The Poisson distribution, abstract fractional difference equations, and stability

We study the initial value problem (∗) { C∆ u(n) = Au(n+ 1), n ∈ N0; u(0) = u0 ∈ X. when A is a closed linear operator with domain D(A) defined on a Banach space X. We introduce a method based on the Poisson distribution to show existence and qualitative properties of solutions for the problem (*), using operator-theoretical conditions on A. We show how several properties for fractional differences, including their own definition, are connected with the continuous case by means of sampling using the Poisson distribution. We prove necessary conditions for stability of solutions, that are only based on the spectral properties of the operator A in case of Hilbert spaces.

[1]  Carlos Lizama,et al.  Spectral criteria for solvability of boundary value problems and positivity of solutions of time-fractional differential equations , 2013 .

[2]  Michael T. Holm,et al.  The Laplace transform in discrete fractional calculus , 2011, Comput. Math. Appl..

[3]  F. Atici,et al.  Modeling with fractional difference equations , 2010 .

[4]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[5]  C. Lubich.  Discretized fractional calculus , 1986 .

[6]  Jiang Wei,et al.  Solving Fractional Difference Equations Using the Laplace Transform Method , 2014 .

[7]  Tomáš Kisela,et al.  Stability regions for linear fractional differential systems and their discretizations , 2013, Appl. Math. Comput..

[8]  Tomáš Kisela,et al.  Stability and asymptotic properties of a linear fractional difference equation , 2012 .

[9]  Vidar Thomée,et al.  Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term , 1996, Math. Comput..

[10]  Christopher S. Goodrich,et al.  Existence of a positive solution to a system of discrete fractional boundary value problems , 2011, Appl. Math. Comput..

[11]  R. Ferreira.  Existence and uniqueness of solution to some discrete fractional boundary value problems of order less than one , 2013 .

[12]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[13]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[14]  K. Tas,et al.  A new transform method in nabla discrete fractional calculus , 2012 .

[15]  J. B. Díaz,et al.  Differences of fractional order , 1974 .

[16]  Michael T. Holm,et al.  The Theory of Discrete Fractional Calculus: Development and Application , 2011 .

[17]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[18]  P. Eloe,et al.  Initial value problems in discrete fractional calculus , 2008 .

[19]  P. Eloe,et al.  A transform method in discrete fractional calculus , 2007 .

[20]  Emilia Bazhlekova,et al.  Fractional evolution equations in Banach spaces , 2001 .

[21]  R. Ferreira.  Calculus of Variations on Time Scales and Discrete Fractional Calculus , 2010, 1007.5087.

[22]  Nien Fan Zhang,et al.  On a new definition of the fractional difference , 1988 .

[23]  Eduardo Cuesta,et al.  ASYMPTOTIC BEHAVIOUR OF THE SOLUTIONS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS AND SOME TIME DISCRETIZATIONS , 2007 .

[24]  Eduardo Cuesta,et al.  A Numerical Method for an Integro-Differential Equation with Memory in Banach Spaces: Qualitative Properties , 2003, SIAM J. Numer. Anal..

[25]  S. Sengul.  Discrete Fractional Calculus and Its Applications to Tumor Growth , 2010 .

[26]  Dumitru Baleanu,et al.  On non-homogeneous singular systems of fractional nabla difference equations , 2014, Appl. Math. Comput..

[27]  M. Meerschaert,et al.  VECTOR GRÜNWALD FORMULA FOR FRACTIONAL DERIVATIVES , 2004 .

[28]  Paul W. Eloe,et al.  DISCRETE FRACTIONAL CALCULUS WITH THE NABLA OPERATOR , 2009 .

[29]  Manuel Duarte Ortigueira,et al.  Discrete-time differential systems , 2015, Signal Process..

[30]  Joseph Lipka,et al.  A Table of Integrals , 2010 .

[31]  Thabet Abdeljawad,et al.  On the Definitions of Nabla Fractional Operators , 2012 .

[32]  J. M. Sanz-Serna,et al.  A numerical method for a partial integro-differential equation , 1988 .

[33]  Rong-Nian Wang,et al.  Abstract fractional Cauchy problems with almost sectorial operators , 2012 .

[34]  Thabet Abdeljawad,et al.  On Riemann and Caputo fractional differences , 2011, Comput. Math. Appl..

[35]  Mark M. Meerschaert,et al.  STOCHASTIC SOLUTIONS FOR FRACTIONAL CAUCHY PROBLEMS , 2003 .

[36]  Christopher S. Goodrich,et al.  Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions , 2011, Comput. Math. Appl..

[37]  Eduardo Cuesta,et al.  Convolution quadrature time discretization of fractional diffusion-wave equations , 2006, Math. Comput..

[38]  E. Bazhlekova.  Subordination principle for fractional evolution equations , 1999 .

[39]  Christopher S. Goodrich,et al.  On a discrete fractional three-point boundary value problem , 2012 .

[40]  C. Goodrich.  On a first-order semipositone discrete fractional boundary value problem , 2012 .

[41]  Paul W. Eloe,et al.  Two-point boundary value problems for finite fractional difference equations , 2011 .

[42]  Shurong Sun,et al.  The existence and uniqueness of solutions to boundary value problems of fractional difference equations , 2012, Mathematical Sciences.

[43]  P. Eloe,et al.  Linear systems of fractional nabla difference equations , 2011 .

[44]  Julia,et al.  Vector-valued Laplace Transforms and Cauchy Problems , 2011 .

[45]  P. Heywood.  Trigonometric Series , 1968, Nature.