Difference methods of the solution of local and non-local boundary value problems for loaded equation of thermal conductivity of fractional order. (English) Zbl 1452.80019

Beshtokov, M. H.; Khudalov, M. Z.

Tarasyev, Alexander (ed.) et al., Stability, control and differential games. Proceedings of the international conference on stability, control and differential games (SCDG2019), Yekaterinburg, Russia, September 16–20, 2019. Cham: Springer. Lect. Notes Control Inf. Sci. – Proc., 187-201 (2020).

Summary: We study local and non-local boundary value problems for a one-dimensional space-loaded differential equation of thermal conductivity with variable coefficients with a fractional Caputo derivative, as well as difference schemes approximating these problems on uniform grids. For the solution of local and non-local boundary value problems by the method of energy inequalities, a priori estimates in differential and difference interpretations are obtained, which implies the uniqueness and stability of the solution from the initial data and the right side, as well as the convergence of the solution of the difference problem to the solution of the corresponding differential problem at the rate of \(O(h^2+\tau^2)\).
For the entire collection see [Zbl 1444.93003].

MSC:

80M20	Finite difference methods applied to problems in thermodynamics and heat transfer
35B45	A priori estimates in context of PDEs
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
35R11	Fractional partial differential equations
26A33	Fractional derivatives and integrals
35Q79	PDEs in connection with classical thermodynamics and heat transfer

Keywords:

thermal conductivity; boundary value problem; finite difference method; a priori error

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