Geodesic
Numerical methods for systems of diffusion and superdiffusion equations with Neumann boundary conditions and with delay
Dalʹnevostočnyj matematičeskij žurnal, Tome 22 (2022) no. 2, pp. 218-224

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A feature of many mathematical models is the presence of two equations of the diffusion type with Neumann boundary conditions and the delay effect, for example, in the model of interaction between a tumor and the immune system. In this paper we construct and study the orders of convergence of analogues of the implicit method and the Crank-Nicolson method. Also, for a system of space fractional superdiffusion-type equations with delay and Neumann boundary conditions, an analogue of the Crank-Nicolson method is constructed. To approximate the two-sided fractional Riesz derivatives, the shifted Grunwald-Letnikov formulas are used; to take into account the delay effect, interpolation and extrapolation of the discrete history of the model are used. 
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     author = {V. G. Pimenov and A. B. Lozhnikov and M. Ibrahim},
     title = {Numerical methods for systems of diffusion and superdiffusion  equations with {Neumann} boundary conditions and with delay},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {218--224},
     publisher = {mathdoc},
     volume = {22},
     number = {2},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2022_22_2_a15/}
}
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V. G. Pimenov; A. B. Lozhnikov; M. Ibrahim. Numerical methods for systems of diffusion and superdiffusion  equations with Neumann boundary conditions and with delay. Dalʹnevostočnyj matematičeskij žurnal, Tome 22 (2022) no. 2, pp. 218-224. http://geodesic.mathdoc.fr/item/DVMG_2022_22_2_a15/