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A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions. (English) Zbl 1227.65075

Fractional differential equations and their novel numerical schemes are investigated. These equations play an important role in applications in physics, chemistry, engineering, finance and other sciences. The model presented here is of one space dimension with fractional sub-diffusion equation with the Riemann-Liouville operator and with Neumann boundary conditions. For deriving new numerical schemes, the authors choose the methodology of order reduction and transform the original problem into an equivalent system of lower order differential equations. For this system a construction of a so-called box-type numerical scheme is done using classical finite differences. Due to the fact that the fractional derivative operator is nonlocal, the authors call it box-type scheme.

First, a new discrete inner product, ${L}^{2}$ norm, ${H}^{1}$ semi-norm and maximum norm are defined. Then the stability estimations for the numerical solution are proved, using the novel technique. Finally, under some smoothing conditions on the continuous solution, and for $0<\alpha <1$ , which represents the fractional derivative of the type $1-\alpha$, the error estimate of the type $C\left({\tau }^{2-\alpha }+{h}^{2}\right)$ in the maximum norm is proved for $\tau$ and $h$ time and space grid mesh size, respectively. Two numerical test examples, which confirm the accuracy and efficiency of the proposed scheme, conclude the paper.

##### MSC:
 65M06 Finite difference methods (IVP of PDE) 65M15 Error bounds (IVP of PDE) 65M12 Stability and convergence of numerical methods (IVP of PDE) 35R11 Fractional partial differential equations