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Population balance equations may be employed to handle a wide variety of particle processes has certainly received unprecedented attention, but the absence of explicit exact solutions necessitates the use of numerical approaches. In this paper, a (2 + 1) dimensional population balance equation with aggregation, nucleation, growth and breakage processes is solved analytically by use of the methods of scaling transformation group, observation and trial function. Symmetries, reduced equations, invariant solutions, exact solutions, existence of solutions, evolution analysis of dynamic behavior for solutions are presented. The exact solutions obtained can be compared with the numerical scheme. The obtained results also show that the method of scaling transformation group can be applied to study integro-partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fractional complex Ginzburg–Landau equation with Kerr law and power law nonlinearity. Using the conformable derivative approach and the bifurcation method, we effectively derived new explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, kink and antikink wave solution, compacton) under different parameter conditions. The quasiperiodic, chaotic behavior and sensitivity analysis of the model is studied for different values of parameters after deploying an external periodic force. Finally, various 2D and 3D simulation figures are plotted to show the physical significance of these exact solutions. [ABSTRACT FROM AUTHOR]

The mathematical analysis on the behavior of the entropy for viscous, compressible, and heat conducting magnetohydrodynamic flows near the vacuum region is a challenging problem as the governing equation for entropy is highly degenerate and singular in the vacuum region. In our previous work [38] , we investigate the global existence of strong solutions to the three-dimensional (3D) compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity. However, it is unknown whether the entropy remains its boundedness or not? Thus, the main novelty of this paper is to give a positive response to this problem. In fact, we show that the uniform boundedness of the entropy and the L 2 regularities of the velocity and temperature can be propagated provided that the initial density decays suitably slow at infinity. Modified De Giorgi type iteration techniques and useful singularly weighted estimates are developed to deduce the lower and upper bounds on the entropy. [ABSTRACT FROM AUTHOR]

The KdV equation has many applications in mechanics and wave dynamics. Therefore, researchers are carrying out work to develop and analyze modified and generalized forms of the standard KdV equation. In this paper, we inspect the KdV-mKdV equation, which is a modified and generalized form of the ordinary KdV equation. We use the fractional operator in the Caputo sense to analyze the equation. We examine some theoretical results concerned with the solution's existence, uniqueness, and stability. We employ a modified Laplace method to extract the numerical results of the considered equation. We use MATLAB-2020 to simulate the results in a few fractional orders. We report the effects of the fractional order on the wave dynamics of the proposed equation. [ABSTRACT FROM AUTHOR]

Population balance equations are essential to scientists and engineers of widely varying discipline, applications have covered a wide range of processes involving bioreactors, catalytic reaction, and cell populations, but the lack of exact solutions necessitates the use of numerical methods. Group classification of Smoluchowski equation with a source is provided by use of the method of preliminary group classification. The determining equation is successfully solved by the method of developed Lie group analysis. All source functions, reduced equations, invariant solutions, explicit exact solutions and dynamic behavior analysis of solutions are presented in this paper. • Population balance equations for aggregation and breakage processes. • Explicit exact solutions of population balance equations with a source term. • Dynamic behavior analysis of solutions. • Preliminary group classification. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the kinetic description of the first-order Cucker-Smale (CS) flocking model on the real line. We reveal the equivalent relation between the measure-valued solution to the first- and second-order kinetic CS equations. The emergent behavior of the first-order kinetic CS equation and the characterization of the asymptotic solution are studied. We also provide the equivalent relation between classical/measure-valued solutions to the first-order kinetic CS equation and a classical solution to the second-order hydrodynamic CS equations, and present the corresponding analysis on the large-time behavior. The numerical experiments support our analysis and provide an efficient algorithm to obtain the asymptotic solution without simulating the model for a long time. [ABSTRACT FROM AUTHOR]

In this paper, we investigate rational solitons and rogue wave (RW) solutions of the nonlocal Lakshmanan-Porsezian-Daniel (LPD) equation by the generalized Darboux transformation method. The exact rational solutions are presented in terms of determinants whose entries are given by the initial eigenfunction and continuous-wave solution. Their dynamics investigated reveals the interesting patterns such as rational solitons, singular and nonsingular RWs depending on the selection of the free parameters. The compression effects of the RWs with respect to the strength of the higher-order nonlinear effects parameter γ are discussed and graphically illustrated. • Construction the rational soliton and higher-order RW solutions for the nonlocal LPD equation by the generalized DT. • Dynamic behavior analysis for the singular and nonsingular rational solutions. • The growing compressed effects of the nonsingular RW solutions are analyzed through numerical plots. [ABSTRACT FROM AUTHOR]