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On the well-posedness of determination of two coefficients in a fractional integrodifferential equation

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Abstract

The authors study an inverse problem for a fractional integrodifferential equation, which aims to determine simultaneously two time varying coefficients, a kernel function and a source function, from the additional integral overdetermination condition. By using the fixed point theorem in suitable Sobolev space, the global existence and uniqueness results of this inverse problem are obtained.

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References

  1. Adams, R. A., Sobolev Spaces, Academic Press, New York, 1975.

    MATH  Google Scholar 

  2. Ahmad, B. and Nieto J. J., Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, Boundary Value Problems, 2009, 2009, 708576.

    MathSciNet  Google Scholar 

  3. Cheng, J., Nakagawa, J., Yamamoto, M. and Yamazaki, T., Uniqueness in an inverse problem for a onedimensional fractional diffusion equation, Inverse Problems, 25, 2009, 115002.

    Article  MathSciNet  Google Scholar 

  4. Colombo, F., An inverse problem for a parabolic integrodifferential model in the theory of combustion, Phys. D, 236, 2007, 81–89.

    Article  MATH  MathSciNet  Google Scholar 

  5. Colombo, F., An inverse problem for the strongly damped wave equation with memory, Nonlinearity, 20, 2007, 659–683.

    Article  MATH  MathSciNet  Google Scholar 

  6. Colombo, F. and Damiano, A., Identification of the memory kernel and the heat source for a phase-field model, Houston J. Math., 32, 2006, 895–920.

    MATH  MathSciNet  Google Scholar 

  7. Colombo, F. and Guidetti, D., A global in time existence and uniqueness result for a semilinear integrodifferential parabolic inverse problem in Sobolev spaces, Math. Models Methods Appl. Sci., 17, 2007, 537–565.

    Article  MATH  MathSciNet  Google Scholar 

  8. Colombo, F. and Guidetti, D., Identification of the memory kernel in the strongly damped wave equation by a flux condition, Commun. Pure Appl. Anal., 8, 2009, 601–620.

    Article  MATH  MathSciNet  Google Scholar 

  9. Colombo, F., Guidetti, D. and Lorenzi, A., Integrodifferential identification problems for thermal materials with memory in non-smooth plane domains, Dynam. Systems Appl., 12, 2003, 533–559.

    MATH  MathSciNet  Google Scholar 

  10. Colombo, F., Guidetti, D. and Vespri, V., Identification of two memory kernels and the time dependence of the heat source for a parabolic conserved phase-field model, Math. Methods Appl. Sci., 28, 2005, 2085–2115.

    Article  MATH  MathSciNet  Google Scholar 

  11. Courant, R. and Hilbert, D., Methods of Mathematical Physics, Interscience, New York, 1953.

    Google Scholar 

  12. D’Aloia, L. and Chanvillard, G., Determining the apparent activation energy of concrete: Eanumerical simulations of the heat of hydration of cement, Cement and Concrete Research, 32, 2002, 1277–1289.

    Article  Google Scholar 

  13. Janno, J. and Lorenzi, A., Recovering memory kernels in parabolic transmission problems, J. Inverse Ill-Posed Probl., 16, 2008, 239–265.

    Article  MATH  MathSciNet  Google Scholar 

  14. Jin, B. T. and Rundell, W., An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems, 28, 2012, 075010.

    Article  MathSciNet  Google Scholar 

  15. Karthikeyan, K. and Trujillo, J. J., Existence and uniqueness results for fractional integrodifferential equations with boundary value conditions, Commun. Nonlinear Sci. Numer. Simulat., 17, 2012, 4037–4043.

    Article  MATH  MathSciNet  Google Scholar 

  16. Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

    MATH  Google Scholar 

  17. Li, F., Liang, L. and Xu, H. K., Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions, J. Math. Anal. Appl., 391, 2012, 510–525.

    Article  MATH  Google Scholar 

  18. Lin, W., Global existence theory and chaos control of fractional differential equation, J. Math. Anal. Appl., 332, 2007, 709–726.

    Article  MATH  MathSciNet  Google Scholar 

  19. Liu, J. J. and Yamamoto, M., A backward problem for the time-fractional diffusion equation, Appl. Anal., 89, 2010, 1769–1788.

    Article  MATH  MathSciNet  Google Scholar 

  20. Lorenzi, A. and Messina, F., An identification problem with evolution on the boundary of parabolic type, Adv. Diff. Equ., 13, 2008, 1075–1108.

    MATH  MathSciNet  Google Scholar 

  21. Lorenzi, A. and Rocca, E., Identification of two memory kernels in a fully hyperbolic phase-field system, J. Inverse Ill-Posed Probl., 16, 2008, 147–174.

    MATH  MathSciNet  Google Scholar 

  22. Pandey, D. N., Ujlayan, A. and Bahuguna, D., On a solution to fractional order integrodifferential equations with analytic semigroups, Nonlinear Anal., 71, 2009, 3690–3698.

    Article  MATH  MathSciNet  Google Scholar 

  23. Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.

    MATH  Google Scholar 

  24. Sakamoto, K. and Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 2011, 426–447.

    Article  MATH  MathSciNet  Google Scholar 

  25. Sakamoto, K. and Yamamoto, M., Inverse source problem with a final overdetermination for a fractional diffusion equation, Math. Control and Related Fields, 1, 2011, 509–518.

    Article  MATH  MathSciNet  Google Scholar 

  26. Tidke, H. L., Existence of global solutions to nonlinear mixed volterra-fredholm integrodifferential equations with nonlocal conditions, Electron J. Diff. Equ., 55, 2009, 1–7.

    MathSciNet  Google Scholar 

  27. Tomovski, Z and Sandev, T., Fractional wave equation with a frictional memory kernel of Mittag-Leffler type, Appl. Math. Comp., 218, 2012, 10022–10031.

    Article  MATH  MathSciNet  Google Scholar 

  28. Waller, V., D’Aloia, L., Cussigh, F. and Lecrux, S., Using the maturity method in concrete cracking control at early ages, Cement and Concrete Composites, 26, 2004, 589–599.

    Article  Google Scholar 

  29. Wang, J., Zhou, Y. and Wei, T., Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation, Appl. Numer. Math., 68, 2013, 39–57.

    Article  MATH  MathSciNet  Google Scholar 

  30. Wang, L. Y. and Liu, J. J., Data regularization for a backward time-fractional diffusion problem, Comp. Math. Appl., 64, 2012, 3613–3626.

    Article  MATH  Google Scholar 

  31. Wu, B. and Liu, J. J., A global in time existence and uniqueness result for an integrodifferential hyperbolic inverse problem with memory effect, J. Math. Anal. Appl., 373, 2011, 585–604.

    Article  MATH  MathSciNet  Google Scholar 

  32. Yang, M. and Liu, J. J., Solving a final value fractional diffusion problem by boundary condition regularization, Appl. Numer. Math., 66, 2013, 45–58.

    Article  MATH  MathSciNet  Google Scholar 

  33. Zhang, S., Positive solutions for boundary-value problems for nonlinear fractional differential equations, Electron. J. Diff. Equ., 36, 2006, 1–12.

    Article  Google Scholar 

  34. Zhang, Y. and Xu, X., Inverse source problem for a fractional diffusion equation, Inverse Problems, 27, 2011, 035010.

    Article  MathSciNet  Google Scholar 

  35. Zhou, Y. and Jiao, F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal: TMA, 11, 2010, 4465–4475.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Haibing Wang

  2. College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Bin Wu

Authors
  1. Haibing Wang
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  2. Bin Wu
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Corresponding author

Correspondence to Haibing Wang.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11201238, 11301075), the Jiangsu Provincial Natural Science Foundation of China (No. BK20130594) and the Fundamental Research Funds for the Central Universities (No. 3207013101).

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Wang, H., Wu, B. On the well-posedness of determination of two coefficients in a fractional integrodifferential equation. Chin. Ann. Math. Ser. B 35, 447–468 (2014). https://doi.org/10.1007/s11401-014-0832-1

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  • DOI: https://doi.org/10.1007/s11401-014-0832-1

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