The case study is not itself a research method, but researchers select methods of data collection and analysis that will generate material suitable for case studies. In design of experiments, single-subject design or single-case research design is a research design most often used in applied fields of psychology, education, and human behavior in which the subject serves as his/her own control, rather than using another individual/group. Multiple-case design, or collective case design, refers to case study research in which several instrumental bounded cases are selected to develop a more in-depth understanding of the phenomena than a single case can provide. A difference between a single case study and a multiple case study is that in multiple case studies, the researcher are studying multiple cases to understand the differences and the similarities between the cases. Another difference is that the researcher is able to analyse the data both within each situation and across situations. Multiple case studies can be used to either augur contrasting results for expected reasons or either augur similar results in the studies (Yin, 2003). Please see the following article for further reference:
Single case studies vs. multiple case studies: A comparative study Johanna Gustafsson, Academy of Business, Engineering and Science Halmstad University