## MURI Publications

- I. C. Tsantili, M. Hyung, W. Cai, G. E. Karniadakis, "Acomputational stochastic methodology for the design of random meta-materials under geometric constraints." SIAM J. Sci. Comput. 40(2), B353-B378, 2018.
- D. Zhang, H. Babaee, G. E. Karniadakis, "Stochastic domain decomposition via moment minimization." SIAM J. Sci. Comput. 40(4), A2152-A2173, 2018
- A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. M. Meerschaert, M. Ainsworth, G. E. Karniadakis, "What is the fractional Laplacian?", https://arxiv.org/pdf/1801.09767.pdf, 2018 (preprint).
- B. Baeumer, M. Kovacs, H. Sankaranarayanan, "Fractional partial differential equations with boundary conditions", https://arxiv.org/pdf/1706.07266.pdf, 2017.
- Z. Zhang, W. Deng, G.E. Karniadakis, "A Riesz basis Galerkin method for the tempered fractional Laplacian", https://arxiv.org/pdf/1709.10415.pdf, 2017.
- M. Ainsworth, C. Glusa, "Hybrid Finite Element - Spectral Method for the Fractional Laplacian: Approximation Theory and Efficient Solver", https://arxiv.org/pdf/1709.01639.pdf, 2017.
- G. Malara, P.D. Spanos, "Nonlinear random vibrations of plates endowed with fractional derivative elements", Probabilistic Engineering Mechanics, Available online 6 July 2017, ISSN 0266-8920, https://doi.org/10.1016/j.probengmech.2017.06.002.
- 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. (To appear.)
- P.D. Spanos, G. Malara, "Random Vibrations of Nonlinear Continua Endowed with Fractional Derivative Elements. Proc. of the X International Conference on Structural Dynamics", EURODYN 2017, Rome, Italy, 10-13 September, 2017. (To appear.)
- 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.
- F. Song, G.E. Karniadakis, “Fractional spectral vanishing viscosity method: Application to the quasi-geostrophic equation”, Chaos, Solitons and Fractals, Vol. 201 (2017), pp. 327-332.
- 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.
- 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”, Computers and Mathematics with Applications, Vol. 73 (2017), pp. 1286-1297.
- 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.
- 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.
- X. Chen, F. Zeng, G.E. Karniadakis, “A tunable finite difference method for fractional differential equations with non-smooth solutions”, Computer Methods in Applied Mechanics and Engineering, Vol. 318 (2017), pp. 193-214.
- 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.
- J. L. Suzuki, M. Zayernouri, M. L. Bittencourt, G. E. Karniadakis, “Fractional-order uniaxial visco-elasto-plastic models for structural analysis”, Computer Methods in Applied Mechanics and Engineering, Vol. 308 (2016), pp. 443-467.
- F. Song, C. Xu and G.E. Karniadakis, “A fractional phase-ﬁeld model for two-phase ﬂows with tunable sharpness: Algorithms and simulations”, Computer Methods in Applied Mechanics and Engineering, Vol. 305 (2016), pp. 376-404.
- 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.
- F. Zeng, Z. Zhang and G.E. Karniadakis, “Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions”, Submitted to Computer Methods in Applied Mechanics and Engineering.
- 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, DOI: 10.1137/140985536.
- 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.
- 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, DOI: 10.1137/141001299.
- 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”, to appear in SIAM J. Numer. Anal.
- 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.
- H. Fu, M. Ng and H. Wang, “A divide-and-conquer fast finite difference method for space time fractional partial differential equation”, Computers and Mathematics with Applications, Vol. 73 (2017), pp. 1233-1242.
- 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.
- 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.
- 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.
- 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.
- 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.
- Z. Liu, A. Cheng, and H. Wang, “An hp-Galerkin method with fast solution for linear peridynamic models in one dimension”, Computers and Mathematics with Applications, Vol. 73 (2017), pp. 1546-1565.
- 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., (2017), DOI:10.1007/s10915-017-0478-8.
- 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., (2017), DOI: 10.1002/mma.4367.
- 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, DOI: 10.1016/j.chaos.2017.04.042.
- H. Fu, H. Wang, and Z. Wang, “POD reduced-order modeling of time-fractional partial differential equations with applications in parameter identification”, J. Sci. Comput., (2017), DOI:10.1007/s10915-017-0433-8.
- 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., (2017), DOI: 10.1016/j.jcp.2017.06.036.
- H. Wang, “Peridynamics and Nonlocal Diffusion Models: Fast Numerical Methods”, to appear
- J. Jia and H. Wang, “A fast finite difference method for space distributed-order fractional diffusion equations on convex domains”, Computers and Mathematics with Applications, submitted.
- M. Zhao, S. He, H. Wang, and G. Qin, “An integrated fractional partial differential equation with molecular dynamics simulation modeling of anomalous diffusive transport in nano porous materials”, J. Comput. Phys., in revision.
- W. Deng, B. Li, Z. Qian and H. Wang, “Time discretization of a tempered fractional Feynman-Kac equation with measured data”, SIAM J. Numer. Anal., submitted.
- S. Duo, H. Wang, and Y. Zhang, “A preliminary investigation on different models approximating the fractional Laplacian”, Disc. Cont. Dyn. Syst. Series B, submitted.
- H. Wang and D. Yang, “Wellposedness of Neumann boundary-value problems of space-fractional differential equations”, Fract. Calc. Appl. Anal., submitted.
- Q. Du, J. Yang and Z. Zhou, “Analysis of a nonlocal-in-time parabolic equation”, Disc. Cont. Dyn. Syst. Series B, 22 (2017), 339-368.
- A. Chen, Q. Du, C. Li, and Z. Zhou. "Asymptotically compatible schemes for space-time nonlocal diffusion equations”, Chaos, Solitons & Fractals (2017), online.
- W. Zhang, J. Yang, J. Zhang, and Q. Du. "Artificial boundary conditions for nonlocal heat equations on unbounded domain”, Communications in Computational Physics 21, no. 1 (2017): 16-39.
- 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.
- Q. Du, J. Yang and W. Zhang, “Uniform Lp-bound of the Allen-Cahn equation and its numerical discretization”, International J. Numer. Anal. Modeling, to appear, (2017).
- 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.
- 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”, Computer Methods in Applied Mechanics and Engineering, Vol. 327 (2017), pp. 4--35, doi:10.1016/j.cma.2017.08.019.
- M. Ainsworth and C. Glusa, “Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains”, Festschrift for 80th birthday of Ian Sloan, 2017.
- 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.
- M. Ainsworth and Z. Mao, “Well-posedness of the Cahn–Hilliard equation with fractional free energy and its Fourier Galerkin approximation”, Chaos, Solitons and Fractals, Vol. 102 (2017), pp. 264-273.
- Z. Zhang, “Error estimates of spectral Galerkin methods for linear fractional reaction-diffusion equation”, submitted.
- Z. Hao, G. Lin, Z. Zhang, “Regularity and spectral methods for two-sided fractional diffusion equations with a low-order term”, submitted.
- M.S. Alrawashdeh, J.F. Kelly, M.M. Meerschaert, and H.-P. Scheffler, “Applications of Inverse Tempered Stable Subordinators”, Computers and Mathematics with Applications, Vol. 73 (2017), No. 6, pp. 892–905. DOI: 10.1016/j.camwa.2016.07.026. Special Issue on Time-fractional PDEs.
- G. Didier, M.M. Meerschaert, and V. Pipiras, “Exponents of operator self-similar random fields”, Journal of Mathematical Analysis and Applications, Vol. 448 (2017), No. 2, pp. 1450–1466.
- M.M. Meerschaert and F. Sabzikar, “Tempered fractional stable motion”, Journal of Theoretical Probability, Vol. 29 (2016), No. 2, pp. 681–706, DOI 10.1007/s10959-014-0585-5.
- Y. Zhang, M.M. Meerschaert, and R.M. Neupauer, “Backward fractional advection dispersion model for contaminant source prediction”, Water Resources Research, Vol. 52 (2016), No. 4, pp. 2462–2473, DOI:10.1002/2015WR018515.
- 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. Special Issue on Challenges in Fractional Dynamics and Control Theory.
- P. Kern, Y. Xiao, and M. M. Meerschaert, “Asymptotic behavior of semistable Lévy exponents and applications to fractal path properties”, Journal of Theoretical Probability, to appear.
- 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 Resources Research, Vol. 53 (2017), No. 3, pp. 1763–2576.
- J. F. Kelly and M. M. Meerschaert, “Space-time duality for the fractional advection dispersion equation”, Water Resources Research, Vol. 53 (2017), No. 4, pp. 3464–3475.
- B. Toaldo and M. M. Meerschaert, “Relaxation patterns and semi-Markov dynamics”, submitted.
- B. Baeumer, T. Luks, M. M. Meerschaert, “Space-time fractional Dirichlet problems”, submitted.
- G. Didier, M. M. Meerschaert, V. Pipiras, “Domain and range symmetries of operator fractional Brownian fields”, to appear in Stochastic Processes and their Applications.
- M. Samiee, M. Zayernouri, M. M. Meerschaert, “A Unified Spectral Method for FPDEs with Two-sided Derivatives; Part I: A Fast Solver”, submitted.
- M. Samiee, M. Zayernouri, M. M. Meerschaert, “A Unified Spectral Method for FPDEs with Two-sided Derivatives; Part II: Stability and Error”, submitted.
- F. Sabzikar, A. I. McLeod, and M. M. Meerschaert, “Parameter Estimation for Tempered Fractional Time Series”, submitted.
- 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.
- J.F. Kelly, C.G, Lee, and M.M. Meerschaert, “Anomalous Diffusion with Ballistic Scaling: A New Fractional Derivative”, submitted.
- B. Baeumer, M. Kovacs, M.M. Meerschaert and H. Sankaranarayanan, “Boundary Conditions for Fractional Diffusion”, submitted.

## Related Publications by MURI investigators and collaborators

- 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.
- 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.
- M. Zayernouri, M. Ainsworth and G.E. Karniadakis, Tempered Fractional Sturm-Liouville eigen-problems. SIAM J. Sci. Comput., 37(4) A1777-A1800, 2015.
- G.E. Karniadakis, J.S. Hesthaven, I. Podlubny, Special Issue on "Fractional PDEs: Theory, Numerics, and Applications". J. Comput. Phys. 293, 1-3 2015.
- 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.
- 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.

- 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.
- 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.

- 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.

- 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

- 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

- 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

- 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.
- M. Zayernouri and G.E. Karnidakis, "Fractional spectral collocation methods for linear and nonlinar variable order FPDEs," J. Comput. Phys. 293, 312-338, 2015.
- 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.
- 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.
- N. Du, H. Wang, and C. Wang, A fast method for a generalized nonlocal elastic model, J. Comput. Phys., 297 (2015), 72–83.
- 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.
- 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.
- 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.
- 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.
- 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

- 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

- 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

- 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.
- 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

- 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.
- 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.

- Paris Perdikaris, G. E. Karniadakis, " Fractional-order viscoelasticity in one-dimensional blood flow models ", Annals of biomedical engineering, 42 (5), 1012-1023, Springer, 2014.
- M. Zayernouri, G. E. Karniadakis, "Exponentially Accurate Spectral and Spectral Element Methods for Fractional ODEs", Journal of Computational Physics, 257, 460–480, 2014.
- 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.
- 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.
- M. Zayernouri, G. E. Karniadakis, "Fractional Spectral Collocation Method", SIAM Journal on Scientific Computing, 36 (1), A40–A62, 2014.
- 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.
- 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.
- H. Wang and N. Du, Fast solution methods for space-fractional diffusion equations, J. Comput. Appl. Math., 255 (2014), 376–383.
- 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.* - 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.
- M. Zheng and G.E. Karniadakis, "Numerical Methods for SPDEs with tempered stable processes",
*SIAM J. Sci. Comput.*, 37(3), A1197–A1217, 2014. - 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.
- M. Zayernouri, G. E. Karniadakis, "Fractional Sturm-Liouville Eigen-Problems: Theory and Numerical Approximations", Journal of Computational Physics, 47-3, 2108–2131, 2013.
- 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.
- 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.

- 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.
- 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.
- H. Wang and N. Du, A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations, J. Comput. Phys., 240 (2013), 49–57.
- H. Wang and D. Yang, Wellposedness of variable-coefficient conservative fractional elliptic differential equations, SIAM J. Numer. Anal., 51 (2013), 1088-1107.
- 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.

- 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.

- 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.

- 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.
- H. Wang and T.S. Basu, A fast finite difference method for two-dimensional space-fractional diffu- sion equations, SIAM J. Sci. Comput., 34 (2012), A2444–A2458.
- 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.
- 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.
- K. Wang and H. Wang, A fast characteristic finite difference method for fractional advection-diffusion equations, Adv. Water Resour., 34 (2011), 810–816.
- 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.
- 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.
- 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.