Journal of Mathematical Chemistry

, Volume 57, Issue 5, pp 1394–1412 | Cite as

An efficient Hamiltonian numerical model for a fractional Klein–Gordon equation through weighted-shifted Grünwald differences

  • Ahmed S. Hendy
  • Jorge E. Macías-DíazEmail author
Original Paper


In this work, we investigate numerically a nonlinear wave equation with fractional derivatives of the Riesz type in space. As opposed to previously published papers which employed fractional centered differences, the present approach is based on the use of weighted and shifted Grünwald difference operators. The mathematical model has an associated energy function which is preserved under suitable parameter conditions. In this manuscript, we propose a discrete energy function that estimates the continuous counterpart and which is preserved under the same conditions. As some of the main result of this work, we show that the method is stable and second-order convergent. Moreover, we establish that the technique is quadratically consistent, and we prove the existence and uniqueness of solutions of the numerical model for arbitrary initial conditions. Some numerical results are provided in order to confirm the quadratic order of convergence of the method.


Fractional wave equation Riesz space-fractional equations Weighted and shifted Grünwald differences Hamiltonian numerical model Convergence and stability 



For the first author, this work was supported by Government of the Russian Federation Resolution 211 of March 16, 2013. The authors wish to thank the anonymous reviewers and the editor in charge of handling this paper for their suggestions and criticisms. Their comments helped substantially in improving the quality of this manuscript.


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceBenha UniversityBenhaEgypt
  2. 2.Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and MathematicsUral Federal UniversityYekaterinburgRussia
  3. 3.Departamento de Matemáticas y FísicaUniversidad Autónoma de AguascalientesAguascalientesMexico

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