Neumann boundary-value problems for a time-fractional diffusion-wave equation in a half-plane

The time-fractional diffusion-wave equation with the Caputo derivative of the order 0<@a<2 is considered in a half-plane. Two types of Neumann boundary condition are examined: the mathematical condition with the prescribed boundary value of the normal derivative and the physical one with the prescribed boundary value of the matter flux.

[1]  A. P. Prudnikov,et al.  Integrals and series of elementary functions , 1981 .

[2]  P. Kythe.  Fundamental Solutions for Differential Operators and Applications , 1996 .

[3]  Igor M. Sokolov,et al.  Physics of Fractal Operators , 2003 .

[4]  M. Gurtin,et al.  A general theory of heat conduction with finite wave speeds , 1968 .

[5]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[6]  Yasuhiro Fujita,et al.  INTEGRODIFFERENTIAL EQUATION WHICH INTERPOLATES THE HEAT EQUATION AND THE WAVE EQUATION I(Martingales and Related Topics) , 1989 .

[7]  I. Podlubny.  Fractional differential equations , 1998 .

[8]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[9]  Yuriy Povstenko,et al.  Time-fractional radial diffusion in a sphere , 2008 .

[10]  V. V. Gafiychuk,et al.  Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems , 2010, Comput. Math. Appl..

[11]  V. Gafiychuk,et al.  Mathematical modeling of time fractional reaction-diffusion systems , 2008 .

[12]  R. Nigmatullin.  To the Theoretical Explanation of the “Universal Response” , 1984 .

[13]  Yuriy Povstenko,et al.  Theory of thermoelasticity based on the space-time-fractional heat conduction equation , 2009 .

[14]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[15]  F. Mainardi.  Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena , 1996 .

[16]  R. Magin.  Fractional Calculus in Bioengineering , 2006 .

[17]  F. Mainardi.  The fundamental solutions for the fractional diffusion-wave equation , 1996 .

[18]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[19]  W. Wyss.  The fractional diffusion equation , 1986 .

[20]  William Alan Day,et al.  The Thermodynamics of Simple Materials with Fading Memory , 1972 .

[21]  Y. Povstenko.  FRACTIONAL HEAT CONDUCTION EQUATION AND ASSOCIATED THERMAL STRESS , 2004 .

[22]  R. Gorenflo,et al.  Fractional Calculus: Integral and Differential Equations of Fractional Order , 2008, 0805.3823.

[23]  R. Gorenflo,et al.  Time Fractional Diffusion: A Discrete Random Walk Approach , 2002 .

[24]  Raoul R. Nigmatullin,et al.  On the Theory of Relaxation for Systems with “Remnant” Memory , 1984 .

[25]  Francesco Mainardi,et al.  Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics , 2012, 1201.0863.

[26]  W. Schneider,et al.  Fractional diffusion and wave equations , 1989 .

[27]  G. Zaslavsky.  Chaos, fractional kinetics, and anomalous transport , 2002 .