Search

The similarity transformation (ST) method is applied to reduce the Lax pair for some nonlinear partial differential equations (NLPDEs) into a system of ordinary differential equations (ODEs) to obtain its similarity solutions. Then the ODE system is considered to find the analytical solutions of the PDE by plotting the acquired similarity solutions. The method is applied to the three different equations named as; Modified Boussinesq (MBQ) equation, Kadomtsev-Petviashvili (KP) equation, and (2+1) - Korteweg-de-Vries breaking type (KDV-BS) equation. The Lie transformation method is utilized to convert the modified Boussinesq equation's Lax Pair into a system of ordinary differential equations and obtain the analytical solutions of this equation. Likewise, this method is used for the KP and (2+1)-dimensional KdV Lax Pairs. The Lie vectors are optimized through the commutation operation. The reduction of the Lax pair instead of the original equation reveals a new solution. The applied method is effective in spreading the solution of NLPDEs. [ABSTRACT FROM AUTHOR]

Nowadays, there is great attention given to solar collectors (SCs) for their important applications based on the advantages of nanotechnology and solar radiation. Hybrid nanofluid (HNF) is our first option due to its thermophysical properties that help in improving the overall performance, unlike other nanofluids. This paper gives a detailed novel analysis of SCs with the existence of Newtonian, power-law HNF in unsteady conditions and a three-dimensional model under the consideration of Brownian motion and thermophoresis parameter. In this research, the group transformation method (GTM) and similarity transformation steady state fluid dynamics are used to transform the mathematical model into a simpler system. This coupled system of ordinary differential equations with the related functions, dimensionless entropy generation and Bejan number is achieved at two cases of power-law index. The impact of involved parameters on velocity profile, temperature distribution, concentration field, entropy output of the system and Bejan number is depicted prominently by various graphs. The fluid velocity shows improvement with higher values of power-law index and shape factor, while it diminishes with magnetic parameter and Prandtl number. Enhancing the values of magnetic field and shape factor, results in increase of temperature characteristic which decreases with Prandtl number and power-law index. Increment in the concentration ratio parameter leads to maximize the entropy generation, whereas entropy generation diminishes with higher values of temperature ratio and magnetic parameter. The obtained results and the previously published work are compared qualitatively and quantitatively to each other to validate that the applied method is more efficient. It is predicted that the Nusselt number improves by 28.18% when the Prandtl number is taken range (4 ≤ Pr ≤ 6). The percentage of increasing in Sherwood number is noted to be 18.61% for range (0. 1 ≤ N b ≤ 0. 5) of Brownian motion parameter. [ABSTRACT FROM AUTHOR]

In this work, the three-dimensional unsteady flow was scrutinized adjacent to a flat plate immersed inside a fluid with electric conductivity in the presence of constant magnetic field. An extended model of group transformation method was exploited, comprising three parameters, to transform the mathematical model into a new system of ordinary differential equations (ODEs). The proposed method is rarely and seldom used in handling higher dimensions coupled equations. Two cases were studied, the stretching and shrinking plate. In the first case, the results showed that both of magnetic parameter, M, and Prandtl number, Pr, affected inversely the fluid velocity, the shear stress and the temperature distribution. On the contrary, the heat flux increased when both M and Pr increase. That could be interpreted by the effect of larger magnetic force and fluid viscosity, respectively. In case of shrinking plate, the attained results were opposite to those of stretching case specially for the effect of magnetic parameter. [ABSTRACT FROM AUTHOR]

In this work, we study the double chain Deoxyribonucleic acid (DNA) model, the study will be carried out by using two analytical methods, namely, Lie transformation method, then combined with the singular manifold method (SMM). Six cases are studied for different vector combinations. Two successive symmetry reductions transform the equation to an ordinary differential equation (ODE). We next apply the singular manifold method to the obtained ordinary differential equations. This will result in a Schwarzian derivative whose solution leads to soliton solutions for longitudinal and transversal motions. The solutions contain different types of wave solutions, N-solitons, kink and periodic wave solutions. Some of the resulting solutions are graphically illustrated. [ABSTRACT FROM AUTHOR]