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It provided a powerful new way for predicting the growth trend of malignant tumor and assisting the treatment of cancer patients. Firstly, a one-dimensional mathematical model for the dynamic proliferation of malignant tumors is established on the premise of related simplification and hypothesis. Secondly, according to the Lie symmetry theory, we deduce the multigroup allowed infinitely small generating elements of partial differential equations and obtain the analytic form of the exact invariant solution. Finally, the influence of the model condition parameters (oxygen concentration and inhibitor concentration) on the tumor multiplication time index T is analyzed and discussed. The results showed that when the concentration of the nutrient substance is higher than the critical concentration, the multiplication time of the tumor region approximately decreased firstly and then increased in the linear form about tumor radius under different oxygen concentrations, and at the same radius, the oxygen concentration is lower, and the multiplication time is longer; the multiplication time of the tumor region approximately decreased in the exponential form about tumor radius under different inhibitor concentrations, and at the same radius, the inhibitor concentration is higher, and the multiplication time is bigger, which are consistent with the experimental and clinical observation.

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The coagulation of aerosol particles plays an important role in the structural morphological changes of suspended particles at any time and in any space. In this study, based on the Smoluchowski equation of population balance, a kinetic model of aerosol coalescence considering Brownian motion collision is established. By applying the developed Lie group method, we derive the allowed infinitesimal symmetries and group-invariant solutions of the integro-differential equation, as well as the exact solution under some special conditions. We also provide detailed steps and a discussion of the properties. The content and results provide an effective analytic solution for the progressive evolution of aerosol particle size considering boundary and initial conditions. This solution reveals the self-conservative phenomena in the process of aerosol coalescence and also provides validation for the numerical algorithms of general dynamics equations.