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Existence and uniqueness results for Cauchy problem of variable-order fractional differential equations

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Abstract

In this paper, the existence and uniqueness results of variable-order fractional differential equations (VOFDEs) are studied. The variable-order fractional derivative is defined in the Caputo sense, and the fractional order is a bounded function. The existence result of Cauchy problem of VOFDEs is obtained by constructing an iteration series which converges to the analytical solution. The uniqueness result is obtained by employing the contraction mapping principle. Since the variable-order fractional derivatives contain classical and fractional derivatives as special cases, many existence and uniqueness results of references are significantly generalized. Finally, we draw some conclusions of variable-order fractional calculus, and two examples are given for demonstrating the theoretical analysis.

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References

  1. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Book  MATH  Google Scholar 

  2. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  3. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  4. Gionaa, M., Romanb, H.E.: Fractional diffusion equation for transport phenomena in random media. Physica A 185, 87–97 (1992)

    Article  Google Scholar 

  5. Bertin, K., Torres, S., Tudor, C.A.: Drift parameter estimation in fractional diffusions driven by perturbed random walks. Stat. Probab. Lett. 81, 243–249 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Müller, S., Kästner, M., Brummund, J., Ulbricht, V.: A nonlinear fractional viscoelastic material model for polymers. Comput. Mater. Sci. 50, 2938–2949 (2011)

    Article  Google Scholar 

  7. Laskin, N.: Fractional market dynamics. Physica A 287, 482–492 (2000)

    Article  MathSciNet  Google Scholar 

  8. Kunsezov, D., Bulagc, A., Dang, G.D.: Quantum Levy processes and fractional kinetics. Phys. Rev. Lett. 82, 1136–1139 (1999)

    Article  Google Scholar 

  9. Gupalo, I.P., Novikov, V.A., Riazantsev, I.S.: Continuous-flow system with fractional order chemical reaction in the presence of axial dispersion. J. Appl. Math. Mech. 45, 213–216 (1981)

    Article  MathSciNet  Google Scholar 

  10. Magin, R.L.: Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl. 59, 1586–1593 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bai, Z.B.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. TMA 72, 916–924 (2010)

    Article  MATH  Google Scholar 

  12. Ahamd, B., Nieto, J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838–1843 (2009)

    Article  MathSciNet  Google Scholar 

  13. Lakshmikantham, V., Leela, S.: A Krasnoselskii-Krein-type uniqueness result for fractional differential equations. Nonlinear Anal. TMA 71(7–8), 3421–3424 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jiao, F., Zhou, Y.: Existence of solutions for a class of fractional boundary value problems via critical point theory. Comput. Math. Appl. 62, 1181–1199 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiao, F., Zhou, Y.: Existence results for fractional boundary value problem via critical point theory. Int. J. Bifurc. Chaos 22(4), 1250086 (2012)

    Article  MathSciNet  Google Scholar 

  16. Samko, S.G.: Fractional integration and differentiation of variable order. Anal. Math. 21, 213–236 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sheng, H., Sun, H.G., Coopmans, C., Chen, Y.Q., Bohannan, G.W.: A physical experimental study of variable-order fractional integrator and differentiator. Eur. Phys. J. 193, 93–104 (2011)

    Google Scholar 

  18. Coimbra, C.: Mechanics with variable-order fractional differential operators. Ann. Phys. 12, 692–703 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sun, H.G., Chen, W., Chen, Y.Q.: Variable-order fractional differential operators in anomalous diffusion modeling. Physica A 388(21), 4586–4592 (2009)

    Article  Google Scholar 

  20. Valério, D., Costa, J.: Variable-order fractional derivatives and their numerical approximations. Signal Process. 91, 470–483 (2011)

    Article  MATH  Google Scholar 

  21. Umarov, S., Steinberg, S.: Variable order differential equations and diffusion processes with changing modes. arXiv:0903.2524v1 [math-ph] (2009)

  22. Cooper, G., Cowan, D.R.: Filtering using variable order vertical derivatives. Comput. Geosci. 30, 455–459 (2004)

    Article  Google Scholar 

  23. Tseng, C.C.: Design of variable and adaptive fractional order FIR differentiators. Signal Process. 86, 2554–2566 (2006)

    Article  MATH  Google Scholar 

  24. Sheng, H., Sun, H.G., Chen, Y.Q., Qiu, T.S.: Synthesis of multifractional Gaussian noises based on variable-order fractional operators. Signal Process. 91, 1645–1650 (2011)

    Article  MATH  Google Scholar 

  25. Razminiaa, A., Dizajib, A.F., Majda, V.J.: Solution existence for non-autonomous variable-order fractional differential equations. Math. Comput. Model. 55, 1106–1117 (2012)

    Article  Google Scholar 

  26. Hartley, T., Lorenzo, C.: Fractional system identification: an approach using continuous order distribution. NASA/TM-1999-209640 (1999)

  27. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

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Acknowledgements

This work was supported by the Scientific Research Innovation Project for Graduate Students in Hunan Province (No. CX2012B109) and the Project of China Scholarship Council.

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  1. Department of Applied Mathematics, Central South University, Changsha, 410083, China

    Yufeng Xu & Zhimin He

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  1. Yufeng Xu
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  2. Zhimin He
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Correspondence to Yufeng Xu.

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Xu, Y., He, Z. Existence and uniqueness results for Cauchy problem of variable-order fractional differential equations. J. Appl. Math. Comput. 43, 295–306 (2013). https://doi.org/10.1007/s12190-013-0664-2

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  • DOI: https://doi.org/10.1007/s12190-013-0664-2

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