Mostly, it is very difficult to obtained the exact solution of fractional-order partial differential equations. However, semi-analytical or numerical methods are considered to be an alternative to handle the solutions of such complicated problems. To extend this idea, we used semi-analytical procedures which are mixtures of Laplace transform, Shehu transform and Homotopy perturbation techniques to solve certain systems with Caputo derivative differential equations. The effectiveness of the present technique is justified by taking some examples. The graphical representation of the obtained results have confirmed the significant association between the actual and derived solutions. It is also shown that the suggested method provides a higher rate of convergence with a very small number of calculations. The problems with derivatives of fractional-order are also solved by using the present method. The convergence behavior of the fractional-order solutions to an integer-order solution is observed. The convergence phenomena described a very broad concept of the physical problems. Due to simple and useful implementation, the current methods can be used to solve problems containing the derivative of a fractional-order.

Classroom teaching evaluation is one of the most important ways to improve the teaching quality of mathematics education in higher education, and it is also a group decision making problems. Meanwhile, there is some uncertain information in the process of evaluation. In order to deal with this uncertainty in classroom teaching quality evaluation and obtain a reliable and accurate evaluation result, an interval analytic hierarchy process (I-AHP) is employed. To begin with, the modern evaluation tool named RTOP is adapted to make it more consistent with the characteristics of the discipline. In addition, the evaluation approach is built by using the I-AHP method, and some details of weights of the criteria and assessors are developed, respectively. Thirdly, a case study has been made to verify the feasibility of the assessment approach for classroom teaching quality evaluation on mathematics. Finally, a comprehensive evaluation of classroom quality under an interval number environment is conducted, and some results analyses and comparisons are also discussed to show that the proposed approach is sound and has a stronger ability to deal with uncertainty.