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The purpose of this paper is the application of the finite difference method (FDM) for numerical modeling of the microscale heat transfer processes occurring in the domain of thin metal film subjected to a laser pulse. The problem discussed is described by the different variants of the second-order dual-phase-lag equation (DPLE). The laser action is taken into account by the introduction of internal volumetric heat source to the governing equation. The capacity of the source is dependent on the geometrical co-ordinates and duration of the laser beam. The modified forms of DPLE presented in the paper, resulting from the certain substitutions introduced to the basic equation.

At the stage of numerical computations, the different variants of the FDM are applied. Both the explicit and implicit FDM schemes are used. The formula determining the capacity of the internal heat source suggests the formulation of the task discussed using the cylindrical co-ordinate system. The in-house programs realizing the numerical computations concern the axially-symmetrical tasks. In this paper, the metal films made of the nickel and gold are considered.

The algorithms presented make possible to analyze the heating/cooling processes occurring in the domain of metal film having a thickness Z for the different laser parameters (laser intensity, characteristic time of laser pulse and laser beam radius) and the different materials (optical penetration depth, reflectivity of irradiated surface, lag times, thermal conductivity and volumetric specific heat).

Not for all metals, one can find information on lag times. In the literature, analytical formulas can be found to calculate these values, but they are strongly approximated. It should be pointed out that there are some limitations concerning the delay times of material considered, which assure the physical correctness of the second-order DPLE.

The FDM algorithm concerns the three-dimensional cylindrical domain while a large majority of the second-order DPLE numerical solutions have been obtained for the one-dimensional tasks. Both the implicit and explicit numerical schemes are proposed and the testing computations confirm the correctness and effectiveness of the algorithms presented.

The knowledge of goals of the successor, who is preparing to take over the business, is extremely important for the succession process and further operation of a family enterprise. The aim of this study is to structure the goals of Polish family enterprises’ successors and to develop a Synthetic Model of the goals of Polish family enterprises' successors with the application of grey clustering evaluation models.

Research into the specifics of the diagnosis and assessment of the goals set for the successors of the first succession in family businesses in Poland was carried out in the third quarter of 2016 at two stages using two research methods: in-depth group interview and individual interview. The main aim of the first stage was the extraction of subjectively identified goals by family enterprises' successors (based on their succession experience). The statements were open and obtained during two in-depth group interviews (2 FGI) with successors being in the process of succession at its various stages (total, n = 14). The respondents presented their experiences connected with the succession process along with emotions that are associated with it. In one of the interview stages, the respondents were asked to enumerate their individual goals they set for themselves in the context of upcoming changes. Next, the group agreed on the most frequently mentioned goals by creating their verbal interpretation. The obtained list of 20 goals was recorded and discussed, and thanks to the application of the elimination rule in the collective decision-making process, that list was reduced to 10 goals, which was approved by all participating successors.

The results show the developed Synthetic Model of the goals of family enterprises’ successors. The study singled out four groups of successors: (1) successors who do not work in the family enterprise yet, (2) successors holding lower-level positions, (3) successors holding managerial positions, (4) successors who manage the entire company. As a result of the calculations, the developed Synthetic Model of the goals of family enterprises' successors was positively verified for successors working in higher-level positions and successors managing the entire family enterprise.

In order to use the results of clustering, e.g. for conducting studies on large samples with the use of statistical tools, a reduced number of goals should be taken into account. A thorough study of three goals may bring results similar to the study of the original ten successors of Polish family enterprises in the process of succession. The aim of future research is to develop a mathematical model using optimization functions that enable selection of elements representing individual clusters in such a way that it leads to the extraction of the elements with the highest value in relation to the accepted criterion for assessing their value.

In the future, conducting family business research in accordance with the developed methodology requires a look at the proposed list of successor goals obtained during the Focus Group Interview (FGI) as it could be shortened using the Cluster of Grey Incidence method. Shortening the list of goals has its analytic and practical justifications. The study of the full list of goals in subsequent (and numerous further studies) could lead to errors related to, for example, different interpretation of goals among the investigated successors. Furthermore, the full list of goals would increase costs and extend research time.