Symmetries and Group-Invariant Solutions for Transonic Pressure-Gradient Equations

Lie symmetry group method is applied to study the transonic pressure-gradient equations in two-dimensionalspace.Its symmetry groups and corresponding optimal systems are determined,and several classes of irrotational groupinvariantsolutions associated to the symmetries are obtained and special case of one-dimensional rarefaction wave isfound.

Symmetry Reductions and Group-Invariant Solutions of (2+1)-Dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada Equation

With the aid of symbolic computation, we present the symmetry transformations of the (2+1)-dimensionalCaudrey-Dodd Gibbon-Kotera-Sawada equation with Lou's direct method that is based on Lax pairs. Moreover, withthe symmetry transformations we obtain the Lie point symmetries of the CDGKS equation, and reduce the equation withthe obtained symmetries. As a result, three independent reductions are presented and some group-invariant solutions ofthe equation are given.

Group-Invariant Solutions for the Generalised Fisher Type Equation

In this paper, we construct the group-invariant (exact) solutions for the generalised Fisher type equation using both classical Lie point and the nonclassical symmetry techniques. The generalised Fisher type equation arises in theory of population dynamics. The diffusion term and coefficient of the source term are given as the power law functions of the spatial variable. We introduce the modified Hopf-Cole transformation to simplify a nonlinear second Order Ordinary Equation (ODE) into a solvable linear third order ODE.

Dynamical Soliton Wave Structures of One-Dimensional Lie Subalgebras via Group-Invariant Solutions of a Higher-Dimensional Soliton Equation with Various Applications in Ocean Physics and Mechatronics Engineering

Having realized various significant roles that higher-dimensional nonlinear partial differ-ential equations(NLPDEs)play in engineering,we analytically investigate in this paper,a higher-dimensional soliton equation,with applications particularly in ocean physics and mechatronics(electrical electronics and mechanical)engineering.Infinitesimal generators of Lie point symmetries of the equation are computed using Lie group analysis of differen-tial equations.In addition,we construct commutation as well as Lie adjoint representation tables for the nine-dimensional Lie algebra achieved.Further,a one-dimensional optimal system of Lie subalgebras is also presented for the soliton equation.This consequently enables us to generate abundant group-invariant solutions through the reduction of the understudy equation into various ordinary differential equations(ODEs).On solving the achieved nonlinear differential equations,we secure various solitonic solutions.In conse-quence,these solutions containing diverse mathematical functions furnish copious shapes of dynamical wave structures,ranging from periodic,kink and kink-shaped nanopteron,soliton(bright and dark)to breather waves with extensive wave collisions depicted.We physically interpreted the resulting soliton solutions by imploring graphical depictions in three dimensions,two dimensions and density plots.Moreover,the gained group-invariant solutions involved several arbitrary functions,thus exhibiting rich physical structures.We also implore the power series technique to solve part of the complicated differential equa-tions and give valid comments on their results.Later,we outline some applications of our results in ocean physics and mechatronics engineering.

GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY

This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.

Localized wave solutions and interactions of the (2+1)-dimensional Hirota-Satsuma-Ito equation

This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional Hirota-Satsuma-Ito equation are obtained,which are all related to the seed solution of the equation.It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons,and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton.Furthermore,the breather wave solution is also obtained by reducing the two-soliton solutions.The trajectory and period of the one-order breather wave are analyzed.The corresponding dynamical characteristics are demonstrated by the graphs.

PDE Standardization Analysis and Solution of TypicalMechanics Problems

A numerical approach is an effective means of solving boundary value problems(BVPs).This study focuses on physical problems with general partial differential equations(PDEs).It investigates the solution approach through the standard forms of the PDE module in COMSOL.Two typical mechanics problems are exemplified:The deflection of a thin plate,which can be addressed with the dedicated finite element module,and the stress of a pure bending beamthat cannot be tackled.The procedure for the two problems regarding the three standard forms required by the PDE module is detailed.The results were in good agreement with the literature,indicating that the PDE module provides a promising means to solve complex PDEs,especially for those a dedicated finite element module has yet to be developed.

THE LIMITING PROFILE OF SOLUTIONS FOR SEMILINEAR ELLIPTIC SYSTEMS WITH A SHRINKING SELF-FOCUSING CORE

In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are bounded given functions whose self-focusing cores{x∈R^(N)|Q_(n)(x)>0} shrink to a set with finitely many points as n→∞.Motivated by the work of Fang and Wang[13],we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points,and we build the localized concentrated bound state solutions for the above equation systems.

Solution approximations for a mathematical model of relativistic electrons with beta derivative

The main aim of this paper is to obtain the exact and semi-analytical solutions of the nonlinear Klein-Fock-Gordon(KFG)equation which is a model of relativistic electrons arising in the laser thermonuclear fusion with beta derivative.For this purpose,both the modified extended tanh-function(mETF)method and the homotopy analysis method(HAM)are used.While applying the mETF the chain rule for beta derivative and complex wave transform are used for obtaining the exact solution.The advantage of this procedure is that discretization or normalization is not required.By applying the mETF,the exact solutions are obtained.Also,by applying the HAM semi-analytical results for the considered equation are acquired.In HAM?curve gives us a chance to find the suitable value of the for the convergence of the solution series.Also,comparative graphical representations are given to show the effectiveness,reliability of the methods.The results show that the m ETF and HAM are reliable and applicable tools for obtaining the solutions of non-linear fractional partial differential equations that involve beta derivative.This study can bring a new perspective for studies on fractional differential equations.On the other hand,it can be said that scientists can apply the considered methods for different mathematical models arising in physics,chemistry,engineering,social sciences and etc.which involves fractional differentiation.Briefly the results may cause a new insight who studies on relativistic electron modelling.

Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation

For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.

Analytic Solutions for Hindered Rotation Quantum Problem

Normalizable analytic solutions of the quantum rotor problem with divergent potential are presented here as solution of the Schrödinger equation. These solutions, unknown to the literature, represent a mathematical advance in the description of physical phenomena described by the second derivative operator associated with a divergent interaction potential and, being analytical, guarantee the optimal interpretation of such phenomena.

Existence of Entropy Solution for Degenerate Parabolic-Hyperbolic Problem Involving p(x)-Laplacian with Neumann Boundary Condition

We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.

Malargüe Site Remediation:A Successful Solution for Uranium Mill Tailings

Comisión Nacional de Energía Atómica (CNEA) has the responsibility for restoring uranium mining facilities once the operations have finished.CNEA,within its Environmental Program and in compliance with its legal responsibilities,decides to implement a restoration project for all sites related to the mining and processing of uranium ores.The Malargüe Site is located within the Province of Mendoza in the city of Malargüe.It is the first site to successfully complete its remediation.The activities consist of relocation of tailings to an engineering repository.The tailings management(encapsulation) and rehabilitation of the area was finished in June 2017.The remediation alternative for the ore tailings was selected after conducting comparative studies and submitted the project to the society for consideration.The objective of the encapsulation of the mineral tails is to isolate them from the environment,also proceeding with the decontamination and rehabilitation of the area (landscaping,post-closure monitoring and 20 years monitoring period).Encapsulation consisted of the construction of a containment cell for the mine tailings,to isolate them and prevent pollutants from entering the environment through the transfer routes.To clean the impacted areas,the soil was removed,it was incorporated into the encapsulation,and the filling was carried out with natural soils from the area.Remediation prevents radon transfer to the environment,as ^(222)Ra is an alpha emitter with a half-life of four days,which produces its own radioactive progeny.Radon progeny are solids,and when a ^(222)Ra nucleus emits an alpha particle into the air,the resulting ^(218)Po nucleus,momentarily electrically charged,adheres to any dust particle.Remediation prevents the discharge into the air containing radon and also containing dust particles charged with intensely radioactive radon progeny.The tasks mentioned make it possible to decrease radon emanation,reduce radiological risks to the public and prevent the entry of rainwater into the system.In addition,the containment system prevents the discharge of contaminated liquids into the environment,avoiding contamination of the groundwater.All these activities are according to the concepts of sustainability.

GLOBAL WEAK SOLUTIONS FOR AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH p-LAPLACIAN DIFFUSION AND LOGISTIC SOURCE

This paper is concerned with the following attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source:■The system here is under a homogenous Neumann boundary condition in a bounded domainΩ ■ R^(n)(n≥2),with χ,ξ,α,β,γ,δ,k_(1),k_(2)> 0,p> 2.In addition,the function f is smooth and satisfies that f(s)≤κ-μs~l for all s≥0,with κ ∈ R,μ> 0,l> 1.It is shown that(ⅰ)if l> max{2k_(1),(2k_(1)n)/(2+n)+1/(p-1)},then system possesses a global bounded weak solution and(ⅱ)if k_(2)> max{2k_(1)-1,(2k_(1)n)/(2+n)+(2-p)/(p-1)} with l> 2,then system possesses a global bounded weak solution.

Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.

On entire solutions of some Fermat type differential-difference equations

On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].

Enhancing Cybersecurity through Cloud Computing Solutions in the United States

This study investigates how cybersecurity can be enhanced through cloud computing solutions in the United States. The motive for this study is due to the rampant loss of data, breaches, and unauthorized access of internet criminals in the United States. The study adopted a survey research design, collecting data from 890 cloud professionals with relevant knowledge of cybersecurity and cloud computing. A machine learning approach was adopted, specifically a random forest classifier, an ensemble, and a decision tree model. Out of the features in the data, ten important features were selected using random forest feature importance, which helps to achieve the objective of the study. The study’s purpose is to enable organizations to develop suitable techniques to prevent cybercrime using random forest predictions as they relate to cloud services in the United States. The effectiveness of the models used is evaluated by utilizing validation matrices that include recall values, accuracy, and precision, in addition to F1 scores and confusion matrices. Based on evaluation scores (accuracy, precision, recall, and F1 scores) of 81.9%, 82.6%, and 82.1%, the results demonstrated the effectiveness of the random forest model. It showed the importance of machine learning algorithms in preventing cybercrime and boosting security in the cloud environment. It recommends that other machine learning models be adopted to see how to improve cybersecurity through cloud computing.

Effect of inorganic salt impurities on seeded precipitation of silica hydrate from sodium silicate solution

To clarify the precipitation of silica hydrate from the real desilication solutions of aluminosilicate solid wastes by adding seeds and improve integrated waste utilization,the seeded precipitation was studied using synthesized sodium silicate solution containing different inorganic salt impurities.The results show that sodium chloride,sodium sulfate,sodium carbonate,or calcium chloride can change the siloxy group structure.The number of high-polymeric siloxy groups decreases with increasing sodium chloride or sodium sulfate concentration,which is detrimental to seeded precipitation.Calcium chloride favors the polymerization of silicate ions,and even the chain groups precipitate with the precipitation of high-polymeric sheet and cage-like siloxy groups.The introduced sodium cations in sodium carbonate render a more open network structure of high-polymeric siloxy groups,although the carbonate ions favor the polymerization of siloxy groups.No matter how the four impurities affect the siloxy group structure,the precipitates are always amorphous opal-A silica hydrate.

ON THE STABILITY OF PERIODIC SOLUTIONS OF PIECEWISE SMOOTH PERIODIC DIFFERENTIAL EQUATIONS

In this paper,we address the stability of periodic solutions of piecewise smooth periodic differential equations.By studying the Poincarémap,we give a sufficient condition to judge the stability of a periodic solution.We also present examples of some applications.

Multi-soliton solutions, breather-like and bound-state solitons for complex modified Korteweg–de Vries equation in optical fibers

Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.