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The implicit midpoint method for the modified anomalous sub-diffusion equation with a nonlinear source term. (English) Zbl 1357.65149

Summary: In this paper, the implicit midpoint method is used to solve the semi-discrete modified anomalous sub-diffusion equation with a nonlinear source term, and the weighted and shifted Grünwald-Letnikov difference operator and the compact difference operator are applied to approximate the Riemann-Liouville fractional derivative and space partial derivative respectively, then the new numerical scheme is constructed. The stability and the convergence of this method are analyzed. Numerical experiment demonstrates the high accuracy of this method and confirm our theoretical results.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35R11 Fractional partial differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs

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[1] Becker-Kern, P.; Meerschaert, M. M.; Scheffler, H. P., Limit theorem for continuous-time random walks with two time scales, J. Appl. Probab., 41, 455-466 (2004) · Zbl 1050.60038
[2] Gorenflo, R.; Mainardi, F.; Scalas, E.; Raberto, M., Fractional calculus and continuous-time finance III: The diffusion limit, (Mathematical Finance. Mathematical Finance, Trends in Mathematics (2001), Birkhäuser: Birkhäuser Basel), 171-180 · Zbl 1138.91444
[3] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier Science Limited · Zbl 1092.45003
[4] Koeller, R. C., Application of fractional calculus to the theory of viscoelasticity, J. Appl. Mech., 51, 229-307 (1984) · Zbl 0544.73052
[5] Meerschaert, M. M.; Scalas, E., Coupled continuous time random walks in finance, Physica A, 370, 114-118 (2006)
[6] Meerschaert, M. M.; Zhang, Y.; Baeumerc, B., Particle tracking for fractional diffusion with two time scales, Comput. Math. Appl., 59, 1078-1086 (2010) · Zbl 1189.65240
[7] Petras, I., Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (2011), Springer · Zbl 1228.34002
[8] Podlubny, I., Fractional Differential Equations (1998), Academic Press · Zbl 0922.45001
[9] Liu, Q.; Liu, F.; Turner, I.; Anh, V., Finite element approximation for a modified anomalous subdiffusion equation, Appl. Math. Model., 35, 4103-4116 (2011) · Zbl 1221.65257
[10] Liu, F.; Yang, C.; Burrage, K., Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term, J. Comput. Appl. Math., 231, 160-176 (2009) · Zbl 1170.65107
[11] Chen, Y.; Chen, C. M., Numerical scheme with high order accuracy for solving a modified fractional diffusion equation, Appl. Math. Comput., 224, 772-782 (2014) · Zbl 1336.65131
[12] Chen, C. M.; Liu, F.; Turner, I.; Anh, V., Numerical schemes and multivariate extrapolation of a two-dimensional anomalous subdiffusion equation, Numer. Algorithms, 54, 1-21 (2010) · Zbl 1191.65116
[13] Chen, C. M.; Liu, F.; Turner, I.; Anh, V., Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation, Math. Comp., 81, 345-366 (2012) · Zbl 1241.65077
[14] Chen, C.; Liu, F.; Anh, V.; Turner, I., Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation, SIAM J. Sci. Comput., 32, 1740-1760 (2010) · Zbl 1217.26011
[15] Cui, M., Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228, 7792-7804 (2009) · Zbl 1179.65107
[16] Huang, H.; Cao, X. N., Numerical method for two dimensional fractional reaction subdiffusion equation, Eur. Phys. J. Spec. Top., 222, 1961-1973 (2013)
[17] Li, Y. F.; Wang, D. L., Improved efficient difference method for the modified anomalous subdiffusion equation with a nonlinear source term, Int. J. Comput. Math. (2016)
[18] Mohebbi, A.; Abbaszadeh, M.; Dehghan, M., A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term, J. Comput. Phys., 240, 36-48 (2013) · Zbl 1287.65064
[19] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[20] Wang, Z. B.; Vong, S. W., Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys., 277, 1-15 (2014) · Zbl 1349.65348
[21] Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103, 16-42 (1992) · Zbl 0759.65006
[22] Tian, W. Y.; Zhou, H.; Deng, W. H., A class of second order difference approximations for solving space fractional diffusion equations, Math. Comp., 84, 1703-1727 (2015) · Zbl 1318.65058
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