A Hermite spectral method for fractional convection diffusion equations on unbounded domains

References

• M. Ainsworth and 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. doi: 10.1137/17M1144696
   Web of Science ®Google Scholar
• A. Akbulut and M. Kaplan, Auxiliary equation method for time-fractional differential equations with conformable derivative, Comput. Math. Appl. 75 (2018), pp. 876–882. doi: 10.1016/j.camwa.2017.10.016
   Web of Science ®Google Scholar
• Y. Chen, X. Wang and W. Deng, Tempered fractional Langevin–Brownian motion with inverse β-stable subordinator, J. Phys. A Math. Theor. 51 (2018), pp. 495001.
   Web of Science ®Google Scholar
• E. Colorado and A. Ortega, The Brezis–Nirenberg problem for the fractional Laplacian with mixed Dirichlet–Neumann boundary conditions, arXiv preprint (2018), Available at arXiv:1805.10093.
   Google Scholar
• N. Cusimano, F. del Teso, L. Gerardo-Giorda and G. Pagnini, Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions, SIAM. J. Numer. Anal. 56 (2018), pp. 1243–1272. doi: 10.1137/17M1128010
   Web of Science ®Google Scholar
• W. Deng, B. Li, W. Tian and P. Zhang, Boundary problems for the fractional and tempered fractional operators, Multiscale Model. Simul. 16 (2018), pp. 125–149. doi: 10.1137/17M1116222
   Web of Science ®Google Scholar
• W. Deng and Z. Zhang, Variational formulation and efficient implementation for solving the tempered fractional problems, Numer. Methods Partial Differ. Equ. 34 (2018), pp. 1224–1257. doi: 10.1002/num.22254
   Web of Science ®Google Scholar
• S. Duo, H.W. van Wyk and Y. Zhang, A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem, J. Comput. Phys. 355 (2018), pp. 233–252. doi: 10.1016/j.jcp.2017.11.011
   Web of Science ®Google Scholar
• S. Duo and Y. Zhang, Finite difference methods for two and three dimensional fractional Laplacian with applications to solve the fractional reaction–diffusion equations, arXiv preprint (2018), arXiv:1804.02718,2018.
   Google Scholar
• C. Hu and S. Tao, A Petrov cgalerkin spectral method for the linearized time fractional kdv equation, Int. J. Comput. Math. 95 (2017), pp. 1–18.
   Web of Science ®Google Scholar
• W. Jiang and H. Li, A space–time spectral collocation method for the two-dimensional variable-order fractional percolation equations, Comput. Math. Appl. 75 (2018), pp. 3508–3520. doi: 10.1016/j.camwa.2018.02.013
   Web of Science ®Google Scholar
• E. Kharazmi and M. Zayernouri, Fractional pseudo-spectral methods for distributed-order fractional PDEs, Int. J. Comput. Math. 95 (2018), pp. 1340–1361. doi: 10.1080/00207160.2017.1421949
   Web of Science ®Google Scholar
• T. Le Minh, T.T. Khieu, T.Q. Khanh and H.-H. Vo, On a space fractional backward diffusion problem and its approximation of local solution, J. Comput. Appl. Math. 346 (2019), pp. 440–455. doi: 10.1016/j.cam.2018.07.016
   Web of Science ®Google Scholar
• H. Li and W. Jiang, A space-time spectral collocation method for the 2-dimensional nonlinear Riesz space fractional diffusion equations, Math. Methods Appl. Sci. 41 (2018), pp. 6130–6144. doi: 10.1002/mma.5124
   Web of Science ®Google Scholar
• A. Lischke, G. Pang, M. Gulian and F. Song, What is the fractional Laplacian?, arXiv preprint (2018), Available at arXiv:1801.09767,2018.
   Google Scholar
• X. Liu and W. Deng, Numerical methods for the two-dimensional Fokker–Planck equation governing the probability density function of the tempered fractional Brownian motion, arXiv preprint (2018), Available at arXiv:1805.03950.
   Google Scholar
• X. Luo, Spectral viscosity method with generalized Hermite functions for nonlinear conservation laws, Appl. Numer. Math. 123 (2018), pp. 256–274. doi: 10.1016/j.apnum.2017.09.014
   Web of Science ®Google Scholar
• H. Luo, Ground state solutions of Pohozaev type for fractional Choquard equations with general nonlinearities, Comput. Math. Appl. 77 (2019), pp. 877–887. doi: 10.1016/j.camwa.2018.10.024
   Web of Science ®Google Scholar
• Z. Mao and J. Shen, Hermite spectral methods for fractional PDEs in unbounded domains, SIAM. J. Sci. Comput. 39 (2017), pp. A1928–A1950. doi: 10.1137/16M1097109
   Web of Science ®Google Scholar
• Y. Shan, W. Liu and B. Wu, Space–time Legendre–Gauss–Lobatto collocation method for two-dimensional generalized Sine-Gordon equation, Appl. Numer. Math. 122 (2017), pp. 92–107. doi: 10.1016/j.apnum.2017.08.003
   Web of Science ®Google Scholar
• J. Shen, T. Tang, L. Wang, Spectral Methods: Algorithms Analysis and Applications, Vol. 41, Springer Science & Business Media, New York, 2011.
   Google Scholar
• T. Tang, H. Yuan and T. Zhou, Hermite spectral collocation methods for fractional PDEs in unbounded domains, arXiv preprint (2018), Available at arXiv:1801.09073,2018.
   Google Scholar
• X. Xiang, A class hermite pseudospectral approximate with ω(x)≡ 1 and application to reaction-diffusion equation, Acta Math. Appl. Sin. 26 (2010), pp. 405–414. doi: 10.1007/s10255-010-0004-3
   Web of Science ®Google Scholar
• H. Yu, B. Wu and D. Zhang, A generalized Laguerre spectral Petrov–Galerkin method for the time-fractional subdiffusion equation on the semi-infinite domain, Appl. Math. Comput. 331 (2018), pp. 96–111. doi: 10.1016/j.cam.2017.09.008
   Web of Science ®Google Scholar
• Z. Zhao, Bäcklund transformations rational, solutions and soliton–cnoidal wave solutions of the modified Kadomtsev–Petviashvili equation, Appl. Math. Lett. 89 (2019), pp. 103–110. doi: 10.1016/j.aml.2018.09.016
   Web of Science ®Google Scholar