An equation with fractional Riesz derivatives with respect to two spatial variables with several delay variables is considered. The problem is characterized by a nonlinear dependence of the diffusion coefficient on the unknown function. Constructions of a nonlinear analog of the alternating directions method with piecewise linear interpolation and extrapolation by continuation are presented. The approximation of the nonlinear diffusion coefficient is performed taking into account the approximate values of the unknown function on the previous time layers. Unique solvability of the algorithm is shown. It is proved that under certain conditions of smoothness of the solution of the original problem, the order of the local error is the second with respect to spatial and temporal discretization steps. The stability of the algorithm is investigated. A theorem on the order of convergence in the energy norm is proved. Result of numerical experiment is presented.
Keywords: fractional derivatives of Riesz, superdiffusion equation, nonlinear diffusion coefficient, delay, alternating direction method, interpolation, extrapolation, stability, convergence
Received November 8, 2025
Revised February 10, 2026
Accepted February 16, 2026
Vladimir Germanovich Pimenov, Dr. Phys.-Math. Sci., Prof., Ural Federal University, Yekaterinburg, 600000 Russia; Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia, e-mail: v.g.pimenov@urfu.ru
Andrey Valentinovich Lekomtsev, Cand. Sci (Phys.-Math.), Associate Prof., Ural Federal University, Yekaterinburg, 600000 Russia, e-mail: avlekomtsev@urfu.ru
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Cite this article as: V.G. Pimenov, A.V. Lekomtsev. Метод переменных направлений для нелинейного супердиффузионного уравнения с запаздыванием. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 2, pp. 186–202.
