In this paper, we present a compact finite difference method for a class of fourth-order nonlinear neutral delay sub-diffusion equations in two-dimensional space. The fourth-order problem is first transformed into a second-order system by a reduced-order method. Next by using compact operator to approximate the second order space derivatives and L2-1 σ formula to approximate the time fractional derivative, the difference scheme which is fourth order in space and second order in time is obtained. Then, the existence and uniqueness of solution, the convergence rate of and the stability of the scheme are proved. Finally, numerical results are given to verify the accuracy and validity of the scheme.
Sokolov, I.M. and Klafter, J. (2005) From Diffusion to Anomalous Diffusion: A Century after Einstein’s Brownian Motion. Chaos: An Interdisciplinary Journal of Nonlinear Science, 15, Article ID: 026103. https://doi.org/10.1063/1.1860472
El-Nabulsi, R.A. (2008) Fractional Field Theories from Multi-Dimensional Fractional Variational Problems. International Journal of Geometric Methods in Modern Physics, 5, 863-892. https://doi.org/10.1142/S0219887808003119
Baleanu, D., Defterli, O. and Agrawal, O.P. (2009) A Central Difference Numerical Scheme for Fractional Optimal Control Problems. Journal of Vibration and Control, 15, 583-597. https://doi.org/10.1177/1077546308088565
Saxena, R.K., Mathai A.M. and Haubold, H.J. (2004) On Generalized Fractional Kinetic Equations. Physica A: Statistical Mechanics and Its Applications, 344, 657-664. https://doi.org/10.1016/j.physa.2004.06.048
Schumer, R., Benson, D.A., Meerschaert, M.M. and Wheatcraft, S.W. (2001) Eulerian Derivation of the Fractional Advection-Dispersion Equation. Journal of Contaminant Hydrology, 48, 69-88. https://doi.org/10.1016/S0169-7722(00)00170-4
Lu, S., Molz, F.J. and Fix, G.J. (2002) Possible Problems of Scale Dependency in Applications of the Three-Dimensional Fractional Advection-Dispersion Equation to Natural Porous Media. Water Resources Research, 38, 4-1-4-7. https://doi.org/10.1029/2001WR000624
Benson, D., Schumer, R., Meerschaert, M. and Wheatcraft, S. (2001) Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests. In: Berkowitz, B., Ed., Dispersion in Heterogeneous Geological Formations, Springer, Dordrecht, 211-240. https://doi.org/10.1007/978-94-017-1278-1_11
Schneider, W.R. and Wyss, W. (1989) Fractional Diffusion and Wave Equations. Journal of Mathematical Physics, 30, 134-144. https://doi.org/10.1063/1.528578
Meerschaert, M.M., Benson, D.A., Scheffler, H.P. and Baeumer, B. (2002) Stochastic Solution of Space-Time Fractional Diffusion Equations. Physical Review E, 65, 1103-1106. https://doi.org/10.1103/PhysRevE.65.041103
Chen, C.M., Liu, F., Turner, I. and Anh, V.A. (2007) Fourier Method for the Fractional Diffusion Equation Describing Sub-Diffusion. Journal of Computational Physics, 227, 886-897. https://doi.org/10.1016/j.jcp.2007.05.012
Tomovski, Z. and Sandev, T. (2013) Exact Solutions for Fractional Diffusion Equation in a Bounded Domain with Different Boundary Conditions. Nonlinear Dynamics, 71, 671-683. https://doi.org/10.1007/s11071-012-0710-x
Langlands, T.A.M. and Henry, B.I. (2005) The Accuracy and Stability of an Implicit Solution Method for the Fractional Diffusion Equation. Journal of Computational Physics, 205, 719-736. https://doi.org/10.1016/j.jcp.2004.11.025
Yuste, S.B. and Acedo, L. (2005) An Explicit Finite Difference Method and a New Von-Neumann Type Stability Analysis for Fractional Diffusion Equations. SIAM Journal on Numerical Analysis, 42, 1862-1874. https://doi.org/10.1137/030602666
Tadjeran, C., Meerschaert, M.M. and Scheffler, H.P. (2006) A Second-Order Accurate Numerical Approximation for the Fractional Diffusion Equation. Journal of Computational Physics, 213, 205-213. https://doi.org/10.1016/j.jcp.2005.08.008
Zhuang, P., Liu, F., Anh, V. and Turner, I. (2008) New Solution and Analytical Techniques of the Implicit Numerical Method for the Anomalous Sub-Diffusion Equation. SIAM Journal on Numerical Analysis, 46, 1079-1095. https://doi.org/10.1137/060673114
Liu, F., Yang, C. and Burrage, K. (2009) Numerical Method and Analytical Technique of the Modified Anomalous Sub-Diffusion Equation with a Nonlinear Source Term. Journal of Computational and Applied Mathematics, 231, 160-176. https://doi.org/10.1016/j.cam.2009.02.013
Chen, J., Liu, F., Anh, V., Shen, S., Liu, Q., et al. (2012) The Analytical Solution and Numerical Solution of the Fractional Diffusion-Wave Equation with Damping. Applied Mathematics and Computation, 219, 1737-1748. https://doi.org/10.1016/j.amc.2012.08.014
Mohammed, O.H., Fadhel, S.F. and AL-Safi. M.G.S. (2015) Numerical Solution for the Time Fractional Diffusion-Wave Equations by Using Sinc-Legendre Collocation Method. Mathematical Theory and Modeling, 5, 49-57.
Zhou, F.Y. and Xu, X.Y. (2017) Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method. Advances in Mathematical Physics, 2017, Article ID: 2610804. https://doi.org/10.1155/2017/2610804
Cui, M. (2009) Compact Finite Difference Method for the Fractional Diffusion Equation. Journal of Computational Physics, 228, 7792-7804. https://doi.org/10.1016/j.jcp.2009.07.021
Chen, C.M., Liu, F., Anh, V. and Turner, I. (2010) Numerical Schemes with High Spatial Accuracy for a Variable Order Anomalous Sub-Diffusion Equations. SIAM Journal on Scientific Computing, 32, 1740-1760. https://doi.org/10.1137/090771715
Gao, G.H. and Sun, Z.Z. (2011) A Compact Difference Scheme for the Fractional Subdiffusion Equations. Journal of Computational Physics, 230, 586-595. https://doi.org/10.1016/j.jcp.2010.10.007
Vong, S., Lyu, P. and Wang, Z. (2015) A Compact Difference Scheme for Fractional Sub-Diffusion Equations with the Spatially Variable Coefficient under Neumann Boundary Conditions. Journal of Scientific Computing, 66, 725-739. https://doi.org/10.1007/s10915-015-0040-5
Zhao, X., Sun, Z.Z. and Karniadakis, G.E. (2015) Second-Order Approximations for Variable Order Fractional Derivatives: Algorithms and Applications. Journal of Computational Physics, 293, 184-200. https://doi.org/10.1016/j.jcp.2014.08.015
Gao, G.H., Sun, H.W. and Sun, Z.Z. (2015) Stability and Convergence of Finite Difference Schemes for a Class of Time-Fractional Sub-Diffusion Equations Based on Certain Superconvergence. Journal of Computational Physics, 280, 510-528. https://doi.org/10.1016/j.jcp.2014.09.033
Alikhanov, A.A. (2015) A New Difference Scheme for the Time Fractional Diffusion Equation. Journal of Computational Physics, 280, 424-438. https://doi.org/10.1016/j.jcp.2014.09.031
Sneddon, I.N. (1951) Fourier Transforms. McGraw Hill, New York.
Oldhan, K.B. and Spainer, J. (1974) The Fractional Calculus. Academic Press, New York.
Myers, T.G., Charpin, J.P.F. and Chapman, S.J. (2002) The Flow and Solidification of a Thin Fluid Film on an Arbitrary Three-Dimensional Surface. Physics of Fluids, 14, 2788-2803. https://doi.org/10.1063/1.1488599
Myers, T.G. and Charpin, J.P.F. (2004) A Mathematical Model for Atmospheric Ice Accretion and Water Flow on a Cold Surface. International Journal of Heat and Mass Transfer, 47, 5483-5500. https://doi.org/10.1016/j.ijheatmasstransfer.2004.06.037
Karpman, V.I. (1996) Stabilization of Soliton Instabilities by Higher-Order Dispersion: Fourth Order Nonlinear Schrödinger-Type Equations. Physical Review E, 53, 1336-1339. https://doi.org/10.1103/PhysRevE.53.R1336
Karpman, V.I. and Shagalov, A.G. (2000) Stability of Soliton Described by Nonlinear Schrödinger-Type Equations with Higher Order Dispersion. Physica D: Nonlinear Phenomena, 144, 194-210. https://doi.org/10.1016/S0167-2789(00)00078-6
Agrawal, O.P. (2001) A General Solution for a Fourth-Order Fractional Diffusion-Wave Equation Defined in A Bounded Domain. Computers and Structures, 79, 1497-1501. https://doi.org/10.1016/S0045-7949(01)00026-8
Hu, X.L. and Zhang, L.M. (2012) On Finite Difference Methods for Fourth-Order Fractional Diffusionwave and Sub-Diffusion Systems. Applied Mathematics and Computation, 218, 5019-5034. https://doi.org/10.1016/j.amc.2011.10.069
Guo, J., Li, C.P. and Ding, H.F. (2014) Finite Difference Methods for Time Subdiffusion Equation with Space Fourth Order. Commun. Applied Mathematics and Computation, 28, 96-108. (In Chinese)
Ji, C.-C., Sun, Z.-Z. and Hao, Z.-P. (2015) Numerical Algorithms with High Spatial Accuracy for the Fourth-Order Fractional Sub-Diffusion Equations with the First Dirichlet Boundary Conditions. Journal of Scientific Computing, 66, 1148-1174. https://doi.org/10.1007/s10915-015-0059-7
Zhang, P. and Pu, H. (2017) A Second-Order Compact Difference Scheme for the Fourth-Order Fractional Sub-Diffusion Equation. Numerical Algorithms, 76, 573-598. https://doi.org/10.1007/s11075-017-0271-7
Nandal, S. and Pandey, D.N. (2019) Numerical Solution of Time Fractional Non-Linear Neutral Delay Differential Equations of Fourth-Order. Malaya Journal of Matematik, 78, 1467-1487. https://doi.org/10.26637/MJM0703/0035
Nandal, S. and Pandey, D.N. (2020) Numerical Solution of Non-Linear Fourth-Order Fractional Subdiffusion Wave Equation with Time Delay. Applied Mathematics and Computation, 369, Article ID: 124900. https://doi.org/10.1016/j.amc.2019.124900
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 Mathematics, 318, 433-443. https://doi.org/10.1016/j.cam.2016.02.039
Sun, Z.Z. (2012) Numerical Methods for Partial Differential Equations. 2nd Edition, Science Press, Beijing. (In Chinese)
Li, Q., Yang, Q. and Chen, H.Z. (2020) Compact Difference Scheme for Two-Dimensional Fourth-Order Nonlinear Hyperbolic Equation. Numerical Methods for Partial Differential Equations, 36, 1938-1961. https://doi.org/10.1002/num.22511
Zhang, Q.F., Ran, M.H. and Xu, D.H. (2017) Analysis of the Compact Difference Scheme for the Semilinear Fractional Partial Differential Equation with Time Delay. Applicable Analysis, 96, 1867-1884. https://doi.org/10.1080/00036811.2016.1197914