A Conservative Scheme with Optimal Error Estimates for a Multidimensional Space–Fractional Gross–Pitaevskii Equation
By: Ahmed S. Hendy and Jorge E. Macías-Díaz
References
- Alikhanov, A.A. (2015). A new difference scheme for the time fractional diffusion equation, Journal of Computational Physics280: 424–438.
- Alves, C.O. and Miyagaki, O.H. (2016). Existence and concentration of solution for a class of fractional elliptic equation in ℝn via penalization method, Calculus of Variations and Partial Differential Equations55(3): 47.
- Antoine, X., Tang, Q. and Zhang, Y. (2016). On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross–Pitaevskii equations with rotation term and nonlocal nonlinear interactions, Journal of Computational Physics325: 74–97.
- Bao, W. and Cai, Y. (2013). Optimal error estimates of finite difference methods for the Gross–Pitaevskii equation with angular momentum rotation, Mathematics of Computation82(281): 99–128.
- Ben-Yu, G., Pascual, P.J., Rodriguez, M.J. and Vázquez, L. (1986). Numerical solution of the sine-Gordon equation, Applied Mathematics and Computation18(1): 1–14.
- Bhrawy, A.H. and Abdelkawy, M.A. (2015). A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations, Journal of Computational Physics294: 462–483.
- El-Ajou, A., Arqub, O.A. and Momani, S. (2015). Approximate analytical solution of the nonlinear fractional KdV–Burgers equation: A new iterative algorithm, Journal of Computational Physics293: 81–95.
- Fei, Z. and Vázquez, L. (1991). Two energy conserving numerical schemes for the sine-Gordon equation, Applied Mathematics and Computation45(1): 17–30.
- Furihata, D. (2001). Finite-difference schemes for nonlinear wave equation that inherit energy conservation property, Journal of Computational and Applied Mathematics134(1): 37–57.
- Glöckle, W.G. and Nonnenmacher, T.F. (1995). A fractional calculus approach to self-similar protein dynamics, Biophysical Journal68(1): 46–53.
- Gross, E.P. (1961). Structure of a quantized vortex in boson systems, Il Nuovo Cimento (1955-1965)20(3): 454–477.
- Iannizzotto, A., Liu, S., Perera, K. and Squassina, M. (2016). Existence results for fractional p-Laplacian problems via Morse theory, Advances in Calculus of Variations9(2): 101–125.
- Kaczorek, T. (2015). Positivity and linearization of a class of nonlinear continuous-time systems by state feedbacks, International Journal of Applied Mathematics and Computer Science25(4): 827–831, DOI: 10.1515/amcs-2015-0059.10.1515/amcs-2015-0059
- Koeller, R. (1984). Applications of fractional calculus to the theory of viscoelasticity, ASME Transactions: Journal of Applied Mechanics51: 299–307.
- Liu, F., Zhuang, P., Turner, I., Anh, V. and Burrage, K. (2015). A semi-alternating direction method for a 2-D fractional FitzHugh–Nagumo monodomain model on an approximate irregular domain, Journal of Computational Physics293: 252–263.
- Macías-Díaz, J.E. (2017). A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives, Journal of Computational Physics351: 40–58.
- Macías-Díaz, J.E. (2018). An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions, Communications in Nonlinear Science and Numerical Simulation59: 67–87.
- Macías-Díaz, J.E. (2019). On the solution of a Riesz space-fractional nonlinear wave equation through an efficient and energy-invariant scheme, International Journal of Computer Mathematics96(2): 337–361.
- Matsuo, T. and Furihata, D. (2001). Dissipative or conservative finite-difference schemes for complex-valued nonlinear partial differential equations, Journal of Computational Physics171(2): 425–447.
- Namias, V. (1980). The fractional order Fourier transform and its application to quantum mechanics, IMA Journal of Applied Mathematics25(3): 241–265.
- Oprz˛edkiewicz, K., Gawin, E. and Mitkowski, W. (2016). Modeling heat distribution with the use of a non-integer order, state space model, International Journal of Applied Mathematics and Computer Science26(4): 749–756, DOI: 10.1515/amcs-2016-0052.10.1515/amcs-2016-0052
- Pimenov, V.G. and Hendy, A.S. (2017). A numerical solution for a class of time fractional diffusion equations with delay, International Journal of Applied Mathematics and Computer Science27(3): 477–488, DOI: 10.1515/amcs-2017-0033.10.1515/amcs-2017-0033
- Pimenov, V.G., Hendy, A.S. and De Staelen, R.H. (2017). On a class of non-linear delay distributed order fractional diffusion equations, Journal of Computational and Applied Mathematics318: 433–443.
- Pitaevskii, L. (1961). Vortex lines in an imperfect Bose gas, Soviet Physics JETP13(2): 451–454.
- Povstenko, Y. (2009). Theory of thermoelasticity based on the space-time-fractional heat conduction equation, Physica Scripta2009(T136): 014017.
- Rakkiyappan, R., Cao, J. and Velmurugan, G. (2015). Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays, IEEE Transactions on Neural Networks and Learning Systems26(1): 84–97.
- Raman, C., Köhl, M., Onofrio, R., Durfee, D., Kuklewicz, C., Hadzibabic, Z. and Ketterle, W. (1999). Evidence for a critical velocity in a Bose–Einstein condensed gas, Physical Review Letters83(13): 2502.
- Scalas, E., Gorenflo, R. and Mainardi, F. (2000). Fractional calculus and continuous-time finance, Physica A: Statistical Mechanics and Its Applications284(1): 376–384.
- Strauss, W. and Vazquez, L. (1978). Numerical solution of a nonlinear Klein–Gordon equation, Journal of Computational Physics28(2): 271–278.
- Su, N., Nelson, P.N. and Connor, S. (2015). The distributed-order fractional diffusion-wave equation of groundwater flow: Theory and application to pumping and slug tests, Journal of Hydrology529(3): 1262–1273.
- Tang, Y.-F., Vázquez, L., Zhang, F. and Pérez-García, V. (1996). Symplectic methods for the nonlinear Schrödinger equation, Computers & Mathematics with Applications32(5): 73–83.
- Tarasov, V.E. (2006). Continuous limit of discrete systems with long-range interaction, Journal of Physics A: Mathematical and General39(48): 14895.
- Tarasov, V.E. and Zaslavsky, G.M. (2008). Conservation laws and HamiltonâĂŹs equations for systems with long-range interaction and memory, Communications in Nonlinear Science and Numerical Simulation13(9): 1860–1878.
- Tian, W., Zhou, H. and Deng, W. (2015). A class of second order difference approximations for solving space fractional diffusion equations, Mathematics of Computation84(294): 1703–1727.
- Wang, P., Huang, C. and Zhao, L. (2016). Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation, Journal of Computational and Applied Mathematics306: 231–247.
- Wang, T., Jiang, J. and Xue, X. (2018). Unconditional and optimal H1 error estimate of a Crank–Nicolson finite difference scheme for the Gross–Pitaevskii equation with an angular momentum rotation term, Journal of Mathematical Analysis and Applications459(2): 945–958.
- Wang, T. and Zhao, X. (2014). Optimal l∞ error estimates of finite difference methods for the coupled Gross–Pitaevskii equations in high dimensions, Science China Mathematics57(10): 2189–2214.
- Ye, H., Liu, F. and Anh, V. (2015). Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains, Journal of Computational Physics298: 652–660.
Language: English
Page range: 713 - 723
Submitted on: Jan 15, 2019
Accepted on: May 28, 2019
Published on: Dec 31, 2019
Published by: University of Zielona Góra
In partnership with: Paradigm Publishing Services
Publication frequency: 4 issues per year
Keywords:
Related subjects:
© 2019 Ahmed S. Hendy, Jorge E. Macías-Díaz, published by University of Zielona Góra
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.