Nonlinear Dynamics

, Volume 93, Issue 4, pp 1911–1922 | Cite as

An analytical framework of 2D diffusion, wave-like, telegraph, and Burgers’ models with twofold Caputo derivatives ordering

  • Imad JaradatEmail author
  • Marwan Alquran
  • Ruwa Abdel-Muhsen
Original Paper


The purpose of the current work is to provide an analytical solution framework based on extended fractional power series expansion to solve 2D temporal–spatial fractional differential equations. For this purpose, we first present a new trivariate expansion endowed with twofold Caputo-fractional derivatives ordering, namely \(\alpha ,\,\beta \in (0,1]\), to study the combined effect of fractional derivatives on both temporal and spatial coordinates. Then, by virtue of this expansion, a parallel scheme of the Taylor power series solution method is utilized to extract both closed-form and supportive approximate series solutions of 2D temporal–spatial fractional diffusion, wave-like, telegraph, and Burgers’ models. The obtained closed-form solutions are found to be in harmony with the exact solutions exist in the literature when \(\alpha =\beta =1\), which exhibits the legitimacy and the validity of the proposed method. Moreover, the accuracy of the approximate series solutions is validated using graphical and tabular tools. Finally, a version of Taylor’s Theorem that associated with our proposed expansion is derived in terms of mixed fractional derivatives.


Fractional derivatives (Fractional)Taylor series expansion Fractional partial differential equations Series solutions 

Mathematics Subject Classification

26A33 41A58 35R11 35C10 


Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.


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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Imad Jaradat
    • 1
    Email author
  • Marwan Alquran
    • 1
  • Ruwa Abdel-Muhsen
    • 1
  1. 1.Department of Mathematics and StatisticsJordan University of Science and TechnologyIrbidJordan

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