M. Li, S. Piskarev and M. M. Meerschaert, “The fractional d’Alembert’s formulas”, J. Funct. Anal., (2019), pp. 108279.
49. J. F. Kelly, H. Sankaranarayanan and M. M. Meerschaert, “Boundary conditions for two-sided fractional diffusion”, J. Comput. Phys., Vol. 376 (2019), pp. 1089-1107.
50. J. F. Kelly, and M. M. Meerschaert, “Space-time duality and high-order fractional diffusion”. Phys. Rev. E, Vol. 99 (2019), No. 2, pp. 022122.
51. M. Ari, R. Teymoori-Faal, M. Zayernouri, (2019) “Vibrations Suppression of Fractionally Damped Plates using Multiple Optimal Dynamic Vibration Absorbers”, Int. J. of Comput. Math., (2019), pp. 1-24.
52. A. Lischke, M. Zayernouri, Z. Zhang, “Spectral and Spectral Element Methods for Fractional Advection–Diffusion–Reaction Equations”, Handbook of Fractional Calculus with Applications, Vol. 3: Numerical Methods, Karniadakis (Ed.), De Gruyter, Berlin, 2019.
53. Z. Zhang, “Error estimates of spectral Galerkin methods for linear fractional reaction-diffusion equation”, J. Sci. Comput., Vol. 78 (2019), No. 2, pp. 1087-1110.
54. Z. Mao, Z. Li and G. E. Karniadakis, “Nonlocal flocking dynamics: Learning the fractional order of PDEs from particle simulations”, Commun. Appl. Math. Comput., Vol. 1 (2019), No. 4, pp. 597-619.
55. N. Wang, Z. Mao, C. Huang, and G. E. Karniadakis, “A spectral penalty method for two-sided fractional differential equations with general boundary conditions”, SIAM J. Sci. Comput., Vol. 41 (2019), No. 3, pp. A1840-A1866.
56. T. Zhao, Z. Mao and G. E. Karniadakis, “Multi-domain Spectral Collocation Method for Variable-Order Nonlinear Fractional Differential Equations”, Comput. Methods Appl. Mech. Engrg., Vol. 348 (2019), pp. 377-395.
57. T. Wang, F. Song, H. Wang, and G. E. Karniadakis, “Fractional Gray-Scott Model: Well-posedness, Discretization, and Simulations“, Comput. Methods Appl. Mech. Engrg., Vol. 347 (2019), pp. 1030-1049.
58. M. Gulian, M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Machine Learning of Space-Fractional Differential Equations”, SIAM J. Sci. Comput., Vol. 41 (2019), No. 4, pp. A2485-A2509.
59. F. Song and G. E. Karniadakis, “Fractional magneto-hydrodynamics: Algorithms and applications”, J. Comput. Phys., Vol. 378 (2019), pp. 44-62.
60. L. Guo, F. Zeng, I. Turner, K. Burrage and G. E. Karniadakis “Efficient Multistep methods for tempered fractional calculus: algorithms and simulations”, SIAM J. Sci. Comput., Vol. 41 (2019), No. 4, pp. A2510–A2535.
61. G Pang, L Lu, GE Karniadakis, “fPINNs: Fractional Physics-Informed Neural Networks”, SIAM J. Sci. Comput., Vol. 41 (2019), No. 4, pp. A2603-A2626.
62. P. Mehta, G. Pang, F. Song and G. E. Karniadakis. “Discovering a universal variable-order fractional model for turbulent Couette flow using a physics-informed neural network.” Fract. Cal. Appl. Anal., Vol. 22 (2019), No. 6, pp. 1675-1688.
63. M. Ainsworth and Z. Mao, “Analysis and Approximation of Gradient Flows Associated with a Fractional Order Gross-Pitaevskii Free Energy”, Commun. Appl. Math. Comput., Vol. 1 (2019), No. 1, pp. 5-19.
64. L. Chen, J, Zhao and H. Wang, “On Power Law Scaling Dynamics for Time-fractional Phase Field Models during Coarsening”, Commun. Nonlinear Sci. Numer. Simul., Vol. 70 (2019), pp. 257-270.
65. H. Fu and H. Wang, “A fast parareal finite difference method for space-time fractional partial differential equation”, J. Sci. Comput., Vol. 78 (2019), No. 3, pp. 1724-1743.
66. S. Duo and H. Wang, “A fractional phase-field model using an infinitesimal generator of α stable Lévy process”, J. Comput. Phys. Vol. 384 (2019), pp. 253-269.
67. H. Wang and X.C. Zheng, “A modified time-fractional diffusion equation and its finite difference method: regularity and error analysis”, Fract. Cal. Appl. Anal., Vol. 22 (2019), pp. 1014-1038.
68. N. Du, H. Sun, and H. Wang, “A preconditioned fast finite difference scheme for space-fractional diffusion equations in convex domains”, Comput. Appl. Math., Vol. 38 (2019), No. 1, pp. 14.
69. X. Zheng, V.J. Ervin and H. Wang, “Spectral approximation of a variable coefficient fractional diffusion equation in one space dimension”, Appl. Math. Comput.  Vol. 361 (2019), pp. 98-111.
70. L. Chen, J. Zhang, J. Zhao, W. Cao, H. Wang, J. Zhang, “An accurate and efficient algorithm for the time-fractional molecular beam epitaxy model with slope selection”, Comput. Phys. Commun., Vol. 245 (2019), pp. 106842.
71. H. Fu, H. Liu, H. Wang, “A finite volume method for two-dimensional Riemann-Liouville space-fractional diffusion equation and its efficient implementation”, J. Comput. Phys., Vol. 388 (2019), pp. 316-334.
72. H. Fu, Y. Sun, H. Wang and X. Zheng, “Stability and convergence of a Crank–Nicolson finite volume method for space fractional diffusion equations”. Appl. Numer. Math., Vol. 139 (2019), pp. 38-51.
73. J. Gao, M. Zhao, N. Du, X. Guo, H. Wang and J. Zhang, “A finite element method for space-time directional fractional diffusion partial differential equations in the plane and its error analysis”, J. Comput. Appl. Math., Vol. 362 (2019), pp. 354-365.
74. J. Jia, H. Wang, “A fast finite volume method for conservative space-time fractional diffusion equations discretized on space-time locally refined meshes”, Comput. Math. Appl., Vol. 78 (2019), pp. 1345-1356.
75. J. Jia, H. Wang, “A fast finite volume method on locally refined meshes for fractional diffusion equations”, East Asian J. Appl. Math., Vol. 9 (2019), pp. 755-779.
76. J. Liu, H. Fu, H. Wang, and X. Chai, “A preconditioned fast quadratic spline collocation method for two-sided space-fractional partial differential equations”, J. Comput. Appl. Math., Vol. 360 (2019), pp. 138-156.
77. F. Wang, H. Chen, H. Wang, “Finite element simulation and efficient algorithm for fractional Cahn–Hilliard equation”, J. Comput. Appl. Math., Vol. 356 (2019), pp. 248-266.
78. F. Wang, H. Chen, H. Wang, “Least squared mixed Galerkin formulation for variable-coefficient fractional differential equations with D-N boundary condition”, Math. Meth. Appl. Sci., Vol. 42 (2019), pp. 4331-4342.
79. H. Wang and X. Zheng, “Analysis and numerical solution of a nonlinear variable-order fractional differential equation”. Adv. Comput. Math., Vol. 45 (2019), pp. 2647–2675.
80. H. Wang, and X. Zheng, “Well-posedness and regularity of the variable-order time-fractional diffusion equations”, J. Math. Anal. Appl., Vol. 475 (2019), pp. 1778-1802.
81. S. Yang, H. Chen, H. Wang, “Least-squared mixed variational formulation based on space decomposition for a kind of variable-coefficient fractional diffusion problems”, J. Sci. Comput., Vol. 78 (2019), pp. 687-709.
82. J. Zhao, L. Chen, H. Wang, “On power law scaling dynamics for time-fractional phase field models during coarsening”. Commun. Nonlinear Sci. Numer. Simul., Vol. 70 (2019), pp. 257-270.
83. M. Zhao, H. Wang, “Fast finite difference methods for space-time fractional partial differential equations in three space dimensions with nonlocal boundary conditions”, Appl. Numer. Math., Vol. 145 (2019), pp.411-428.
84. X. Zheng, J. Cheng, H. Wang, “Uniqueness of determining the variable fractional order in variable-order time-fractional diffusion equations”, Inverse Probl., Vol. 35 (2019), pp. 125002.
85. X. Zheng, H. Liu, H. Wang, H. Fu, “An efficient finite volume method for nonlinear distributed-order space-fractional diffusion equations in three space dimensions”, J. Sci. Comput., Vol. 80 (2019), pp. 1395-1418.
86. X. Zheng, H. Wang, “Well-posedness and regularity of a nonlinear variable-order fractional wave equation”, Appl. Math. Lett., Vol. 95 (2019), pp. 29-35.
87. C. Chen, H. Liu, X. Zheng, H. Wang, “A two-grid MMOC finite element method for nonlinear variable-order time-fractional mobile/immobile advection–diffusion equations”, Comput. Math. Appl., available on line 23 December 2019.
88. H. Wang, “Fast numerical methods for space-fractional partial differential equations”, Handbook of Fractional Calculus with Applications. Vol. 3, Numerical Methods, Karniadakis (Ed.), De Gruyter, Berlin, 2019, pp. 307-328.
89. Q. Du, L. Ju, X. Li and Z. Qiao, “Maximum Principle Preserving Exponential Time Differencing Schemes for the Nonlocal Allen-Cahn Equation”, SIAM J. Numer. Anal, Vol. 57 (2019), pp. 875-898.
90. Q. Du, L. Ju and L. Lu, “A discontinuous Galerkin method for one-dimensional time-dependent nonlocal diffusion problems”, Math. of Comput., Vol. 88 (2019), pp. 123-147.
91. Q. Du, L. Ju, J. Lu and X. Tian, “A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Diffusion Problems”, Commun. Appl. Math. Comput., Vol. 2 (2019), No.1, pp. 31-55.
92. Q Du, L Ju, and J Lu. “Analysis of fully discrete approximations for dissipative systems and application to time‐dependent nonlocal diffusion problems”, J. Sci. Comput., Vol. 78 (2019), pp. 1438-1466.
93. Q. Du, Y. Tao, X. Tian, and J. Yang. “Asymptotically compatible discretization of multidimensional nonlocal diffusion models and approximations of nonlocal Green’s functions”, IMA J. Numer. Anal., Vol. 39 (2019), pp. 607-625.
94. Q. Du. “Nonlocal Modeling, Analysis, and Computation”, volume CBMS-NSF conference series in applied mathematics 94. SIAM, 2019.
95. Q. Du and X. Tian. “Mathematics of smoothed particle hydrodynamics via a nonlocal stokes equation”. Found. Comput. Math., (2019), pp. 1-26.
96. Q. Du and Xiaobo Yin. “A conforming DG method for linear nonlocal models with integrable kernels”. J. Sci. Comput., Vol. 80 (2019), pp. 1913-1935.
97. Q. Du, J. Zhang, and C. Zheng. “On uniform second order nonlocal approximations to linear two-point boundary value problems”, Commun. Math. Sci., Vol. 17(6), (2019), No. 6, pp.1737–1755.
98. H. Lee and Q Du. “Asymptotically compatible SPH-like particle discretizations of one-dimensional linear advection models”, SIAM J. Numer. Anal., Vol. 57 (2019), pp. 127–147.
99. B. Baeumer, T. Luks and M. M. Meerschaert, “Space‐time fractional Dirichlet problems”. Math. Nachr., Vol. 291(2018), pp. 2516-2535.
100. J. F. Kelly, C. G. Li and M. M. Meerschaert, “Anomalous diffusion with ballistic scaling: A new fractional derivative”, J. Comput. Appl. Math., Vol. 339 (2018), pp. 161-178.
101. P. Kern, Y. Xiao, and M. M. Meerschaert, “Asymptotic behavior of semistable Lévy exponents and applications to fractal path properties”, J. Theoret. Probab., Vol. 31 (2018), No. 1, pp. 598–617.
102. G. Didier, M. M. Meerschaert, V. Pipiras, “Domain and range symmetries of operator fractional Brownian fields”, Stoch. Process. Their Appl., Vol. 128 (2018), No. 1, pp. 39–78.
103. B. Baeumer, M. Kovacs, M.M. Meerschaert and H. Sankaranarayanan, “Reprint: Boundary Conditions for Fractional Diffusion”, J. Comput. Appl. Math., Vol. 339 (2018), pp. 414–430.
104. G. Malara, P.D. Spanos, “Nonlinear random vibrations of plates endowed with fractional derivative elements”, Probabilist. Eng. Mech., Vol. 54 (2018), pp. 2-8.
105. Z. Mao and G. E. Karniadakis, “A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative”, SIAM J. Numer. Anal., 56 (2018), No.1 pp. 24–49.
106. Z. Zhang, W. Deng, G.E. Karniadakis, “A Riesz basis Galerkin method for the tempered fractional Laplacian“, SIAM J. Numer. Anal., 56 (2018), No. 5, pp. 3010–3039.
107. F. Song and G. E. Karniadakis, “A Universal Fractional Model of Wall-Turbulence”, https://arxiv.org/pdf/1808.10276.pdf, 2018.
108. F. Zeng, I. Turner, K. Burrage, and G. E. Karniadakis. “A new class of semi-implicit methods with linear complexity for nonlinear fractional differential equation“, SIAM J. Sci. Comput., Vol. 40 (2018), No. 5, pp. A2986-A3011.
109. 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, (2018), pp. 17-57.
110. M. Ainsworth, C. Glusa, “Hybrid Finite Element – Spectral Method for the Fractional Laplacian: Approximation Theory and Efficient Solver”, SIAM J. Sci. Comput., 40 (2018), pp. A2383–A2405.
111. H. Fu, H. Wang, and Z. Wang, “POD/DEIM reduced-order modeling of time-fractional partial differential equations with applications in parameter identification”, J. Sci. Comput., Vol. 74 (2018), No. 1, pp. 220-243.
112. M. Zhao, S. He, H. Wang, and G. Qin, “An integrated fractional partial differential equation and molecular dynamics model of anomalously diffusive transport in heterogeneous nano-pore structures”, J. Comput. Phys., Vol. 373 (2018), pp. 1000-1012.
113. W. Deng, B. Li, Z. Qian and H. Wang, “Time discretization of a tempered fractional Feynman-Kac equation with measure data”, SIAM J. Numer. Anal., Vol. 56 (2018), No. 6, pp. 3249-3275.
114. S. Duo, H. Wang, and Y. Zhang, “A comparative study on nonlocal diffusion operators related to the fractional Laplacian”, Disc. Cont. Dyn. Syst. Series B, (2018), pp. 439-467.
115. J. Jia and H. Wang, “A fast finite difference method for distributed-order space-fractional partial differential equations on convex domains”, Comput. Math. Appl., Vol. 75 (2018), pp. 2031-2043.
116. X. Guo, Y. Li, and H. Wang, “A fourth order scheme for space fractional diffusion equations”, J. Comput. Phys., Vol. 373 (2018), pp. 410-424.
117. X. Guo, Y. Li and H. Wang, “A fast finite difference method for tempered fractional diffusion equations”, Commun. Comput. Phys, Vol. 24 (2018), pp. 531-556.
118. M. Zhao, A. Cheng, and H. Wang, “A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations with fractional derivative boundary conditions”, J. Sci. Comput., Vol. 74 (2018), pp. 1009-1033.
119. H. Liu, H. Wang, and A. Cheng, “A fast discontinuous Galerkin method for a bond-based linear peridynamic model discretized on a locally refined composite mesh”, J. Sci. Comput., Vol. 76 (2018), pp. 913-942.
120. H. Liu, A. Cheng, H. Yan, Z. Liu, and H. Wang, “A fast compact finite difference method for quasilinear time fractional parabolic equation without singular kernel”, Int. J. Comput. Math., (2018), pp. 1-19.
121. H. Liu, A. Cheng, H. Wang, and J. Zhao, “Time-fractional Allen–Cahn and Cahn–Hilliard phase-field models and their numerical investigation”, Comput. Math. Appl., Vol. 76 (2018), pp. 1876-1892.
122. X. Guo, Y. Li and H. Wang, “Tempered fractional diffusion equations for pricing multi-asset options under CGMYe process”, Comput. Math. Appl., Vol. 76 (2018), No.6, pp 1500-1514.
123. S Yang, H Chen, H Wang, “Least-squared mixed variational formulation based on space decomposition for a kind of variable-coefficient fractional diffusion problems”, J. Sci. Comput., (2018), pp. 1-23.
124. X. Guo, Y. Li and H. Wang, “A high order finite difference method for tempered fractional diffusion equations with applications to the CGMY model”, SIAM J Sci Comput. Vol. 40 (2018), pp. A3322-A3343.
125. Qiang Du and Xiaochuan Tian. “Heterogeneously localized nonlocal operators, boundary traces and variational problems”. In Proceedings of the international congress of Chinese mathematicians, Vol. 43 of the ALM series (2018), pp. 217–236. International Press.
126. Q. Du, J. Yang and W. Zhang, “Uniform Lp-bound of the Allen-Cahn equation and its numerical discretization”, International J. Numer. Anal. Modeling, Vol. 15 (2018), pp. 213–227.
127. RM Slevinsky, H Montanelli, Q Du, “A spectral method for nonlocal diffusion operators on the sphere”, J. Comput. Phys., Vol. 372 (2018), pp. 893-911.
128. Q. Du, H. Li, J. Lu and X. Tian, “A quasinonlocal coupling method for nonlocal and local diffusion models”, SIAM J. Numer. Anal., Vol. 56 (2018), No. 3, pp. 1386-1404.
129. H. Liang, Q. Sun and Q. Du, “Data-driven compressive sensing and applications in uncertainty quantification”, J. Comp. Phys., Vol. 372 (2018), pp. 893-911.
130. Q. Du and X. Tian, “Stability of nonlocal Dirichlet integrals and implications for peridynamic correspondence material modeling”, SIAM J. Appl. Math, Vol. 78 (2018), No. 3, pp. 1536-1552.
131. Q. Du, H. Han, J. Zhang and C. Zheng, “Numerical solution of a two-dimensional nonlocal wave equation on unbounded domains”, SIAM J. Sci. Comp., Vol. 40 (2018), No. 3, pp. A1430-A1445.
132. Q. Du, J. Zhang and C. Zheng, “Nonlocal wave propagation in unbounded multi-scale media”, Comm. Comp. Phys., Vol. 24 (2018), pp. 1049-1072.
133. Q. Du, L. Ju, X. Li and Z. Qiao, “Stabilized linear semi-implicit schemes for the nonlocal Cahn–Hilliard equation”, J. Comput. Phys., Vol. 363 (2018), pp. 39-54.
134. Yali Qiao, Zhi Zhou, Zhixing Chen, Sicen Du, Qian Cheng, Haowei Zhai, Nathan Joseph Fritz, Qiang Du, and Yuan Yang. “Visualizing ion diffusion in battery systems by fluorescence microscopy: a case study on the dissolution of limn2o4”. Nano Energy,Vol. 45 (2018), pp. 68-74.
135. M.S. Alrawashdeh, J.F. Kelly, M.M. Meerschaert, and H.-P. Scheffler, “Applications of Inverse Tempered Stable Subordinators”, Comput. Math. Appl., Vol. 73 (2017), No. 6, pp. 892–905.
136. G. Didier, M.M. Meerschaert, and V. Pipiras, “Exponents of operator self-similar random fields”, J. Math. Anal. Appl., Vol. 448 (2017), No. 2, pp. 1450–1466.
137. J. F. Kelly, D. Bolster, J. D. Drummond, A. I. Packman, M. M. Meerschaert, “FracFit: A Robust Parameter Estimation Tool for Fractional Calculus Models”, Water Resour. Res., Vol. 53 (2017), No. 3, pp. 1763–2576.
138. J. F. Kelly and M. M. Meerschaert, “Space-time duality for the fractional advection dispersion equation”, Water Resour. Res., Vol. 53 (2017), No. 4, pp. 3464–3475.
139. P.D. Spanos, “Randomly excited nonlinear dynamic systems endowed with fractional derivatives elements”, Proc. of the 6th International Congress of Serbian Society of Mechanics, Mountain Tara, Serbia, June 19-21, 2017.
140. P.D. Spanos, G. Malara, “Random Vibrations of Nonlinear Continua Endowed with Fractional Derivative Elements“, Procedia engineering, Vol. 199 (2017), pp. 18-27.
141. F. Song, C. Xu, G.E. Karniadakis, “Computing fractional Laplacians on complex-geometry domains: algorithms and simulations“, SIAM J. Sci. Comput., Vol. 39 (2017), No. 4, pp. A1320-A1344.
142. F. Song, G. E. Karniadakis, “Fractional spectral vanishing viscosity method: Application to the quasi-geostrophic equation”, Chaos, Solitons & Fractals, Vol. 201 (2017), pp. 327-332.
143. F. Zeng, Z. Mao, G.E. Karniadakis, “A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities”, SIAM J. Sci. Comput., Vol. 39 (2017), No. 1, pp. A360-A383.
144. F. Song, F. Zeng, W. Cai, W. Chen, G. E. Karniadakis, “Efficient two-dimensional simulations of the fractional Szabo equation with different time-stepping schemes”, Comput. Math. Appl., Vol. 73 (2017), pp. 1286-1297.
145. Z. Mao, G.E. Karniadakis, “Fractional Burgers equation with nonlinear non-locality: Spectral vanishing viscosity and local discontinuous Galerkin methods”, J. Comput. Physics, Vol. 336 (2017), pp. 143-163.
146. A. Lischke, M. Zayernouri, G. E. Karniadakis, “A Petrov-Galerkin spectral method of linear complexity for fractional multi-term ODEs on the half line”, SIAM J. Sci. Comput., Vol. 39 (2017), No. 3, pp. A922-A946.
147. X. Chen, F. Zeng, G.E. Karniadakis, “A tunable finite difference method for fractional differential equations with non-smooth solutions”, Comput. Methods Appl. Mech. Engrg., Vol. 318 (2017), pp. 193-214.
148. F. Zeng, Z. Zhang and G.E. Karniadakis, “Second-order numerical methods for multi-term fractional differential equa­tions: Smooth and non-smooth solutions”, Comput. Methods Appl. Mech. Engrg., Vol. 327 (2017), pp. 478-502.
149. E. Kharazmi, M. Zayernouri, G.E. Karniadakis, “A Petrov–Galerkin spectral element method for fractional elliptic problems”, Comput. Methods Appl. Mech. Engrg., Vol. 324 (2017), pp. 512–536.
150. G. Pang, P. Perdikaris, W. Cai, and G. E. Karniadakis, “Discovering variable fractional orders of advection–dispersion equations from field data using multi-fidelity Bayesian optimization”, J. Comput. Phys., 348 (2017), 694-714.
151. M. Ainsworth and C. Glusa, “Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver”, Comput. Methods Appl. Mech. Engrg., 327 (2017), pp. 4–35.
152. M. Ainsworth and Z. Mao, “Analysis and Approximation of a Fractional Cahn–Hilliard Equation”, SIAM J. Numer. Anal., Vol. 55 (2017), No. 4, pp. 1689–1718.
153. M. Ainsworth and Z. Mao, “Well-posedness of the Cahn–Hilliard equation with fractional free energy and its Fourier Galerkin approximation”, Chaos, Solitons & Fractals, Vol. 102 (2017), pp. 264-273.
154. H. Fu, M. Ng and H. Wang, “A divide-and-conquer fast finite difference method for space time fractional partial differential equation”, Comput. Math. Appl., Vol. 73 (2017), pp. 1233-1242.
155. H. Fu and H. Wang, “A fast space-time finite difference method for space-time fractional diffusion equations”, ‎Fract. Calc. Appl. Anal., Vol. 20 (2017), pp. 88-116.
156. M. Zhao, A. Cheng, and H. Wang, “A preconditioned fast Hermite finite element method for space-fractional diffusion equations”, Disc. Cont. Dyn. Syst. Series B, Vol. 22 (2017), pp. 3529-3545.
157. L. Jia, H. Chen and H. Wang, “Mixed-type Galerkin variational principle and numerical simulation for a generalized nonlocal elastic model”, J. Sci. Comput., Vol. 71 (2017), pp. 660-681.
158. C. Wang and H. Wang, “A fast collocation method for a variable-coefficient nonlocal diffusion model”, J. Comput. Phys., Vol. 330 (2017), pp. 114-126.
159. Z. Liu, A. Cheng, and H. Wang, “An hp-Galerkin method with fast solution for linear peridynamic models in one dimension”, Comput. Math. Appl., Vol. 73 (2017), pp. 1546-1565.
160. M. Zhao, H. Wang, and A. Cheng, “A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations with fractional derivative boundary conditions”, J. Sci. Comput., Vol. 74 (2017), No. 2, pp. 1009–1033.
161. Y. Li, H. Chen and H. Wang, “A mixed-type Galerkin variational formulation and fast algorithms for variable-coefficient fractional diffusion equations”, Math. Methods Appl. Sci., Vol. 40 (2017), No. 14, pp. 5018-5034.
162. Z. Li, H. Wang, R. Xiao, and S. Yang, “A variable-order fractional differential equation model of shape memory polymers”, Chaos, Solitons & Fractals, Vol. 102 (2017), pp. 473-485.
163. Z. Li, H. Wang and D. Yang, “A space-time fractional phase-field model with tunable sharpness and decay behavior and its efficient numerical simulation”, J. Comput. Phys., Vol. 347 (2017), pp. 20-38.
164. H. Wang, “Peridynamics and Nonlocal Diffusion Models: Fast Numerical Methods”, Handbook of Nonlocal Continuum Mechanics for Materials and Structures (2017), pp. 1-23.
165. H. Wang and D. Yang, “Well-posedness of Neumann boundary-value problems of space-fractional differential equations”, ‎ Fract. Calc. Appl. Anal., 20 (2017), pp. 1356-1381.
166. Z. Liu, A. Cheng, X. Li, and H. Wang, “A fast solution technique for finite element discretization of the space–time fractional diffusion equation”, Appl. Numer. Math., Vol. 119 (2017), pp. 146-163.
167. Q. Du, J. Yang and Z. Zhou, “Analysis of a nonlocal-in-time parabolic equation”, Disc. Cont. Dyn. Syst. Series B, 22 (2017), pp. 339-368.
168. A. Chen, Q. Du, C. Li, and Z. Zhou. “Asymptotically compatible schemes for space-time nonlocal diffusion equations”, Chaos, Solitons & Fractals, Vol. 102 (2017), pp. 361-371.
169. W. Zhang, J. Yang, J. Zhang, and Q. Du. “Artificial Boundary Conditions for Nonlocal Heat Equations on Unbounded Domain”, Commun. Comput. Phys., Vol. 21 (2017), No. 1, pp. 16-39.
170. Q. Du and J. Yang. “Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications”, J. Comput. Phys., Vol. 332 (2017), pp. 118-134.
171. X. Tian and Q. Du, “Trace theorems for some nonlocal energy spaces with heterogeneous localization”, SIAM J. Math. Anal., Vol. 49 (2017), No. 2, pp. 1621-1644.
172. C. Zheng, J. Hu, Q. Du, and J. Zhang, “Numerical solution of nonlocal diffusion equation on the real line”, SIAM J. Sci. Comp., Vol. 39 (2017), No. 5, pp. A1951-A1968.
173. Q. Du, Z. Huang and P. LeFloch, “Nonlocal conservation laws. I. A new class of monotonicity-preserving models”, SIAM J. Numer. Analysis, Vol. 55 (2017), No. 5, pp. 2465-2489.
174. M. D’Elia, Q. Du, M. Gunzburger, and R. Lehoucq, “Nonlocal convection-diffusion problems on bounded domains and finite-range jump processes”, Comput. Meth. in Appl. Math, Vol. 17 (2017), pp. 707-722.
175. Y. Qiao, Z. Zhou, Z. Chen, S. Du, Q. Cheng, H. Zhai, N.J. Fritz, Q. Du, Y Yang, “Visualizing Ion Diffusion in Battery Systems by Fluorescence Microscopy: A Case Study on the Dissolution of LiMn2O4”, Nano Energy, Vol. 45 (2017), pp. 68-74.
176. H. Tian, L. Ju and Q. Du, “A conservative nonlocal convection-diffusion model and asymptotically compatible finite difference discretization”, Comput. Meth. Applied Mech. Eng., Vol. 320 (2017), pp. 46-67.
177. Q. Du, Y. Tao and X. Tian, “Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations”, Appl. Math. Comp., Vol. 305 (2017), pp. 282-298.
178. M.M. Meerschaert and F. Sabzikar, “Tempered fractional stable motion”, J. Theoret. Probab., Vol. 29 (2016), No. 2, pp. 681–706.
179. Y. Zhang, M.M. Meerschaert, and R.M. Neupauer, “Backward fractional advection dispersion model for contaminant source prediction”, Water Resour. Res., Vol. 52 (2016), No. 4, pp. 2462–2473.
180. M.M. Meerschaert, R.L. Magin, and A.Q. Ye, “Anisotropic fractional diffusion tensor imaging”, Journal of Vibration and Control, Vol. 22 (2016), No. 9, pp. 2211–2221.
181. W. Cao, Z. Zhang and G.E. Karniadakis, “Implicit-explicit difference schemes for nonlinear fractional differential equations with non-smooth solutions”, SIAM J. Sci. Comput., Vol. 38 (2016), No. 5, pp. A3070-A3093.
182. J. L. Suzuki, M. Zayernouri, M. L. Bittencourt, G. E. Karniadakis, “Fractional-order uniaxial visco-elasto-plastic models for structural analysis”, Comput. Methods Appl. Mech. Engrg., Vol. 308 (2016), pp. 443-467.
183. F. Song, C. Xu and G.E. Karniadakis, “A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations”, Comput. Methods Appl. Mech. Engrg., Vol. 305 (2016), pp. 376-404.
184. F. Zeng, Z. Zhang and G.E. Karniadakis, “Fast difference schemes for solving high-dimensional time-fractional subdif­fusion equations”, J. Comput. Phys., Vol. 307 (2016), pp. 15-33.
185. X. Zhang and H. Wang, “A fast method for a steady-state bond-based peridynamic model”, Comput. Methods Appl. Mech. Engrg., Vol. 311 (2016), pp. 280-303.
186. J. Jia and H. Wang, “A fast finite volume method for conservative space-fractional diffusion equations in convex domains”, J. Comput. Phys., Vol. 310 (2016), pp. 63-84.
187. Q. Du and J. Yang, “Asymptotically compatible Fourier spectral approximations of nonlocal Allen-Cahn equations”, SIAM J. Numer. Anal., Vol. 54 (2016), No. 3, pp. 1899-1919.
188. M. Zayernouri, M. Ainsworth and G.E. Karniadakis, “Tempered Fractional Sturm-Liouville eigen-problems”, SIAM J. Sci. Comput., Vol. 37 (2015), No. 4, pp. A1777-A1800.
189. X. Zhao, Z.Z. Sun, and G.E. Karniadakis, “Second-order approximations for variable order fractional derivatives: Algorithms and applications”, J. Comput. Phys., Vol. 293 (2015), pp. 184-200.
190. F. Zeng, Z. Zhang and G.E. Karniadakis, “A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations”, SIAM J. Sci. Comput., Vol. 37 (2015), No. 6, pp. A2710-A2732.

Related Publications by MURI investigators and collaborators

1. B. Baeumer, M. Kovács, M.M. Meerschaert, R.L. Schilling, and P. Straka, Reflected spectrally negative stable processes and their governing equations, Transactions of the American Mathematical Society, Vol. 368 (2016), No. 1, pp. 227–248.
2. H. Chen and H. Wang, Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation, J. Comput. Appl. Math., 296 (2016), 480–498.
3. M. Zayernouri, M. Ainsworth and G.E. Karniadakis, Tempered Fractional Sturm-Liouville eigen-problems. SIAM J. Sci. Comput., 37(4) A1777-A1800, 2015.
4. G.E. Karniadakis, J.S. Hesthaven, I. Podlubny, Special Issue on “Fractional PDEs: Theory, Numerics, and Applications“. J. Comput. Phys. 293, 1-3 2015.
5. F. Sabzikar, M.M. Meerschaert, and J. Chen, Tempered Fractional Calculus, Journal of Computational Physics, Vol. 293 (2015), pp. 14–28, Special Issue on Fractional Partial Differential Equations.
6. X. Tian, Q. Du, “Nonconforming discontinuous Galerkin methods for nonlocal variational problems“, SIAM J. Numer. Anal., 53, 762-781, 2015.
   • Highlight: Nonconforming discontinuous Galerkin methods for nonlocal problems with singular kernels. Convergence analysis via theory of AC schemes.
7. O. Defterli, M. D’Elia, Q. Du, M. Gunzburger, R. Lehoucq, and M.M. Meerschaert, Fractional diffusion on bounded domains, Fractional Calculus and Applied Analysis, Vol. 18 (2015), No. 2, pp. 342–360.
8. Q. Du, R. Lehoucq, and A. Tartakovsky, “Integral approximations to classical diffusion and smoothed particle hydrodynamics, Comp. Meth. Appl. Mech. Engr, 286, 216-229, 2015.
   • Highlight: Explore the relation between PD & SPH, in particular, for Neumann data.
9. Q. Du, H. Tian and L. Ju, “Nonlocal convection-diffusion problems and finite element approximation“, Comp. Meth. Appl. Mech. Engr, 289, 60-78, 2015.
   • Highlight: A maximum-principle preserving formulation of nonlocal convection-diffusion.
10. Y.X. Zhao, J.K. Wang, Y.P. Ma, and Q. Du, “Generalized local and nonlocal master equations for some stochastic processes“, Comp. Math Appl., online 2015.
    • Highlight: nonlocal master equations for various processes
11. T. Mengesha, Q. Du, “On the variational limit of some nonlocal convex functionals of vector field”, Nonlinearity, online 2015.
    • Highlight: Nonlocal convex functional of vector fields, existence of minimizers, local limit
12. Q. Du, R. Lipton, and T. Mengesha, “Multiscale analysis of an abstract evolution equation with applications to nonlocal models for heterogeneous media“, ESIAM: M2AN, online, 2015.
    • Highlight: Homogenization of time-dependent linear nonlocal models
13. H. Wang and X. Zhang, A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations, J. Com- put. Phy., 281 (2015), 67-81.
14. M. Zayernouri and G.E. Karnidakis, “Fractional spectral collocation methods for linear and nonlinar variable order FPDEs,” J. Comput. Phys. 293, 312-338, 2015.
15. J. Jia and H. Wang, A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh, J. Comput. Phys., 299 (2015), 842–862.
16. N. Du and H. Wang, A fast finite element method for space-fractional dispersion equations on bounded domains in R2, SIAM J. Sci. Comput., 37 (2015), A1614-A1635.
17. N. Du, H. Wang, and C. Wang, A fast method for a generalized nonlocal elastic model, J. Comput. Phys., 297 (2015), 72–83.
18. H. Wang, D. Yang, and S. Zhu, A Petrov-Galerkin finite element method for variable-coefficient fractional diffusion equations, Comput. Methods Appl. Mech. Engrg., 290 (2015), 45–56.
19. H. Wang, A. Cheng, and K. Wang, Fast finite volume methods for space-fractional diffusion equations, Disc. Cont. Dyn. Syst. Series B, 20 (2015), 1427–1441.
20. J. Jia and H. Wang, Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions, J. Comput. Phys., 293 (2015), 359–369.
21. A. Cheng, H. Wang, and K. Wang, A Eulerian-Lagrangian control volume method for solute transport with anomalous diffusion, Numer. Methods Partial Diff. Eqns., 31 (2015), 253–267.
22. X. Tian, Q. Du, and M. Gunzburger, “Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains“, submitted, 2015.
    • Highlight: Application of the framework of AC schemes to approximations of fractional diffusion problems on bounded domains by nonlocal diffusion models
23. X. Tian, Q. Du, “A class of high order nonlocal operators“, Submitted, 2015.
    • Highlight: Extensions of nonlocal operators and spaces involving higher order differences and applications to theory of shell/beam/plate
24. Q. Du, J. Yang, “Asymptotic compatible Fourier spectral approximations of nonlocal Allen-Cahn equations“, Submitted, 2015.
    • Highlight: Spectral methods for nonlocal Allen-Cahn equations and their local limit
25. F. Sabzikar, M.M. Meerschaert, and J. Chen, Tempered Fractional Calculus, Journal of Computational Physics, Vol. 293 (2015), pp. 14–28, Special Issue on Fractional Partial Differential Equations. Winner of the Riemann-Liouville Award for the Best Theory Paper at the International Symposium on Fractional Differentiation and Its Applications, Catania, Sicily, Italy, June 2014.
26. X. Tian, Q. Du, “Asymptotically compatible schemes and applications to robust discretization of nonlocal models“, SIAM J. Numer. Anal., 52, 1641-1665, 2014.
    • Highlight: General framework of AC schemes for approximating parametrized variational problems, with applications to nonlocal models and their local limit
27. H. Wang, D. Yang, S. Zhu, Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations, SIAM J. Numer. Anal., 52 (2014), 1292–1310.
28. Q. Du, Z. Huang, and R. Lehoucq, “Nonlocal convection-diffusion volume-constrained problems & jump processes“, Disc. Cont. Dyn. Sys. B, 19, 373-389, 2014.
    • Highlight: formulation & simulations of a nonlocal model with both convection & diffusion effects, as well as connections to general nonsymmetric jump processes.
29. Paris Perdikaris, G. E. Karniadakis, ” Fractional-order viscoelasticity in one-dimensional blood flow models “, Annals of biomedical engineering, 42 (5), 1012-1023, Springer, 2014.
30. M. Zayernouri, G. E. Karniadakis, “Exponentially Accurate Spectral and Spectral Element Methods for Fractional ODEs“, Journal of Computational Physics, 257, 460–480, 2014.
31. M. Zayernouri, W. Cao, Z. Zhang, G. E. Karniadakis, “Spectral and Discontinuous Spectral Element Methods for Fractional Delay Equations“, SIAM Journal on Scientific Computing, Vol. 36 (6), B904–B929, 2014.
32. M. Zayernouri, G. E. Karniadakis, “Discontinuous Spectral Element Methods for Time- and Space-Fractional Advection Equations“, SIAM Journal on Scientific Computing, 36 (4), B684–B707, 2014.
33. M. Zayernouri, G. E. Karniadakis, “Fractional Spectral Collocation Method“, SIAM Journal on Scientific Computing, 36 (1), A40–A62, 2014.
34. M. Zayernouri, G. E. Karniadakis, “Fractional Spectral Collocation Methods for Linear and Nonlinear Variable Order Fractional PDEs“, Journal Computational Physics, Special Issue on Fractional PDEs, 2015.
35. M. Zayernouri, M. Ainsworth, and G. E. Karniadakis, “A Unified Petrov-Galerkin Spectral Method for Fractional PDEs“, Computer Methods in Applied Mechanics and Engineering (CMAME), 2015.
36. H. Wang and N. Du, Fast solution methods for space-fractional diffusion equations, J. Comput. Appl. Math., 255 (2014), 376–383.
37. J. Jia, C. Wang, and H. Wang, A fast locally refined method for a space-fractional diffusion equation, # IE0147, ICFDA’14 Catania, 23 – 25 June 2014 Copyright 2014 IEEE ISBN 978-1- 4799-2590-2. 
38. H. Wang and H. Tian, A fast and faithful collocation method with efficient matrix assembly for a two dimensional nonlocal diffusion model, Comput. Methods Appl. Mech. Engrg., 273C (2014), 19–36.
39. M. Zheng and G.E. Karniadakis, “Numerical Methods for SPDEs with tempered stable processes“, SIAM J. Sci. Comput., 37(3), A1197–A1217, 2014.
40.  Xuan Zhao, Zhi-zhong Sun, and George Em Karniadakis, “Second order approximations for variable order fractional derivatives: Algorithms and applications“, Journal of Computational Physics, Special Issue on Fractional PDEs, 2014.
41. M. Zayernouri, G. E. Karniadakis, “Fractional Sturm-Liouville Eigen-Problems: Theory and Numerical Approximations“, Journal of Computational Physics, 47-3, 2108–2131, 2013.
42. H. Wang and N. Du, Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations, J. Comput. Phys., 258 (2013), 305–318.
43. Q. Du, L. Tian, and X. Zhao, “A convergent adaptive finite element algorithm for nonlocal diffusion models“, SIAM J. Numer. Anal., 51, 1211-1234, 2013.
    • Highlight: Convergence proof of an adaptive finite element approximation of nonlocal diffusion for weakly singular kernels.
44. H. Tian, H. Wang, and W. Wang, An efficient collocation method for a non-local diffusion model, Int’l J. Numer. Anal. Modeling, 10 (2013), 815–825.
45. H. Wang and N. Du, A fast finite difference method for three-dimensional time-dependent space- fractional diffusion equations and its efficient implementation, J. Comput. Phys., 253 (2013), 50–63.
46. H. Wang and N. Du, A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations, J. Comput. Phys., 240 (2013), 49–57.
47. H. Wang and D. Yang, Well posedness of variable-coefficient conservative fractional elliptic differential equations, SIAM J. Numer. Anal., 51 (2013), 1088-1107.
48. Q. Du, M. Gunzburger,R. Lehoucq, and K. Zhou, “A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws“, Math. Mod. Meth. Appl. Sci.,23, 493-540, 2013.
    • Highlight: Develop a basic nonlocal vector calculus framework for nonlocal vector fields and nonlocal balance laws.
49. Q. Du, J. Kamm, R. Lehoucq, and M. Parks, “A new approach for a nonlocal, nonlinear conservation law“, SIAM J. Appl. Math., 72, 467-487, 2012.
    • Highlight: analysis and simulation of a nonlinear nonlocal conservation law.
50. Q. Du, M. Gunzburger, R. Lehoucq, and K. Zhou, “Analysis and approximation of nonlocal diffusion problems with volume constraints“, SIAM Review, 54, 667-696, 2012.
    • Highlight: Discuss various issues related to nonlocal diffusion problems with nonlocal constrained values, based on the nonlocal vector calculus.
51. M.M. Meerschaert and Alla Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics Vol. 43, 2012, ISBN 978-3-11-025869-1.
52. H. Wang and T.S. Basu, A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2444–A2458.
53. T.S. Basu and H. Wang, A fast second-order finite difference method for space-fractional diffusion equations, Int’l J. Numer. Anal. Modeling, 9 (2012), 658–666.
54. L. Su, W. Wang, and H. Wang, A characteristic finite difference method for transient fractional advection-diffusion equations, Appl. Numer. Math., 61 (2011), 946–960.
55. K. Wang and H. Wang, A fast characteristic finite difference method for fractional advection-diffusion equations, Adv. Water Resour., 34 (2011), 810–816.
56. H. Wang and K. Wang, An O(N log2 N) alternating-direction finite difference method for two-dimensional fractional diffusion equations, J. Comput. Phys., 230 (2011), 7830-7839.
57. H. Wang, K. Wang, and T. Sircar, A direct O(N log2 N) finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010), 8095-8104.
58. M.M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations Journal of Computational and Applied Mathematics, Vol. 172 (2004), No. 1, pp. 65-77.