Volume 18, Issue 2

We introduce a method for the evaluation of singular integrals arising in the discretisation of integral equations. The method is based on the idea of repeated subdivision of domains. The integrals defined on these subdomains are classified. Each class can be expressed as a sum of regular integrals and representatives of other classes. A system of equations describes the relations between the classes. Therefore the approximate value of the singular integral only depends on the accuracy of the calculation of regular integrals.

This paper is concerned with computable estimates of the difference between exact solutions of initial-boundary value problems generated by the Stokes equation and an arbitrary function from the corresponding energy space. They provide guaranteed upper bounds of errors in terms of weighted norms defined on the space-time cylinder where the exact solution is considered. Estimates are derived with the help of techniques based on a transformation of integral identities that was earlier applied (see [Repin, A posteriori Estimates for Partial Differential Equations, Walter de Gruyter, 2008] and the references therein) to the stationary Stokes problem. In this paper, two types of error majorants are derived. They are explicitly computable and contain only global constants. It is proved that the estimates vanish if and only if the functions considered coincide with exact solutions.

Performing numerical analysis in elastoplasticity, in several situations one observes plastic regions to cause difficulties in deriving nearly optimal error estimates under adequate regularity assumptions. In this note, for a typical model problem we propose an alternative estimate for the discretisation error localised to these critical parts. The discretisation error, measured locally in terms of stresses, is controlled by an a priori interpolation result and an a posteriori consistency estimate. The interpolation part possesses optimal order convergence in terms of the mesh size together with an adequate regularity assumption on the stresses. The consistency part is fully computable and does not contain heuristics.

Demand-forecasting problems frequently arise in logistics and supply chain management. The Newsboy problem is one such problem. In this paper, we present an improved solution method by application of the Black–Scholes model incorporating a stochastic process used in financial engineering for option pricing. The proposed model is shown to be effective through numerical experiments using real-world data.