Semilinear ordinary differential equation coupled with distributed order fractional differential equation

Nonlinear Analysis, Theory, Methods and Applications

Volumn 72, Issue 11, 2010, Pages 4101-4114

  Atanackovic, Teodor M ^a   Oparnica, Ljubica ^b   Pilipović, Stevan ^a  

^a UNIVERSITY OF NOVI SAD   (Serbia)

^b MATHEMATICAL INSTITUTE   (Serbia)

Author keywords

Fractional differentiation; Schauder fixed point theorem; Tempered distributions

Indexed keywords

CLASSICAL SOLUTIONS; EXISTENCE AND UNIQUENESS; FRACTIONAL DIFFERENTIAL EQUATIONS; FRACTIONAL DIFFERENTIATION; LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS; SCHAUDER FIXED-POINT THEOREM; SEMILINEAR; SUFFICIENT CONDITIONS; TEMPERED DISTRIBUTIONS;

ORDINARY DIFFERENTIAL EQUATIONS;

MATHEMATICAL OPERATORS;

EID: 77949542095     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2010.01.042     Document Type: Article

References (30)

1. Truesdell C., and Noll W. The Non-Linear Field Theories of the Mechanics. Handbuch der Physik Band III/3 (1965), Springer-Verlag, Berlin 1-602
2. Atanackovic T.M. A generalized model for the uniaxial isothermal deformation of a viscoelastic body. Acta Mech. 30 (2002) 1-10
3. Bagley R.L., and Torvik P.J. On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. (1984) 294-298
4. Caputo M., and Mainardi F. A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91 (1971) 134-147
5. Caputo M. Linear models of disipation whose q is almost frequency independent. Geophys. J. R. Astron. Soc. 13 (1967) 529-539
6. Atanackovic T.M. On a distributed derivative model of viscoelastic body. CRAS Mech. 331 (2003) 687-692
7. Atanackovic T.M., Budimčević M., and Pilipović S. On a fractional distributed-order oscilator. J. Phys. A: Math. Gen. 38 (2005) 6703-6713
8. Atanackovic T.M., and Pilipović S. On a class of equations arising in linear viscoelasticity theory. Z. Angew. Math. Mech. 85 10 (2005) 748-754
9. Caputo M. Distributed order differential equation modeling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4 (2001) 421-442
10. Atanackovic T.M., Pilipović S., and Zorica D. Time distributed diffusion-wave equation I. Volterra-type equation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009) 1869-1891
11. Atanackovic T.M., Pilipović S., and Zorica D. Time distributed diffusion-wave equation II. Applications of Laplace and Fourier transformations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009) 1893-1917
12. Chechkin A.V., Gorenflo R., and Sokolov I.M. Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equation. Phys. Rev. E 66 046129 (2002) 1-6
13. Chechkin A.V., Gorenflo R., Sokolov I.M., and Gonchar V.Y. Distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal. 6 (2003) 259-279
14. F. Mainardi, G. Pagnini, The role of the fox-wright function in fractional sub-diffusion of distributed order, J. Comput. Appl. Math., 207, pp. 245-257.
15. F. Mainardi, G. Pagnini, R. Gorenflo, Some aspects of fractional diffusion equation of single and distributed order, Appl. Math. Comput., 187, pp. 295-305.
16. Bagely R.L., and Torvik P.J. On the existence of the order domain and the solution of distributed order equations I and II. Int. J. Appl. Math. 2 (2000) 865-882 pp. 965-987
17. Hartley T.T., and Lorenzo C.F. Fractional-order system identification based on continuous order-distributions. Appl. Anal. 86 (2007) 1347-1363
18. Kilbas A.A., Srivastava H.M., and Trujillo J.J. Theory and Applications of Fractional Differential Equations (2006), Elsevier, Amsterdam
19. S. Kempfle, I. Schäfer, H. Beyer, Fractional calculus via functional calculus: Theory and applications, Nonlinear Dynam., 29, pp. 99-127.
20. Diethelm K., and Ford N.J. Numerical analysis for distributed-order differential equations. J. Comput. Appl. Math. 225 1 (2009) 96-104
21. A.N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340, pp. 252-281.
22. Nakhushev A.M. Fractional Calculus and its Applications (2003), Fizmatlit, Moscow
23. Atanackovic T.M., Oparnica Lj., and Pilipović S. On a nonlinear distributed order fractional differential equation. J. Math. Anal. Appl. 328 (2007) 590-608
24. Delbosco D., and Rodino L. Existence and uniqueness for nonlinear fractional differential equation. J. Math. Anal. Appl. 204 (1995) 609-625
25. Podlubny I. Fractional Differential Equations (1999), Academic Press, San Diego
26. Samko S.G., Kilbas A.A., and Marichev O.I. Fractional Integrals and Derivatives, Theory and Applications (1993), Gordon and Breach Science Publishers, Amsterdam
27. Vladimirov V.S. Generalized Function in Mathematical Physics (1979), Mir, Moscow
28. Atanackovic T.M., Oparnica Lj., and Pilipović S. Distributional framework for solving fractional differential equation. Integral Transforms Spec. Funct. 20 (2009) 215-222. arXiv:0902.0496
29. Vladimirov V.S., Drozinov Yu.N., and Zavlyalov B.I. Tauberian Theorems for Generalized Functions (1987), Nauka, Moscow
30. Christensen R. Theory of Viscoelasticity (1982), Academic Press, New York