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Financial mathematics is one of the most rapidly evolving fields in today’s banking and cooperative industries. In the current study, a new fractional differentiation operator with a nonsingular kernel based on the Robotnov fractional exponential function (RFEF) is considered for the Black–Scholes model, which is the most important model in finance. For simulations, homotopy perturbation and the Laplace transform are used and the obtained solutions are expressed in terms of the generalized Mittag-Leffler function (MLF).

The homotopy perturbation method (HPM) with the help of the Laplace transform is presented here to check the behaviours of the solutions of the Black–Scholes model. HPM is well known for its accuracy and simplicity.

In this attempt, the exact solutions to a famous financial market problem, namely, the BS option pricing model, are obtained using homotopy perturbation and the LT method, where the fractional derivative is taken in a new YAC sense. We obtained solutions for each financial market problem in terms of the generalized Mittag-Leffler function.

The Black–Scholes model is presented using a new kind of operator, the Yang-Abdel-Aty-Cattani (YAC) operator. That is a new concept. The revised model is solved using a well-known semi-analytic technique, the homotopy perturbation method (HPM), with the help of the Laplace transform. Also, the obtained solutions are compared with the exact solutions to prove the effectiveness of the proposed work. The different characteristics of the solutions are investigated for different values of fractional-order derivatives.

With the continuous increase of users, resources and tags, social tagging systems gradually present the characteristics of “big data” such as large number, fast growth, complexity and unreliable quality, which greatly increases the complexity of recommendation. The contradiction between the efficiency and effectiveness of recommendation service in social tagging is increasingly becoming prominent. The purpose of this study is to incorporate topic optimization into collaborative filtering to enhance both the effectiveness and the efficiency of personalized recommendations for social tagging.

Combining the idea of optimization before service, this paper presents an approach that incorporates topic optimization into collaborative recommendations for social tagging. In the proposed approach, the recommendation process is divided into two phases of offline topic optimization and online recommendation service to achieve high-quality and efficient personalized recommendation services. In the offline phase, the tags' topic model is constructed and then used to optimize the latent preference of users and the latent affiliation of resources on topics.

Experimental evaluation shows that the proposed approach improves both precision and recall of recommendations, as well as enhances the efficiency of online recommendations compared with the three baseline approaches. The proposed topic optimization–incorporated collaborative recommendation approach can achieve the improvement of both effectiveness and efficiency for the recommendation in social tagging.

With the support of the proposed approach, personalized recommendation in social tagging with high quality and efficiency can be achieved.