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Approximations of Solutions of a Neutral Fractional Integro-Differential Equation

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Abstract

In the present work, we consider a fractional integro-differential equation in an arbitrary separable Hilbert space H. An associated integral equation and a sequence of approximate integral equations is studied. The existence and uniqueness of solutions to every approximate integral equation is obtained by using analytic semigroup and Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. We show the convergence of the solutions using Faedo–Galerkin approximation and demonstrate some convergence results. Finally, an example is considered to show the effectiveness of the obtained theory.

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References

  1. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  2. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publisher, Yverdon (1993)

    MATH  Google Scholar 

  3. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  4. Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  5. Hino, Y., Murakami, S., Naito, T.: Functional differential equations with infinite delay. In: Lecture Notes in Mathematics, vol. 1473. Springer, Berlin (1991)

  6. Hale, J.K., Kato, J.: Phase space for retarded equations with infinite delay. Funkcial. Ekvac. 21, 11–41 (1978)

    MathSciNet  MATH  Google Scholar 

  7. Muslim, M., Agarwal, R.P., Nandakumaran, A.K.: Existence, uniqueness and convergence of approximate solutions of impulsive neutral differential equations. Funct. Differ. Equ. 16, 529–544 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Dubey, R.S.: Approximations of solutions to abstract neutral functional differential equations. Numer. Funct. Anal. Optim. 32, 286–308 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Muslim, M.: Approximation of solutions to history-valued neutral functional differential equations. Int. J. Comput. Math. Appl. 51, 537–550 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hernández, E., Henríquez, H.R.: Existence results for partial neutral functional differential equations with bounded delay. J. Math. Anal. Appl. 221, 452–475 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Muslim, M., Nandakumaran, A.K.: Existence and approximations of solutions to some fractional order functional integral equations. J. Integral Equ. Appl. 22, 95–114 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bahuguna, D., Srivastava, S.K.: Approximation of solutions to evolution integrodifferential equations. J. Appl. Math. Stoch. Anal. 9, 315–322 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bahuguna, D., Srivastava, S.K., Singh, S.: Approximations of solutions to semilinear integrodifferential equations. Numer. Funct. Anal. Optim. 22, 487–504 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bahuguna, D., Shukla, R.: Approximations of solutions to nonlinear Sobolev type evolution equations. Electron. J. Differ. Equ. 31, 1–16 (2003)

    MathSciNet  MATH  Google Scholar 

  15. Kumar, P., Pandey, D.N., Bahuguna, D.: Approximations of solutions to a fractional differential equations with a deviating argument. Differ. Equ. Dyn. Syst. 2013, 20 (2013)

    MathSciNet  Google Scholar 

  16. Chaddha, A., Pandey, D.N.: Approximations of solutions for a Sobolev type fractional order differential equation. Nonlinear Dyn. Syst. Theory 14, 11–29 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Miletta, P.D.: Approximation of solutions to evolution equations. Math. Methods Appl. Sci. 17, 753–763 (1994)

  18. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

  19. Chadha, A., Pandey, D.N.: Existence and approximation of solution to neutral fractional differential equation with nonlocal conditions. Comput. Math. Appl. 69, 893–908 (2015)

  20. Chadha, A., Pandey, D.N.: Faedo-Galerkin approximation of solution for a nonlocal neutral fractional differential equation. Mediterr. J. Math. 1–27. (2016). doi:10.1007/s00009-015-0671-7

  21. Gelfand, I.M., Shilov, G.E.: Generalized Functions, vol. 1. Nauka, Moscow (1959)

    Google Scholar 

  22. El-Borai, M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fract. 14, 433–440 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mainardi, F.: On a Special Function Arising in the Time Fractional Diffusion-Wave Equation: Transform Methods and Special Functions, pp. 171–183. Science Culture Technology, Singopore (1994)

    Google Scholar 

  24. Pollard, H.: The representation of \(e^{-x^\lambda }\) as a Laplace integral. Bull. Am. Math. Soc. 52, 908–910 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. Real World Appl. 11, 4465–4475 (2010)

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Pin-247667, Uttarakhand, India

    Alka Chadha & Dwijendra N. Pandey

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  1. Alka Chadha
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  2. Dwijendra N. Pandey
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Correspondence to Alka Chadha.

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Chadha, A., Pandey, D.N. Approximations of Solutions of a Neutral Fractional Integro-Differential Equation. Differ Equ Dyn Syst 25, 117–133 (2017). https://doi.org/10.1007/s12591-016-0286-x

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  • DOI: https://doi.org/10.1007/s12591-016-0286-x

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