A Time-Fractional Borel–Pompeiu Formula and a Related Hypercomplex Operator Calculus

  • M. Ferreira
  • M. M. Rodrigues
  • N. VieiraEmail author


In this paper we develop a time-fractional operator calculus in fractional Clifford analysis. Initially we study the \(L_p\)-integrability of the fundamental solutions of the multi-dimensional time-fractional diffusion operator and the associated time-fractional parabolic Dirac operator. Then we introduce the time-fractional analogues of the Teodorescu and Cauchy–Bitsadze operators in a cylindrical domain, and we investigate their main mapping properties. As a main result, we prove a time-fractional version of the Borel–Pompeiu formula based on a time-fractional Stokes’ formula. This tool in hand allows us to present a Hodge-type decomposition for the forward time-fractional parabolic Dirac operator with left Caputo fractional derivative in the time coordinate. The obtained results exhibit an interesting duality relation between forward and backward parabolic Dirac operators and Caputo and Riemann–Liouville time-fractional derivatives. We round off this paper by giving a direct application of the obtained results for solving time-fractional boundary value problems.


Fractional Clifford analysis Fractional derivatives Time-fractional parabolic Dirac operator Fundamental solution Borel–Pompeiu formula 

Mathematics Subject Classification

Primary 30G35 Secondary 35R11 26A33 35A08 30E20 45P05 



The work of M. Ferreira, M.M. Rodrigues and N. Vieira was supported by Portuguese funds through CIDMA-Center for Research and Development in Mathematics and Applications, and FCT–Fundação para a Ciência e a Tecnologia, within project UID/MAT/04106/2019. The work of the authors was supported by the project New Function Theoretical Methods in Computational Electrodynamics / Neue funktionentheoretische Methoden für instationäre PDE, funded by Programme for Cooperation in Science between Portugal and Germany (“Programa de Ações Integradas Luso-Alemãs 2017” - DAAD-CRUP - Acção No. A-15/17 / DAAD-PPP Deutschland-Portugal, Ref: 57340281). N. Vieira was also supported by FCT via the FCT Researcher Program 2014 (Ref: IF/00271/2014).


  1. 1.
    Ahmad, B., Alhothuali, M.S., Alsulami, H.H., Kirane, M., Timoshin, S.: On a time fractional reaction diffusion equation. Appl. Math. Comput. 257, 199–204 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 10th printing. National Bureau of Standards, Wiley, New York (1972)zbMATHGoogle Scholar
  3. 3.
    Boyadjiev, L., Luchko, Y.: Multi-dimensional \(\alpha \)-fractional diffusion-wave equation and some properties of its fundamental solution. Comput. Math. Appl. 73(12), 2561–2572 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boyadjiev, L., Luchko, Y.: Mellin integral transform approach to analyze the multidimensional diffusion-wave equations. Chaos Solitons Fractals 102, 127–134 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    de Carvalho-Neto, P.M., Planas, G.: Mild solutions to the time fractional Navier–Stokes equations in \({\mathbb{R}}^n\). J. Differ. Equ. 259(7), 2948–2980 (2015)CrossRefzbMATHGoogle Scholar
  6. 6.
    Cerejeiras, P., Vieira, N.: Regularization of the non-stationary Schrödinger operator. Math. Methods Appl. Sci. 32(5), 535–555 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cerejeiras, P., Kähler, U., Sommen, F.: Parabolic Dirac operators and the Navier–Stokes equations over time-varying domains. Math. Methods Appl. Sci. 28(14), 1715–1724 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, Z.Q., Kim, K.H., Kim, P.: Fractional time stochastic partial differential equations. Stoch. Processes Appl. 125(4), 1470–1499 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Delanghe, R., Sommen, F., Souc̆ek, V.: Clifford Algebras and Spinor-Valued Functions. A Function Theory for the Dirac Operator. Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (1992)CrossRefGoogle Scholar
  10. 10.
    Ferreira, M., Vieira, N.: Fundamental solutions of the time fractional diffusion-wave and parabolic Dirac operators. J. Math. Anal. Appl. 447(1), 329–353 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gürlebeck, K., Sprößig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Mathematical Methods in Practice. Wiley, Chichester (1997)zbMATHGoogle Scholar
  12. 12.
    Hanyga, A.: Multidimensional solutions of time-fractional diffusion-wave equations. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 458(2020), 933–957 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hanyga, A.: Multi-dimensional solutions of space-time-fractional diffusion equations. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 458(2018), 429–450 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Idczak, D., Walczak, S.: Fractional Sobolev spaces via Riemann-Liouville derivatives. J. Funct. Spaces Appl. (2013).
  15. 15.
    Kilbas, A.A., Srivastava, H.M.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  16. 16.
    Kilbas, A., Saigo, M.: H-transforms. Theory and Applications. Analytical Methods and Special Functions, vol. 9. Chapman & Hall, Boca Raton (2004)zbMATHGoogle Scholar
  17. 17.
    Kim, K.H., Lim, S.: Asymptotic behaviours of fundamental solution and its derivatives to fractional diffusion-wave equations. J. Korean Math. Soc. 53(4), 929–967 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lin, S., Azaïez, M., Xu, C.: Fractional Stokes equation and its spectral approximation. Int. J. Numer. Anal. Model. 15(1–2), 170–192 (2018)MathSciNetGoogle Scholar
  19. 19.
    Luchko, Y.: On some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation. Mathematics 5(4), Article ID: 76 (2017)Google Scholar
  20. 20.
    Luchko, Y.: Multi-dimensional fractional wave equation and some properties of its fundamental solution. Commun. Appl. Ind. Math., 6(1), Article ID 485 (2014)Google Scholar
  21. 21.
    Luchko, Y.: Fractional wave equation and damped waves. J. Math. Phys., 54(3), Article ID: 031505 (2013)Google Scholar
  22. 22.
    Lorenzi, A., Sinestrari, E.: An inverse problem in the theory of materials with memory. Nonlinear Anal. Theory Methods Appl. 12(12), 1317–1335 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A, Math. Gen. 37(31), R161–R208 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Meerschaert, M.M., Sikorskii, A.: Stochastic Models for Fractional Calculus. Gruyter Studies in Mathematics, vol. 43. de Gruyter, Berlin (2012)zbMATHGoogle Scholar
  25. 25.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mikhlin, S.G., Prössdorf, S.: Singular Integral Operators. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  27. 27.
    Roscani, S.D., Tarzia, D.A.: A generalized Neumann solution for the two-phase fractional Lamé-Clapeyron-Stefan problem. Adv. Math. Sci. Appl. 24(2), 237–249 (2014)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382(1), 426–447 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York, NY (1993)zbMATHGoogle Scholar
  30. 30.
    Schumer, R., Benson, D.A., Meerschaert, M.M., Wheatcraft, S.W.: Eulerian derivation of the fractional advection-dispersion equation. J. Contam. Hydrol. 48(1–2), 69–88 (2001)CrossRefGoogle Scholar
  31. 31.
    Tarasov, V.E.: Fractional Dynamics. Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Nonlinear Physical Science. Springer, Berlin (2010)zbMATHGoogle Scholar
  32. 32.
    Wyss, W.: The fractional diffusion equation. J. Math. Phys. 27(11), 2782–2785 (1986)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Technology and ManagementPolytechnic Institute of LeiriaLeiriaPortugal
  2. 2.CIDMA - Center for Research and Development in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal

Personalised recommendations