In this paper, we introduce an efficient method for solving the quadratic Riccati differential equation. In this technique, combination of Laplace trans-form and new homotopy perturbation methods (LTNHPM) are considered as an algorithm to the exact solution of the nonlinear Riccati equation. Un-like the previous approach for this problem, so-called NHPM, the present method, does not need the initial approximation to be defined as a power series. Four examples in different cases are given to demonstrate simplicity and efficiency of the proposed method.

In this study, by Laplace transform method, the analytical solution of the pollution transport equation in the limited domain for the river network to the upstream and downstream Dirichlet boundary conditions, and the initial condition of zero was extracted, and simulation was performed for two branch and loop networks with fixed and variable boundary conditions. After naming the nodes, by forming matrices of how to connect, flow characteristics and geometry of the river for each network as input to the problem, the diffusion matrix is created based on a function of the Laplace variable, which The value of the concentration in each node is calculated by solving the complex device created and using the inverse Laplace transform. Then, using the analytical solution extracted in a branch of the river for the pollution transfer equation, the analytical solution can calculate the value of pollution concentration at any desired location and time along with the river network. Finally, for validation, the analytical solution was compared with the numerical solution, and then the statistical error indices were calculated. The results indicate the optimal performance and high ability of analytical solution in modeling the two networks and its good adaptation to the numerical solution, which can replace numerical solution due to high accuracy and speed of calculations. Generally, due to common errors in numerical solutions such as numerical dispersion error, Round-off error, Truncation error of Taylor expansion mathematical sentences, analytical solutions, if any, for the river network are recommended over numerical solutions. Also, in the performed simulations, due to the change of the inlet flow and the flow cross-section to each, changes in the pollution concentration occur in the areas where the branches connect to each other, increasing or decreasing. The proposed analytical solution for river networks can model more complex river networks and can be considered a criterion for the validation of numerical solutions. Also, the existing analytical solution can be used as a tool to validate other analytical solutions in the river network.

In this work, we investigate a non-linear differential equation relevant to the computation of entanglement entropy in interacting quantum systems. We do this, inspired by field-theoretic and holographic models, we introduce a simplified equation to describe the entropy as a function of subsystem size and coupling strength. By comparing analytical approximations with numerical solutions, we uncover a crossover behavior between area-law and volume-law regimes, which are key features of entanglement dynamics in quantum systems. This minimal model captures the essential non-linear structure underlying entanglement growth, offering insights into the interplay between subsystem size and interaction strength. Our findings provide a framework for understanding the transition between distinct entanglement scaling laws, bridging theoretical predictions and numerical observations. This work highlights the utility of simplified models in exploring complex quantum phenomena and contributes to the broader understanding of entanglement dynamics in interacting systems, with potential applications in quantum information and condensed matter physics.

In this work an analytical equation of state has been employed to calculate the PVT properties of ternary refrigerant mixtures. The theoretical EoS is that of Ihm, Song and Mason, which is based on statistical-mechanical perturbation theory, and the two constants are enthalpy of vaporization DHvap and molar density rnb, both at the normal boiling temperature. The following three temperature dependent parameters are needed to use the EoS: the second virial coefficient, B2(T), an effective van der Waals covolume, b(T), and a scaling factor, a(T). The second virial coefficients are calculated from a correlation based on the heat of vaporization, DHvap, and the liquid density at the normal boiling point, rnb. a(T) and b(T) can also be calculated from second virial coefficients by scaling rules. This procedure predicts liquid densities of ternary mixtures at saturated state with a temperature range from 173 K to 373 K and pressure up to  4.0MPa, with very good results.

In this present study analytical method based on Riccati Equation as for converting the Nonlinear Lakshmanan-Porsezian-Daniel (LPD) equation into the nonlinear ODE and finding soliton solutions of this sustem discused. Obtaining solutions are new and obtained from wave transformation. The obtained results show that the presented method is effective and appropriate for solving nonlinear differential equations.

Nonlinear equations frequently arise in effective models of particle physics, particularly in the description of bound states, mass renormalization, and self-consistent field theories. In this paper we investigate a representative nonlinear integral, differential equation motivated by self-energy corrections in scalar field theory. We provide both approximate analytical solutions and numerical analysis, uncovering stability properties and physical implications for mass shifts and resonance structures. Our results indicate the existence of multiple solution branches, with physical relevance determined by energy minimization. This study highlights the subtle interplay between nonlinearities and renormalized observables in particle dynamics.