On Mathematical Aspects of Evolution of Dislocation Density in Metallic Materials

This paper deals with the solution of delay differential equations describing evolution dislocation density in metallic materials. Hardening, restoration, and recrystallization characterizing populations provide essential equation model. The last term transforms ordinary (ODE) into (DDE) strong (in general, H\"older) nonlinearity. We prove upper error bounds for explicit Euler method, under assumption that right-hand side function is H\"older continuous monotone which allows us to compare accuracy other numerical methods our model (e.g. Runge-Kutta), particular when formulas solutions are not known. Finally, we test above results simulations real industrial process.