A Class of Shock Solutions for the Semilinear Singularly Perturbed Boundary Value Problem

The shock solution for the semilinear singularly perturbed two-point boundary value problem was studied. Under suitable conditions and using the theory of differential inequalities, the existence and asymptotic behavior of the solution for the original boundary value problems are discussed. The uniformly effective asymptotic expansion and estimation of solution u(x, ε) were obtained.

A SIMPLE FAST METHOD IN FINDING PARTICULAR SOLUTIONS OF SOME NONLINEAR PDE

The 'trial function method' ( TFM for short) and a routine way in finding traveling,wave solutions to some nonlinear partial differential equations( PDE for short), wer explained. Two types of evolution equations are studied, one is a generalized Burgers or KdV equation, the other is the Fisher equation with special nonlinear forms of its reaction rate term. One can see that this method is simple, fast and allowing further extension.

Uniqueness of Transonic Shock Solutions to Euler-Poisson System with Varying Background Charges

In this paper,we investigate a one-dimensional Euler-Poisson sys-tem with varying background charges,which are two different constants when the flow speed is less than and greater than the sound speed.Using the shock matching method,we derive the properties of the solution trajectories and es-tablish a monotonic relationship between the density at the nozzle exit and the shock position.This relationship demonstrates the existence and uniqueness of a transonic shock solution under suitable boundary conditions.

Local Existence of the Shock Front Solution to the Axi-symmetrical Piston Problem in Compressible Flow

In this paper,we study a kind of 2-dimensional axi-symmetrical piston problem in com- pressible flow.The corresponding mathematical model is the well-known Euler system.With the Newton iteration procedure and energy estimate,we give the local existence of the shock front solution to this problem.

Symmetries,Integrability and Exact Solutions to the (2+1)-Dimensional Benney Types of Equations

This paper is concerned with the (2+1)-dimensional Benney types of equations. By the complete Lie group classification method, all of the point symmetries of the Benney types of equations are obtained, and the integrable condition of the equation is given. Then, the symmetry reductions and exact solutions to the (2+1)-dimensional nonlinear wave equations are presented. Especially, the shock wave solutions of the Benney equations are investigated by the symmetry reduction and trial function method.

Shock relations in gases of heterogeneous thermodynamic properties

Shock relations usually found in literatures are derived theoretically under the assumption of homogeneous thermodynamic properties, i.e., constant ratio of specific heats, γ. However, high temperature effects post a strong shock wave may result in thermodynamic heterogeneities and failure to the original shock relations. In this paper, the shock relations are extended to take account of high-temperature effects. Comparison indicates that the present approach is more feasible than other analytical approaches to reflect the influence of γ heterogeneity on the post-shock parameters.

Spontaneous anchoring Cl intoα-Co(OH)_(2) as efficient and stable oxygen reduction electrocatalysts for seawater battery

Seawater battery is an advanced energy storage system that enables conversion of chemical energy to electricity by consuming metals,dissolved oxygen and seawater in anode,cathode and electrolyte,respectively.However,the oxygen reduction reaction(ORR)activity and stability of electrocatalysts can be easily deactivated due to the severe Cl~-permeation and corrosion in seawater electrolyte.Herein,we developed a structural buffer engineering strategy by spontaneously anchoring Cl~-intoα-Co(OH)_(2) as efficient and stable ORR electrocatalysts,in which the ultrathinα-Co(OH)_(2) nanosheets were synthesized using an ultrafast solution high-temperature shock(SHTS)strategy.The large lattice space(~0.8 nm)of layeredα-Co(OH)_(2) ensured the spontaneously penetration of Cl~-into the lattice structure and replaced part of OH~-to formα-Co(OH)_(2-x)Cl_x.The continuous leaching and compensating of saturated Cl inα-Co(OH)_(2-x)Cl_x could enhance the Cl~-corrosion resistance and modulate electronic structure of Co metallic sites,thus improving the ORR electrocatalytic activity and stability in seawater electrolyte.Theα-Co(OH)_(2-x)Cl_x seawater batteries display superior onset and half-wave potentials of 0.71 and 0.66 V,respectively,which are much better than the counterparts ofα-Co(OH)_(2) and ofβ-Co(OH)_(2) with no Cl~-penetrating and no buffer structure.Theα-Co(OH)_(2-x)Cl_x-based seawater batteries display stable open-circuit potential of 1.69 V and outstanding specific capacity of 1345 mAh·g^(-1).

On the Jacobi Elliptic Function Expansion Method

The main idea of this method is to take full advantage of the elliptic equation that Jacobi elliptic functions satisfy and use its solutions to replace Jacobi elliptic functions in Jacobi elliptic function method. Some illustrative equations are investigated by this means.

The Space-Time CE/SE Method for Solving Reduced Two-Fluid Flow Model

The space-time conservation element and solution element(CE/SE)method is proposed for solving a conservative interface-capturing reducedmodel of compressible two-fluid flows.The flow equations are the bulk equations,combined with mass and energy equations for one of the two fluids.The latter equation contains a source term for accounting the energy exchange.The one and two-dimensional flow models are numerically investigated in this manuscript.The CE/SE method is capable to accurately capture the sharp propagating wavefronts of the fluids without excessive numerical diffusion or spurious oscillations.In contrast to the existing upwind finite volume schemes,the Riemann solver and reconstruction procedure are not the building block of the suggested method.The method differs from the previous techniques because of global and local flux conservation in a space-time domain without resorting to interpolation or extrapolation.In order to reveal the efficiency and performance of the approach,several numerical test cases are presented.For validation,the results of the current method are compared with other finite volume schemes.

Soliton solution of the generalized modified BBM equation and the generalized Boussinesq equation

In this paper,we obtained the topological soliton solution of the(1+1)-dimensional generalized modified Benjamin-Bona-Mahony equation and shock wave solution of the generalized Boussinesq equation.We get that solutions by using solitary wave ansatz in terms of tanh^(p) functions.The velocity and the free parameters are the physical parameters in the soliton solutions.They can be obtained as functions of the dependent model coefficients.The domain restriction were also identified in the process.we hope that in nonlinear dynamical system these solutions will be explain some nonlinear physical phenomena.