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Original Articles

Quenching phenomenon in a fractional diffusion equation and its numerical simulation

Pages 98-113 | Received 21 Feb 2017, Accepted 22 May 2017, Published online: 29 Jun 2017

References

  • O.P. Agrawal, Formulation of Euler–Lagrange equations for fractional variational problems, J. Math. Anal. Appl. 272(1) (2002), pp. 368–379. doi: 10.1016/S0022-247X(02)00180-4
  • R.P. Agarwal, M. Benchohra, and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109 (2010), pp. 973–1033. doi: 10.1007/s10440-008-9356-6
  • V.E. Arkhincheev, Anomalous diffusion and charge elaxation on comb model: exact solutions, Phys. A: Stat. Mech. Appl. 280(3-4) (2000), pp. 304–314. doi: 10.1016/S0378-4371(99)00593-2
  • V.E. Arkhincheev and É.M. Baskin, Anomalous diffusion and drift in a comb model of percolation clusters, Sov. Phys. JETP 73(1) (1991), pp. 161–300.
  • R.L. Bagley and P.J. Torvik, On the fractional calculus model of viscoelastic behavior, J. Rheo. 30(1) (1986), pp. 133–155. doi: 10.1122/1.549887
  • M.A. Beauregard and Q. Sheng, A semi-adaptive compact splitting method for the numerical solution of 2-dimensional quenching problems, Appl. Math. Comput. 218 (2012), pp. 11240–11254.
  • J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Applied Mathematical Sciences, Vol. 83, Springer, New York, 1989.
  • P.H. Chang and H.A. Levine, The quenching of solutions of semilinear hyperbolic equations, SIAM J. Math. Anal. 12(6) (1981), pp. 893–903. doi: 10.1137/0512075
  • C.P. Chen, A. Chen, and J. Ye, Numerical approaches to fractional calculus and fractional ordinary differential equation, J. Comput. Phys. 230(9) (2011), pp. 3352–3368. doi: 10.1016/j.jcp.2011.01.030
  • Q.Y. Dai and Y.G. Gu, A short note on quenching phenomena for semilinear parabolic equations, J. Diff. Equa. 137(2) (1997), pp. 240–250. doi: 10.1006/jdeq.1996.3249
  • V.J. Ervin and J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods PDEs 22(3) (2006), pp. 558–576. doi: 10.1002/num.20112
  • K.S. Fa, Generalized Langevin equation with fractional derivative and long-time correlation function, Phys. Rev. E 73 (2006), pp. 061104-1–061104-4. doi: 10.1103/PhysRevE.73.061104
  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
  • H. Kawarada, On solutions of the initial-boundary problem for ut=uxx+11−u, Publ. Res. Inst. Math. Sci. 10 (1975), pp. 729–736. doi: 10.2977/prims/1195191889
  • A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
  • M. Klimek and O.P. Agrawal, Fractional Sturm-Liouville problem, Comput. Math. Appl. 66(5) (2013), pp. 795–812. doi: 10.1016/j.camwa.2012.12.011
  • H.A. Levine, The phenomenon of quenching: A survey, Trends Theory Pract. Non-Linear Anal. 110 (1985), pp. 275–286.
  • H.A. Levine and J.T. Montgomery, The quenching of solutions of some nonlinear parabolic equations, SIAM J. Math. Anal. 11(5) (1980), pp. 842–847. doi: 10.1137/0511075
  • X.J. Li and C.J. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal. 47 (2009), pp. 2108–2131. doi: 10.1137/080718942
  • X.J. Li and C.J. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys. 8 (2010), pp. 1016–1051.
  • C.P. Li, Z. Zhao, and Y. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl. 62(3) (2011), pp. 855–875. doi: 10.1016/j.camwa.2011.02.045
  • Y.M. Lin, X.J. Li, and C.J. Xu, Finite difference/spectral approximations for the fractional cable equation, Math. Comput. 275 (2011), pp. 1369–1396.
  • F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.
  • R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Report. 339 (2000), pp. 1–77. doi: 10.1016/S0370-1573(00)00070-3
  • M.C. Néel, A. Abdennadher, and M. Joelson, Fractional Fick's law: The direct way, J. Phys. A: Math. Theor. 40 (2007), pp. 8299–8314. doi: 10.1088/1751-8113/40/29/007
  • P. Paradisi, R. Cesari, F. Mainardi, and F. Tampieri, The fractional Fick's law for non-local transport processes, Phys. A: Stat. Mech. Appl. 293(1/2) (2001), pp. 130–142. doi: 10.1016/S0378-4371(00)00491-X
  • J. Petersson, A note on quenching for parabolic equations with dynamic boundary conditions, Nonlinear Anal. 58 (2004), pp. 417–423. doi: 10.1016/j.na.2004.01.006
  • J.P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2, J. Comput. Appl. Math. 193(1) (2006), pp. 243–268. doi: 10.1016/j.cam.2005.06.005
  • Q. Sheng, Adaptive decomposition finite difference methods for solving singular problems, Front. Math. China 4 (2009), pp. 599–626. doi: 10.1007/s11464-009-0038-0
  • Q. Sheng and H. Cheng, An adaptive grid method for degenerate semi-linear quenching problems, Comput. Math. Appl. 39 (2000), pp. 57–71. doi: 10.1016/S0898-1221(00)00086-9
  • Q. Sheng and A. Khaliq, A compound adaptive approach to degenerate nonlinear quenching problems, Numer. Meth. PDEs 15 (1999), pp. 29–47. doi: 10.1002/(SICI)1098-2426(199901)15:1<29::AID-NUM2>3.0.CO;2-L
  • Y. Tian and Z.B. Bai, Existence results for the three-point impulsive boundary value problem involving fractional differential equations, Comput. Math. Appl. 59(8) (2010), pp. 2601–2609. doi: 10.1016/j.camwa.2010.01.028
  • J.R. Wang, Y. Zhou, and M. Medved, Existence and stability of fractional differential equations with Hadamard derivative, Topol. Methods Nonlinear Anal. 41(1) (2013), pp. 113–133.
  • H. Zhang, F. Liu, and I. Turner, Numerical solution of the time fractional Black–Scholes model governing European options, Comput. Math. Appl. 71(9) (2016), pp. 1772–1783. doi: 10.1016/j.camwa.2016.02.007
  • L. Zhang, H.W. Sun, and H.K. Pang, Fast numerical solution for fractional diffusion equations by exponential quadrature rule, J. Comput. Phys. 299 (2015), pp. 130–143. doi: 10.1016/j.jcp.2015.07.001
  • S.M. Zhou, C.L. Mu, Q.L. Du, and R. Zeng, Quenching for a reactiondiffusion equation with nonlinear memory, Commun. Nonlinear Sci Numer Simulat. 17(2) (2012), pp. 754–763. doi: 10.1016/j.cnsns.2011.05.043
  • S.M. Zhou, C.L. Mu, and R. Zeng, Quenching for a non-local diffusion equation with a singular absorption term and Neumann boundary condition, J. Appl. Math. Phys. (ZAMP) 62(3) (2011), pp. 483–493. doi: 10.1007/s00033-010-0103-2

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