Abstract
Two alternating direction implicit difference schemes are established for solving a class of two-dimensional time distributed-order wave equations. The schemes are proved to be unconditionally stable and convergent in the maximum norm with the convergence orders \(O(\tau ^2+h_1^2+h_2^2+\Delta \gamma ^2)\) and \(O(\tau ^2+h_1^4+h_2^4+\Delta \gamma ^4),\) respectively, where \(\tau , h_i\; (i=1,2)\) and \(\Delta \gamma \) are the step sizes in time, space and distributed order. Also, several numerical experiments are carried out to validate the theoretical results.
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References
Metzler, R., Klafter, J.: The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)
Eab, C.H., Lim, S.C.: Fractional Langevin equations of distributed order. Phys. Rev. E 83, 031136 (2011)
Meerschaert, M., Scheffler, H.: Stochastic model for ultraslow diffusion. Stoch. Process. Appl. 116, 1215–1235 (2006)
Mainardi, F., Pagnini, G., Mura, A., Gorenflo, R.: Time-fractional diffusion of distributed order. J. Vib. Control 14, 1267–1290 (2008)
Atanackovic, T.M., Pilipovic, S., Zorica, D.: Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations. Proc. R. Soc. A 465, 1893–1917 (2009)
Gorenflo, R., Luchko, Y., Stojanoveć, M.: Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density. Fract. Calc. Appl. Anal. 16, 297–316 (2013)
Li, Z., Luchko, Y., Yamamoto, M.: Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations. Fract. Calc. Appl. Anal. 17, 1114–1136 (2014)
Ford, N.J., Morgado, M.L., Rebelo, M.: An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time. Electron. Trans. Numer. Anal. 44, 289–305 (2015)
Hu, X., Liu, F., Anh, V., Turner, I.: A numerical investigation of the time distributed-order diffusion model. ANZIAM J. 55, C464–C478 (2014)
Ye, H., Liu, F., Anh, V., Turner, I.: Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains. IMA J. Appl. Math. 80, 825–838 (2015)
Morgado, M.L., Rebelo, M.: Numerical approximation of distributed order reaction-diffusion equations. J. Comput. Appl. Math. 275, 216–227 (2015)
Alikhanov, A.A.: Numerical methods of solutions of boundary value problems for the multi-term variable-distributed order diffusion equation. Appl. Math. Comput. 268, 12–22 (2015)
Gao, G.H., Sun, Z.Z.: Two unconditionally stable and convergent difference schemes with the extrapolation method for the one-dimensional distributed-order differential equations. Numer. Methods Partial Differ. Equ. 32, 591–615 (2016)
Gao, G.H., Sun, H.W., Sun, Z.Z.: Some high-order difference schemes for the distributed-order differential equations. J. Comput. Phys. 298, 337–359 (2015)
Ye, H., Liu, F., Anh, V.: Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains. J. Comput. Phys. 298, 652–660 (2015)
Gao, G.H., Sun, Z.Z.: Two difference schemes for solving the one-dimensional time distributed-order fractional wave equations (under review)
Gao, G.H., Sun, Z.Z.: Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations. Comput. Math. Appl. 69, 926–948 (2015)
Gao, G.H., Sun, Z.Z.: Two alternating direction implicit difference schemes for two-dimensional distributed-order fractional diffusion equations. J. Sci. Comput. 66, 1281–1312 (2016)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection–dispersion equations. J. Comput. Appl. Math. 172, 65–77 (2004)
Tadjeran, C., Meerschaert, M.M., Scheffler, H.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213, 205–213 (2006)
Tian, W.Y., Zhou, H., Deng, W.H.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703–1727 (2015)
Wang, Z., Vong, S.: Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J. Comput. Phys. 277, 1–15 (2014)
Samarskiĭ, A.A.: The Theory of Difference Schemes. Marcel Dekker Inc., New York (2001)
Liao, H.L., Sun, Z.Z.: Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer. Methods Partial Differ. Equ. 26, 37–60 (2010)
Sun, Z.Z.: The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations. Science Press, Beijing (2009)
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The authors would like to thank the anonymous referees for their valuable comments and suggestions to greatly improve this work.
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The research is supported by the research Grant 11401319, 11271068, 11501301 from National Natural Science Foundation of China and BK20130860 from Natural Science Youth Foundation of Jiangsu Province of China.
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Gao, Gh., Sun, Zz. Two Alternating Direction Implicit Difference Schemes for Solving the Two-Dimensional Time Distributed-Order Wave Equations. J Sci Comput 69, 506–531 (2016). https://doi.org/10.1007/s10915-016-0208-7
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DOI: https://doi.org/10.1007/s10915-016-0208-7