Publications

MURI Publications

  1. B. Baeumer, M. Kovacs, H. Sankaranarayanan, "Fractional partial differential equations with boundary conditions", https://arxiv.org/pdf/1706.07266.pdf, 2017.
  2. 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.
  3. 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.
  4. 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. 
  5. 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.)
  6. 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.)
  7. 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.
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. 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.
  14. 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.
  15. 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.
  16. 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.
  17. 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.
  18. 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.
  19. 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.
  20. 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.
  21. 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.
  22. 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.
  23. 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.
  24. 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.
  25. 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.
  26. 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.
  27. 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.
  28. 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.
  29. 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.
  30. 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.
  31. 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.
  32. 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.
  33. 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.
  34. 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.
  35. 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.
  36. H. Wang, “Peridynamics and Nonlocal Diffusion Models: Fast Numerical Methods”, to appear in Handbook of Nonlocal Continuum Mechanics for Materials and Structures, Springer.
  37. 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.
  38. 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.
  39. 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.
  40. S. Duo, H. Wang, and Y. Zhang, “A preliminary investigation on different models approximating the fractional Laplacian”, Disc. Cont. Dyn. Syst. Series B, submitted.
  41. H. Wang and D. Yang, “Wellposedness of Neumann boundary-value problems of space-fractional differential equations”, ‎Fract. Calc. Appl. Anal., submitted.
  42. 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.
  43. A. Chen, Q. Du, C. Li, and Z. Zhou. "Asymptotically compatible schemes for space-time nonlocal diffusion equations”, Chaos, Solitons & Fractals (2017), online.
  44. 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.
  45. 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.
  46. 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).
  47. 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.
  48. 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”, to appear in Computer Methods in Applied Mechanics and Engineering.
  49. 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.
  50. 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.
  51. 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.
  52. Z. Zhang, “Error estimates of spectral Galerkin methods for linear fractional reaction-diffusion equation”, submitted.
  53. Z. Hao, G. Lin, Z. Zhang, “Regularity and spectral methods for two-sided fractional diffusion equations with a low-order term”, submitted.
  54. 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.
  55. 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.
  56. 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.
  57. 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.
  58. 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.
  59. 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.
  60. 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.
  61. 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.
  62. B. Toaldo and M. M. Meerschaert, “Relaxation patterns and semi-Markov dynamics”, submitted.
  63. B. Baeumer, T. Luks, M. M. Meerschaert, “Space-time fractional Dirichlet problems”, submitted.
  64. G. Didier, M. M. Meerschaert, V. Pipiras, “Domain and range symmetries of operator fractional Brownian fields”, to appear in Stochastic Processes and their Applications.
  65. M. Samiee, M. Zayernouri, M. M. Meerschaert, “A Unified Spectral Method for FPDEs with Two-sided Derivatives; Part I: A Fast Solver”, submitted.
  66. M. Samiee, M. Zayernouri, M. M. Meerschaert, “A Unified Spectral Method for FPDEs with Two-sided Derivatives; Part II: Stability and Error”, submitted.
  67. F. Sabzikar, A. I. McLeod, and M. M. Meerschaert, “Parameter Estimation for Tempered Fractional Time Series”, submitted.
  68. 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.
  69. J.F. Kelly, C.G, Lee, and M.M. Meerschaert, “Anomalous Diffusion with Ballistic Scaling: A New Fractional Derivative”, submitted.
  70. B. Baeumer, M. Kovacs, M.M. Meerschaert and H. Sankaranarayanan, “Boundary Conditions for Fractional Diffusion”, submitted.

 

 

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, Wellposedness 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 diffu- sion 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.