On the stability and convergence of the time-fractional variable order telegraph equation

Journal of Computational Physics

Volumn 293, Issue , 2015, Pages 104-114

  Atangana, Abdon ^a  

^a UNIVERSITY OF THE FREE STATE   (South Africa)

Author keywords

Convergence; Crank Nicholson scheme; Fractional variable order derivative; Stability; Telegraph equation

Indexed keywords

COMPUTATIONAL METHODS;

APPROXIMATE SOLUTION; CONVERGENCE; CRANK-NICHOLSON SCHEME; FRACTIONAL TELEGRAPH EQUATIONS; FRACTIONAL VARIABLE ORDER DERIVATIVE; GENERALIZED EQUATIONS; NUMERICAL SOLUTION; STABILITY AND CONVERGENCE; TELEGRAPH EQUATION; VARIABLES ORDERING;

TELEGRAPH;

EID: 84934288557     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2014.12.043     Document Type: Article

References (43)

1. Samko S.G. Fractional integration and differentiation of variable order. Anal. Math. 1995, 21:213-236.
2. Valério D., Sá da Costa J. Variable-order fractional derivatives and their numerical approximations. Signal Process. 2011, 91(3):470-483.
3. Grainger J.J., Stevenson W.D. Power Systems Analysis 1994, McGraw-Hill, International.
4. Joubert S.V., Greeff J.C. Accuracy estimates for computer algebra systems IVP solvers. South Afr. J. Sci. 2006, 102:46-50.
5. Tomasi W. Electronic Communication Systems 2004, Prentice Hall, New Jersey.
6. Wylie C. Advance Engineering Mathematics 1995, McGraw-Hill, International.
7. Cascaval R.C., Eckstein E.C., Frota L., Goldstein J.A. Fractional telegraph equations. J. Math. Anal. Appl. 2002, 276:145-159.
8. Orssingher E., Zhao X. The space fractional telegraph equation and the related telegraph process. Chin. Ann. Math. 2003, 24B:45-56.
9. Chen J., Liu F., Anh V. Analytical solution for the time-fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl. 2008, 338:1364-1377.
10. Orsingher E., Beghin L. Time fractional telegraph equation and telegraph process with brownian time. Probab. Theory Relat. Fields 2004, 128:141-160.
11. Momani S. Analytic and approximate solutions of the space and time fractional telegraph equations. Appl. Math. Comput. 2005, 170:1126-1134.
12. Dehghan M., Yousefi S.A., Lotfi A. The use of He's variational iteration method for solving the telegraph and fractional telegraph equations. Int. J. Numer. Methods Biomed. Eng. 2011, 27:219-231.
13. Pedro H.T.C., Kobayashi M.H., Pereira J.M.C., Coimbra C.F.M. Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere. J. Vib. Control 2008, 14:1569-1672.
14. Ramirez L.E.S., Coimbra C.F.M. On the selection and meaning of variable order operators for dynamic modelling. Int. J. Differ. Equ. 2010, 2010:846107. 16 pp.
15. Ramirez L.E.S., Coimbra C.F.M. On the variable order dynamics of the nonlinear wake caused by a sedimenting particle. Physica D 2011, 240:1111-1118.
16. Ross B., Samko S.G. Fractional integration operator of a variable order in the Holder spaces H_(x). Int. J. Math. Math. Sci. 1995, 18:777-788.
17. Umarov S., Steinberg S. Variable order differential equations and diffusion with changing modes. Z. Anal. Anwend. 2009, 28:431-450.
18. Magin R.L., Abdullah O., Baleanu D., Zhou X.J. Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation. J. Magn. Reson. 2008, 190:255-270.
19. Miller K.S., Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations 1993, Wiley, New York.
20. Lorenzo C.F., Hartley T.T. Variable order and distributed order fractional operators. Nonlinear Dyn. 2002, 29:57-98.
21. Liu Yang, Fang Zhichao, Li Hong, He Siriguleng A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl. Math. Comput. 2014, 243:703-717.
22. Atangana Abdon, Baleanu Dumitru Numerical solution of a kind of fractional parabolic equations via two difference schemes. Abstr. Appl. Anal. 2013, 2013:828764. 8 pp.
23. Meerschaert M.M., Tadjeran C. Finite difference approximations for fractional advection dispersion equations. J. Comput. Appl. Math. 2004, 172:65-77.
24. Zhang Y. A finite difference method for fractional partial differential equation. Appl. Math. Comput. 2009, 215:524-529.
25. Tadjeran C., Meerschaert M.M., Scheffler H.P. A second order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 2006, 213:205-213.
26. Chen C.M., Liu F., Turner I., Anh V. A Fourier method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 2007, 227:886-897.
27. Yuste S.B., Acedo L. An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 2005, 42:1862-1874.
28. Podlubny I., Chechkin A., Skovranek T., Chen Y.Q., Vinagre Jara B.M. Matrix approach to discrete fractional calculus II: partial fractional differential equations. J. Comput. Phys. 2009, 228:3137-3153.
29. Hanert E. On the numerical solution of space time fractional diffusion models. Comput. Fluids 2011, 46:33-39.
30. Zhuang P., Liu F., Anh V., Turner I. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 2009, 47:1760-1781.
31. Lin R., Liu F., Anh V., Turner I. Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl. Math. Comput. 2009, 212:435-445.
32. Crank J., Nicolson P. A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. Proc. Camb. Philol. Soc. 1947, 43:50-67.
33. Diethelm K., Ford N.J., Freed A.D. Detailed error analysis for a fractional Adams method. Numer. Algorithms 2004, 36:31-52.
34. Li C.P., Tao C.X. On the fractional Adams method. Comput. Math. Appl. 2009, 58:1573-1588.
35. Cooper G.R.J., Cowan D.R. Filtering using variable order vertical derivatives. Comput. Geosci. 2004, 30:455-459.
36. Tseng C.-C. Design of variable and adaptive fractional order FIR differentiators. Signal Process. 2006, 86:2554-2566.
37. Umarov S., Steinberg S. Variable order differential equations and diffusion with changing modes. Z. Anal. Anwend. 2009, 28:431-450.
38. Sun H.G., Chen W., Li C., Chen Y.Q. Fractional differential models for anomalous diffusion. Physica A 2010, 389:2719-2724.
39. Koksal Mehmet Emir An operator-difference method for telegraph equations arising in transmission lines. Discrete Dyn. Nat. Soc. 2011, 2011:561015. 17 pp.
40. Sun H.G., Chen W., Wei H., Chen Y.Q. A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. Eur. Phys. J. Spec. Top. 2011, 193:185-192.
41. Shyu J.-J., Pei S.-C., Chan C.-H. An iterative method for the design of variable fractional-order FIR differintegrators. Signal Process. 2009, 89:320-327.
42. Coimbra C. Mechanics with variable-order differential operators. Ann. Phys. 2003, 12(11-12):692-703.
43. Valério D., Vinagre G., Domingues J., Sá da Costa J. Variable-order fractional derivatives and their numerical approximations I-real orders. Symposium on Fractional Signals and Systems 2009.