The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal reactors find importance. Using F α -fractal calculus, in this paper, we derive exact F α -differential forms of an ideal gas. Depending on the dimensionality of space, we should first obtain the integral staircase function and mass function of our geometry. When gases expand inside the fractal structure because of changes from the i + 1 iteration to the i iteration, in fact, we are faced with fluid mixing inside our fractal structure, which can be described by physical quantities P, V, and T. Finally, for the ideal gas equation, we calculate volume expansivity and isothermal compressibility.

Sound insulation of building elements such as floors plays a vital role in noise control in buildings. When the incident sound wave hits the floor surface, part of it passes through the floor in the form of airborne and percussion and the other part is reflected or absorbed by the floor material. In the latter case, the measurement of sound wave energy variation is difficult and time-consuming and requires simulation or on-site tests. Hence, estimating sound reduction has always been considered by experts and engineers in the field of building engineering. In the present study, using Artificial Neural Network (ANN), Adaptive Neuro-Fuzzy Inference System (ANFIS), and linear regression, the sound reduction is estimated in floor materials in buildings with acceptable results. The required data for machine learning methods were obtained by simulation of different floor systems with varying material and thickness in the INSUL software. From the 252 data, 80% were randomly selected and used as training data for modeling and training the networks, and the other 20% was employed as test data to investigate the accuracy of the defined models. The results showed that ANFIS with correlation coefficients of 0.982 and 0.974, respectively for train and test data, is a better and more accurate tool compared to ANN and linear regression for estimating the sound reduction in common building floor systems.