Nonlinear Dynamics

, Volume 92, Issue 2, pp 395–413 | Cite as

Local discontinuous Galerkin method for distributed-order time and space-fractional convection–diffusion and Schrödinger-type equations

  • Tarek Aboelenen
Original Paper


We develop a local discontinuous Galerkin finite element method for the distributed-order time and Riesz space-fractional convection–diffusion and Schrödinger-type equations. The stability of the presented schemes is proved and optimal order of convergence \(\mathcal {O}(h^{N+1}+(\Delta t)^{1+\frac{\theta }{2}}+\theta ^{2})\) for the Riesz space-fractional diffusion and Schrödinger-type equations with distributed order in time, an order of convergence of \(\mathcal {O}(h^{N+\frac{1}{2}}+(\Delta t)^{1+\frac{\theta }{2}}\) \(+\theta ^{2})\) is provided for the Riesz space-fractional convection–diffusion equations with distributed order in time where h, \(\theta \) and \(\Delta t\) are space step size, the distributed-order variables and the step sizes in time, respectively. Finally, the performed numerical examples confirm the optimal convergence order and illustrate the effectiveness of the method.


Fractional convection–diffusion equations with distributed order in time Fractional Schrödinger-type equations with distributed order in time Local discontinuous Galerkin method Stability Optimal convergence 


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

Authors and Affiliations

  1. 1.Department of MathematicsAssiut UniversityAssiutEgypt

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