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The fractional Dodson diffusion equation: a new approach

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Abstract

In this paper, after a brief review of the general theory concerning regularized derivatives and integrals of a function with respect to another function, we provide a peculiar fractional generalization of the \((1+1)\)-dimensional Dodson’s diffusion equation. For the latter we then compute the fundamental solution, which turns out to be expressed in terms of an M-Wright function of two variables. Then, we conclude the paper providing a few interesting results for some nonlinear fractional Dodson-like equations.

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Notes

  1. For further details on the M-Wright function, and its applications, we refer the interested reader to a very interesting surveys [21, 23]. From an historical perspective, it is also worth remarking that M-Wright function was first introduced in the occasion of the 7th WASCOM conference, see [19].

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Acknowledgements

The work of the authors has been carried out in the framework of the activities of the National Group of Mathematical Physics (GNFM, INdAM). Moreover, the work of A.G. has been partially supported by GNFM/INdAM Young Researchers Project 2017 “Analysis of Complex Biological Systems”.

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Authors and Affiliations

  1. Department of Statistical Sciences, Sapienza University of Rome, P.le Aldo Moro, 5, Rome, Italy

    Roberto Garra

  2. Department of Physics and Astronomy, University of Bologna and INFN, Via Irnerio 46, Bologna, Italy

    Andrea Giusti & Francesco Mainardi

  3. Arnold Sommerfeld Center, Ludwig-Maximilians-Universität, Theresienstraße 37, Munich, 80333, Germany

    Andrea Giusti

Authors
  1. Roberto Garra
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  2. Andrea Giusti
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  3. Francesco Mainardi
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Correspondence to Francesco Mainardi.

Additional information

Dedicated to Professor Tommaso Ruggeri on the occasion of his 70th anniversary.

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Garra, R., Giusti, A. & Mainardi, F. The fractional Dodson diffusion equation: a new approach. Ricerche mat 67, 899–909 (2018). https://doi.org/10.1007/s11587-018-0354-3

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