Difference methods for solving nonlocal boundary value problems for fractional-order differential convection-diffusion equations with memory effect

M. KH. Beshtokov ; M. Z. Khudalov

Dalʹnevostočnyj matematičeskij žurnal, Tome 21 (2021) no. 1, pp. 3-25

Résumé

In the present paper, in a rectangular domain, we study nonlocal boundary value problems for one-dimensional in space differential equations of convection-diffusion of fractional order with a memory effect, in which the unknown function appears in the differential expression and at the same time appears under the integral sign. The emergence of the integral term in the equation is associated with the need to take into account the dependence of the instantaneous values of the characteristics of the described object on their respective previous values, i.e. the effect of its prehistory on the current state of the system. For the numerical solution of nonlocal boundary value problems, two-layer monotone difference schemes are constructed that approximate these problems on a uniform grid. Estimates of solutions of problems in differential and difference interpretations are derived by the method of energy inequalities. The obtained a priori estimates imply the uniqueness, as well as the continuous and uniform dependence of the solution on the input data of the problems under consideration and, due to the linearity of the problem under consideration, the convergence of the solution of the difference problem to the solution of the corresponding differential problem with the rate $O(h^2+\tau^2)$.