In this paper, fractional order calculus is more widely used in teaching evaluation systems due to its non-local application. In this paper, the principle and algorithm of Caputo error estimation of fractional order calculus are proposed, the positive solutions of Caputo-type fractional order differential functions are examined, and then the error estimation of its numerical solution is studied. Then, the quality evaluation model and questionnaire of modern apprenticeship training that incorporates the cultivation of ancient literacy are designed in terms of five influencing factors, namely, the support of enterprises, the cultivation of masters, the support of schools, the importance of government and the cooperation of apprentice parents, and five quality dimensions, namely, knowledge, personality traits, mental skills, motor skills and general abilities. The Caputo-type fractional order calculus estimation method was used to evaluate and analyze the influencing factors. The experimental results showed that school support, government attention, and apprentice-parent cooperation were found to positively influence the cultivation quality, with the t -value and partial correlation coefficient values of apprentice-parent cooperation being 8.407 and 0.484, respectively, and having the greatest influence on the cultivation of modern apprenticeship talents. This study provides a guiding reference for the cultivation practice of modern apprenticeship in ancient literature, thus leading to the development of the whole vocational education with practical significance. © 2023 Xu Zhang, published by Sciendo.
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