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Mathematical Analysis of a Cauchy Problem for the Time-Fractional Diffusion-Wave Equation with \( \alpha \in \left( 0,2\right) \)

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Abstract

This paper deals with a theoretical mathematical analysis of a Cauchy problem for the time-fractional diffusion-wave equation in the upper half-plane, \(x\in \mathbb {R}\), \(t\in \mathbb {R}^+\), where the Caputo fractional derivative of order \(\alpha \in \left( 0,2\right) \) is considered. An explicit solution to this Cauchy problem is obtained via separation of variables. A first proof of the validity of the obtained results is provided for a certain kind of initial conditions. Throughout this work a new expression of the solution to this problem and its utility for carrying out rigurous proofs are presented. Finally, several new properties of the solution are obtained.

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Notes

  1. The Beta function is defined as \(\mathcal {B}\left( x,y\right) :=\int \limits _0^{1}(1-r)^{x-1}r^{y-1}dr\), convergent for \(x,y>0\).

References

  1. Agrawal, O.: Fractional variational calculus and the transversality conditions. J. Phys. A 39, 10375–10384 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Almeida, R., Torres, D.: Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1490–1500 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brezis, H.: Analyse Fonctionnelle. Dunod, Paris (2005)

    MATH  Google Scholar 

  4. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. Int. 13(5), 529–539 (1967)

    Article  MathSciNet  Google Scholar 

  5. Chen, W., Ye, K., Sun, H.: Fractional diffusion equations by the Kansa method. Comput. Math. Appl. 59, 1614–1620 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Daftardar-Gejji, V., Bhalekar, S.: Solving fractional diffusion-wave equations using a new iterative method. Fract. Calc. Appl. Anal. 11(2), 193–202 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Heidelberg (2004)

    Google Scholar 

  8. Eidelman, S., Kochubei, A.: Cauchy problem for fractional diffusion equations. J. Differ. Equ. 199(2), 211–255 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ferreira, R., Torres, D.: Isoperimetric problems of the calculus of variations on time scales. Nonlinear Anal. Optim. II Contemp. Math. 514, 123–131 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Goos, D., Reyero, G., Roscani, S., Santillan Marcus, E.: On the initial–boundary–value problem for the time–fractional diffusion equation on the real positive semiaxis. Int. J. Differ. Equ. 2015, 439419 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Gorenflo, R., Loutchko, J., Luchko, Y.: Computation of the Mittag-Leffler function \(E_{\alpha,\beta }\) and its derivative. Fract. Calc. Appl. Anal. 5(4), 491–518 (2002)

    MathSciNet  MATH  Google Scholar 

  12. Gorenflo, R., Luchko, Y., Mainardi, F.: Analytical Properties and Applications of the Wright Function. Fract. Calc. Appl. Anal. 2(4), 383–414 (1999)

    MathSciNet  MATH  Google Scholar 

  13. Hanyga, A.: Multi-dimensional solutions of space-time-fractional diffusion equations, proceedings: mathematical. Phys. Eng. Sci. 458(2018), 429–450 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag–Leffler functions and their applications. J. Appl. Math. 2011, 298628 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Junyi, L., Mingyu, X.: Some exact solutions to Stefan problems with fractional differential equations. J. Math. Anal. Appl. 351, 536–542 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kemppanien, J.: Existence and uniqueness of the solution for a time–fractional diffusion equation with Robin boundary condition. Abstr. Appl. Anal. 2011, 321903 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Kochubei, A.N.: A Cauchy problem for evolution equations of fractional order. Differ. Equ. 25, 967–974 (1989)

    MathSciNet  Google Scholar 

  18. Luchko, Y.: Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Fract. Calc. Appl. Anal. 15, 141–160 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Luchko, Y.: Maximum principle and its application for the time-fractional diffusion equations. Fract. Calc. Appl. Anal. 14(1), 110–124 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Luchko, Y.: Multi–dimensional fractional wave equation and some properties of its fundamental solution. Commun. Appl. Ind. Math. 6, e-485 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Luchko, Y., Matrínez, H., Trujillo, J.: Fractional Fourier transform and some of its applications. Fract. Calc. Appl. Anal. 11(4), 457–470 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Luchko, Y., Rivero, M., Trujillo, J., Velasco, M.: Fractional models, non-locality, and complex systems. Comput. Math. Appl. 59(3), 1048–1056 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)

    Book  MATH  Google Scholar 

  24. Mainardi, F., Luchko, Y., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4(2), 153–192 (2001)

    MathSciNet  MATH  Google Scholar 

  25. Mainardi, F., Mura, A., Pagnini, G.: The M–Wright function in time–fractional diffusion processes: a tutorial survey. Int. J. Differ. Equ. 2010, 104505 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Mainardi, F., Pagnini, G.: The Wright functions as solutions of the time-fractional diffusion equation. Appl. Math. Comput. 141(1), 51–62 (2002)

    MathSciNet  MATH  Google Scholar 

  27. Mainardi, F., Tomirotti, M.: On a special function arising in the time fractional diffusion-wave equation. Transform Methods Spec. Funct., 171–183 (1995)

  28. Masaeva, O.: Dirichlet problem for the generalized Laplace equation with the Caputo derivative. Differ. Equ. 48(3), 449–454 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Meerschaert, M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mentrelli, A., Pagnini, G.: Front propagation in anomalous diffusive media governed by time-fractional diffusion. J. Comput. Phys. 293, 427–441 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Murio, D.: Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 56, 1138–1145 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nane, E.: Fractional Cauchy problems on bounded domains: survey of recent results. Fract. Dyn. Control, 185–198 (2011)

  33. Povstenko, Y.: Fractional heat conduction in a semi-infinite composite body. Commun. Appl. Ind. Math., e-482 (2014)

  34. Al-Refai, M., Luchko, Y.: Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications. Fract. Calc. Appl. Anal. 17(2), 483–498 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Roscani, S., Santillan Marcus, E.: Two equivalent Stefans’s problems for the time fractional diffusion equation. Fract. Calc. Appl. Anal. 16(4), 802–815 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Roscani, S.: A Generalization of the Hopf’s Lemma for the 1-D Moving-Boundary Problem for the Fractional Diffusion Equation and its Application to a Fractional Free–Boundary Problem. arXiv:1502.01209 (2015)

  37. Rossato, R., Lenzi, M.K., Evangelista, L.R., Lenzi, E.K.: Fractional diffusion equation in a confined region: surface effects and exact solutions. Phys. Rev. E76, 032102 (2007)

    Google Scholar 

  38. Schneider, W.R., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 134(30), 129–139 (1989)

    MathSciNet  MATH  Google Scholar 

  39. Wang, J., Lv, L., Zhou, Y.: Boundary value problems for fractional differential equations involving Caputo derivative in Banach spaces. J. Appl. Math. Comput. 38, 209–224 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to express our gratitude towards the referees who reviewed this paper for their useful suggestions and helpful comments. This paper has been sponsored by Project ING349 “Problemas de frontera libre con ecuaciones diferenciales fraccionarias”, from Universidad Nacional de Rosario, Argentina and “Beca Estímulo a las Vocaciones Científicas en el marco del Plan de Fortalecimiento de la Investigación Científica, el Desarrollo Tecnológico y la Innovación en las Universidades Nacionales” from Consejo Interuniversitario Nacional, Argentina.

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Authors and Affiliations

  1. CONICET – Departamento de Matemática – ECEN Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, Avenida Pellegrini 250, (2000), Rosario, Argentina

    Demian Nahuel Goos

  2. Instituto de Estudios Nucleares y Radiaciones Ionizantes Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, Riobamba y Berutti, (2000), Rosario, Argentina

    Demian Nahuel Goos

  3. Departamento de Matemâtica - ECEN Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, Avenida Pellegrini 250, (2000), Rosario, Argentina

    Demian Nahuel Goos & Gabriela Fernanda Reyero

Authors
  1. Demian Nahuel Goos
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  2. Gabriela Fernanda Reyero
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Correspondence to Demian Nahuel Goos.

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Communicated by Luis Vega.

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Goos, D.N., Reyero, G.F. Mathematical Analysis of a Cauchy Problem for the Time-Fractional Diffusion-Wave Equation with \( \alpha \in \left( 0,2\right) \) . J Fourier Anal Appl 24, 560–582 (2018). https://doi.org/10.1007/s00041-017-9527-9

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  • DOI: https://doi.org/10.1007/s00041-017-9527-9

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