The Maximum Principle for Time-Fractional Diffusion Equations and Its Application

REFERENCES

• H. Brunner , H. Han , and D. S. Yin ( 2014 ) Artificial boundary conditions and finite difference approximations for a time-fractional diffusion-wave equation on a two-dimensional unbounded spatial domain. J. Comput. Phys. 276:541–562. .
   Google Scholar
• P. Clément and S.-O. Londen ( 2000 ). Regularity aspects of fractional evolution equations . Rend. Istit. Mat. Univ. Trieste 31 ( 2 ): 19 – 30 .
   Google Scholar
• E. Cuesta , C. Lubich , and C. Palencia ( 2006 ). Convolution quadrature time discretization of fractional diffusion-wave equations . Math. Comp. 75 : 673 – 696 .
   Web of Science ®Google Scholar
• K. Diethelm ( 1997 ). An algorithm for the numerical solution of differential equations of fractional order . Electr. Trans. Numer. Anal. 5 : 1 – 6 .
   Google Scholar
• K. Diethelm ( 2010 ). The Analysis of Fractional Differential Equations . Lecture Notes in Math., 2004. Springer-Verlag, Berlin .
   Google Scholar
• K. Diethelm , N. J. Ford , A. D. Freed , and Y. Luchko ( 2005 ). Algorithms for the fractional calculus: A collection of numerical methods . Comput. Methods Appl. Mech. Eng. 194 : 743 – 773 .
   Web of Science ®Google Scholar
• H. Ding and Y. Zhang ( 2011 ). Notes on implicit finite difference approximation for time fractional diffusion equations . Comput. Math. Appl. 61 : 2924 – 2938 .
   Web of Science ®Google Scholar
• Y. Fujita ( 1990 ). Integrodifferential equations which interpolates the heat equation and the wave equation . Osaka J. Math. 27 : 309 – 321 .
   Web of Science ®Google Scholar
• Y. Fujita ( 1990 ). Integrodifferential equation which interpolates the heat equation and the wave equation (II) . Osaka J. Math. 27 : 797 – 804 .
   Web of Science ®Google Scholar
• R. Gorenflo and S. Vessella ( 1991 ). Abel Integral equations: Analysis and Applications . Lecture Notes in Mathematics, Vol. 1461. Springer-Verlag, Berlin .
   Google Scholar
• Y. Luchko ( 2009 ). Maximum principle for the generalized time-fractional diffusion equation . J. Math. Anal. Appl. 351 : 218 – 223 .
   Web of Science ®Google Scholar
• Y. Luchko (2010). Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Comput. Math. Appl. 59: 1766–1772 .
   Google Scholar
• F. Mainardi ( 2010 ). Fractional Calculus and Waves in Viscoelasticity . Imperial College Press , London .
   Google Scholar
• W. McLean ( 2010 ). Regularity of solutions to a time-fractional diffusion equation . ANZIAM J. 52 : 123 – 138 .
   Web of Science ®Google Scholar
• D. A. Murio (2008). Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 56:1138–1145.
   Web of Science ®Google Scholar
• K. Mustapha and W. McLean ( 2013 ). Superconvergence of a discontinuous Galerkin method for fractional order diffusion and wave equations . SIAM J. Numer. Anal. 57 : 491 – 515 .
   Web of Science ®Google Scholar
• J. J. Nieto ( 2010 ). Maximum principles for fractional differential equations derived from Mittag-Leffler functions . Appl. Math. Lett. 23 : 1248 – 1251 .
   Web of Science ®Google Scholar
• J. C. Ren and Z. Z. Sun ( 2013 ). Numerical algorithm with spatial accuracy for the fractional diffusion-wave equation with Neumann boundary conditions . J. Sci. Comput. 56 : 381 – 408 .
   Web of Science ®Google Scholar
• Z. Z. Sun and X. Wu ( 2006 ). A fully discrete difference scheme for a diffusion-wave system . Appl. Numer. Math. 56 : 193 – 209 .
   Web of Science ®Google Scholar
• S. B. Yuste and J. Quintana-Murillo ( 2012 ). A finite difference method with non-uniform time-steps for fractional diffusion equations . Comput. Phys. Commun. 183 : 2594 – 2600 .
   Web of Science ®Google Scholar