Distributed-order fractional wave equation on a finite domain: creep and forced oscillations of a rod

We study waves in a rod of finite length with a viscoelastic constitutive equation of distributed fractional order type for the special choice of weight functions. Prescribing boundary conditions on displacement and stress, we obtain, as special solutions, cases corresponding to creep and forced oscillations. In solving system of differential and integro-differential equations, we use the Laplace transformation in the time domain.

[1]  Jia Guo Liu,et al.  Higher-order fractional constitutive equations of viscoelastic materials involving three different parameters and their relaxation and creep functions , 2007 .

[2]  T. Atanacković,et al.  On a class of equations arising in linear viscoelasticity theory , 2005 .

[3]  G. Weiss,et al.  Equations of Mathematical Physics , 1967 .

[4]  S. Welch,et al.  Application of Time-Based Fractional Calculus Methods to Viscoelastic Creep and Stress Relaxation of Materials , 1999 .

[5]  Teodor M. Atanackovic,et al.  Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[6]  Alexander Lion,et al.  On the thermodynamics of fractional damping elements , 1997 .

[7]  Teodor M. Atanackovic,et al.  Distributional framework for solving fractional differential equations , 2009, 0902.0496.

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

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

[10]  Teodor M. Atanackovic,et al.  On a distributed derivative model of a viscoelastic body , 2003 .

[11]  Teodor M. Atanackovic,et al.  Distributed-order fractional wave equation on a finite domain. Stress relaxation in a rod , 2010, 1005.3379.

[12]  Stevan Pilipović,et al.  On a fractional distributed-order oscillator , 2005 .

[13]  R. Bagley,et al.  On the Fractional Calculus Model of Viscoelastic Behavior , 1986 .

[14]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[15]  T. Atanacković.  A modified Zener model of a viscoelastic body , 2002 .

[16]  A. Morro,et al.  Mathematical problems in linear viscoelasticity , 1987 .

[17]  Marina V. Shitikova,et al.  Analysis of Dynamic Behaviour of Viscoelastic Rods Whose Rheological Models Contain Fractional Derivatives of Two Different Orders , 2001 .

[18]  Marina V. Shitikova,et al.  A new method for solving dynamic problems of fractional derivative viscoelasticity , 2001 .

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

[20]  Teodor M. Atanackovic,et al.  Time distributed-order diffusion-wave equation. I. Volterra-type equation , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[21]  A. Hanyga.  Fractional-order relaxation laws in non-linear viscoelasticity , 2007 .

[22]  N. M. Grahovac,et al.  Modelling of the hamstring muscle group by use of fractional derivatives , 2010, Comput. Math. Appl..

[23]  A. Hanyga,et al.  Hamiltonian and Lagrangian theory of viscoelasticity , 2008 .

[24]  Nicole Heymans.  Constitutive equations for polymer viscoelasticity derived from hierarchical models in cases of failure of time-temperature superposition , 2003, Signal Process..

[25]  H. Schiessel,et al.  Applications to Problems in Polymer Physics and Rheology , 2000 .

[26]  T. Atanacković.  A generalized model for the uniaxial isothermal deformation of a viscoelastic body , 2002 .

[27]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[28]  Tom T. Hartley,et al.  Fractional-order system identification based on continuous order-distributions , 2003, Signal Process..