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Solution to fractional pseudoparabolic equation with fractional integral condition

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Abstract

In this paper we are concerned with the application of Rothe time discretization scheme to find an approximate solution for certain time fractional pseudoparabolic equation with fractional integral condition. Existence and uniqueness of weak solution as well as some regularity results are obtained.

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References

  1. Agrawal, O.P.: Solution for fractional diffusion wave equation defined in bounded domain. Nonlinear Dyn. 29, 145–155 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anh, V.V., Leonenko, N.N.: Spectral analysis of fractional kinetic equations with random data. J. Stat. Phys. 104, 1349–1387 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bagley, R.L.: Theorical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27(03), 201–210 (1983)

    Article  MATH  Google Scholar 

  4. Bahuguna, D., Abbas, S., Dabas, J.: Partial functional differential equation with an integral condition and applications to population dynamics. Nonliear Anal. 69, 2623–2635 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bahuguna, D., Raghavendra, V.: Rothe’s method to parabolic integrodiferential equation via abstact integrodifferential equation. Appl. Anal 33, 153–167 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bouziani, A., Merazga, N.: Solution to a semilinear pseudoparabolic problem with integral conditions. EJDE 115, 1–18 (2006)

    MATH  Google Scholar 

  7. Chaoui, A., Guezane Lakoud, A.: Rothe-Galerkin’s method for a nonlinear integrodifferential equation. Bound. Value Probl. 2012, 10 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chaoui, A., Guezane, A.: Lakoud, solution to an integrodifferential equation with integral codition. Appl. Math. Comput. 266, 903–908 (2015)

    MathSciNet  Google Scholar 

  9. Coleman, B.D., Noll, W.: Approximation theorem for functionals, with applications in continuum mechanics. Arch. Anal. 6, 355–370 (1960)

    MathSciNet  MATH  Google Scholar 

  10. El-Azab, M.S.: Solution of nonlinear transport diffusion problem by linearisation. Appl. Math. Comput. 192, 205–2015 (2007)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Engeita, N.: On fractional calculus and fractional mulipoles in electromagnetism. IEEE Trans. 44(4), 554–566 (1996)

    Article  MathSciNet  Google Scholar 

  13. Evans, L.C.: Weak convergence methods for nonlinear partial differential equations. In: Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, vol. 74, American Mathematical Society, Providence, RI (1990)

  14. Guezane-Lakoud, A., Belakroum, D.: Rothe’s method for a telegraph equation with integral conditions. Nonlinear Anal. 70, 3842–3853 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guezane-Lakoud, A., Jasmati, M.S., Chaoui, A.: Rothe’s method for an integrodifferential equation with integral conditions. Nonlinear Anal. 72, 1522–1530 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hilfer, R.: Application of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  17. Kacur, J.: Method of Roth in evolution equations, In: Teubner Texte zur Mathematik, vol. 80, Teubner, Leipzig (1985)

  18. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. ElsevierScience B. V, Amesterdam (2006)

    MATH  Google Scholar 

  19. Kuliev, K., Petersson, L.-E.: An extension of Rothe’s method to non-cylindrical domains. Appl. Math. 52(5), 365–389 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ladyzenskaja, O.A.: On solution of nonstationary operator equations. Math. Sb. 39(4), 491–524 (1956)

    Google Scholar 

  21. Ladyzenskaja, O.A., Ural’ceva, N.N.: Boundary problems for linear and quasilinear parabolic equations. Am. Math. Soc. Transl. Ser. 2 47, 217–299 (1956)

    Google Scholar 

  22. Magin, R.: Fractional calculus in bioengeenering. Crit. Rev. Biomed. Eng. 32(1), 1–104 (2004)

    Article  Google Scholar 

  23. Mainardi, F.: Fractal and Fractional Calculus in Cintinuum Mechanics. Springer, New York (1997)

    Google Scholar 

  24. Mainardi, F., Paradisi, P.: Model of diffusion waves in viscoelasticity based on fractal calculus. In: Gonzales, O.R. (eds.) Proceedings of IEEE Conference of decision and Control, vol. 5, pp. 4961–4966. IEEE, New york (1997)

  25. Metzler, R., Klafter, J.: The random walk’s guide to anoumalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MATH  Google Scholar 

  26. Mophou, Gisèle M., N’Guérékata, Gaston M.: On class of fractional differential equations in Sobolev space. Appl. Anal.: Int. J. 91(1), 15–34 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nishimoto, K.: Fractional Calculus and Its Applications. Nihon University, Koriyama (1990)

    Google Scholar 

  28. Oldham, K.B.: Fractional Differential equations in electrochemistry. Adv. Eng. Softw. 41(1), 9–12 (2010)

    Article  MATH  Google Scholar 

  29. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic press, New york (1974)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  31. Rektorys, K.: On application of direct variational methods to the solution of parabolic boundary value problems of arbitrary order in space variables. Czechoslov. Math. J. 21, 318–339 (1971)

    MathSciNet  MATH  Google Scholar 

  32. Rektorys, K.: The Method of Discretization in Time and Partial Differential Equations. D. Reidel Publishing Company, Dordrecht (1982)

    MATH  Google Scholar 

  33. Rothe, E.: Zweidimensionale parabolische Randwertaufgaben als Grenz fall eindimonsionaler Randwertaufgaben. Math. Ann. 102, 650–670 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sabatier, J., Agrawl, O.P., Machado, J.A.T.: Advances in Fractional Calculus. Springer, New York (2007)

    Book  Google Scholar 

  35. Ting, T.W.: Certain non-steady flows of second order fluids. Arch. Ration. Mech. Anal. 14, 1–26 (1963)

    Article  MathSciNet  MATH  Google Scholar 

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Chaoui, A., Rezgui, N. Solution to fractional pseudoparabolic equation with fractional integral condition. Rend. Circ. Mat. Palermo, II. Ser 67, 205–213 (2018). https://doi.org/10.1007/s12215-017-0306-x

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