THE POINTWISE ESTIMATES OF SOLUTIONS FOR A NONLINEAR CONVECTION DIFFUSION REACTION EQUATION

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal Lp,1≤ p ≤ ＋∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.

DIFFUSION REACTIONS BETWEEN Cu AND NbTi-ALLOY

Diffusion reaction of superconductig composite NbTi/Cu assembled both mechanically and metallurgically,diffusion couples Ti/Cu and Nb/Cu have/been studied by means of EPMA.The results indicate that the initial interface state of the composite NbTi/Cu significantly affects the forming process of intermetallic compounds at the interface.In comparison with the metallurgically bound composite,the intermetallic com- pounds forms at a higher temperature in mechanically bound one.Therefore.the mechanical hinding process is beneficial to the production of NbTi/Cu superconducting composite. No intermetallic compounds have been observed in Nb/Cu system.The morphology of intermetallic phases and the sequence in which they form in Ti/Cu system are somewhat different from that in NbTi/Cu.The relationship between the thickness of compound layer and annealing time obeys the rule of y^(1.5) ∝ t.

Diffusion and reaction mechanism of limestone and quartz in fluxed iron ore pellet roasting process

The increase to the proportion of fluxed pellets in the blast furnace burden is a useful way to reduce the carbon emissions in the ironmaking process.In this study,the interaction between calcium carbonate and iron ore powder and the mineralization mechanism of fluxed iron ore pellet in the roasting process were investigated through diffusion couple experiments.Scanning electron microscopy with energy dispersive spectroscopy was used to study the elements’diffusion and phase transformation during the roasting process.The results indicated that limestone decomposed into calcium oxide,and magnetite was oxidized to hematite at the early stage of preheating.With the increase in roasting temperature,the diffusion rate of Fe and Ca was obviously accelerated,while the diffusion rate of Si was relatively slow.The order of magnitude of interdiffusion coefficient of Fe_(2)O_(3)-CaO diffusion couple was 10^(−10) m^(2)·s^(−1) at a roasting temperature of 1200℃for 9 h.Ca_(2)Fe_(2)O_(5) was the initial product in the Fe_(2)O_(3)-CaO-SiO_(2) diffusion interface,and then Ca_(2)Fe_(2)O_(5) continued to react with Fe_(2)O_(3) to form CaFe_(2)O_(4).With the expansion of the diffusion region,the sillico-ferrite of calcium liquid phase was produced due to the melting of SiO_(2) into CaFe_(2)O_(4),which can strengthen the consolidation of fluxed pellets.Furthermore,andradite would be formed around a small part of quartz particles,which is also conducive to the consolidation of fluxed pellets.In addition,the principle diagram of limestone and quartz diffusion reaction in the process of fluxed pellet roasting was discussed.

Exponential Time Differencing Method for a Reaction-Diffusion System with Free Boundary

For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary.It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties.Applying an implicit scheme may be able to remove the stability constraints on the time step,however,it usually requires solving a large global system of nonlinear equations for each time step,and the computational cost could be significant.Integration factor(IF)or exponential time differencing(ETD)methods are one of the popular methods for temporal partial differential equations(PDEs)among many other methods.In our paper,we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries.In particular,we rewrite all ETD schemes into a linear combination of specificФ-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications,which offers significant computational advantages with adaptive Krylov subspaces.In addition,we extend this method by incorporating the level set method to solve the free boundary problem.The accuracy,stability,and efficiency of the developed method are demonstrated by numerical examples.

Turing Instability of Gray-Scott Reaction-Diffusion Model with Time Delay Effects

The reaction diffusion Gray-Scott model with time delay is put forward with the assumption of Neumann boundary condition is satisfied. Based on the Turing bifurcation condition, the Turing curves on two parameter plane are discussed without time delay. The normal form is computed via applying Lyapunov-Schmidt reduction method in system of PDE, and the bifurcating direction of pitchfork bifurcation underlying codimension-1 singularity of Turing point is computed. The continuation of Pitchfork bifurcation is simulated with varying free parameter continuously near the turing point, which is in coincidence with the theoritical analysis results. The wave pattern formation in the case of turing instability is also simulated which discover Turing oscillation phenomena from periodicity to irregularity.

Preparation of TiAl alloy powder by high-energy ball milling and diffusion reaction at low temperature

In this paper, TiAl alloy powders were prepared successfully by high-energy ball milling and diffusion reaction in vacuum at low temperature. The titanium powder, aluminum powder, and titanium hydride powder were used as raw materials. The samples were characterized by scanning electron microscopy（SEM）, X-ray diffraction（XRD）, field-emission scanning electron microscopy（FESEM）, and differential thermal analysis（DTA）. The results show that the alloy powders with the main intermetallic compounds of TiAl are obtained using Ti-Al powders and TiH2-Al powders after heated for 2 h at 500 ℃,3 h at 600 ℃,and 3 h at 750 ℃,respectively.The average grain sizes of alloy powder are about 45 and20 μm with irregular shape, respectively. The prepared TiAl alloy powders are relatively pure, and the average quality content of oxygen in the alloy powders is0.33 wt%. The forming process of alloy powder contains both the diffusion reaction of Ti and Al,which gives priority to the diffusion reaction of aluminum.

L^(p) Decay Rate for a Nonlinear Convection Diffusion Reaction Equation in R~n

This paper studies the asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in R^(n).Firstly,the global existence and uniqueness of classical solutions for small initial data are established.Then,we obtain the L^(p),2≤p≤+∞decay rate of solutions.The approach is based on detailed analysis of the Green function of the linearized equation with the technique of long wave-short wave decomposition and the Fourier analysis.

THE NONLINEAR SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH TIME DELAY

A class of nonlinear for singularly perturbed problems for reaction diffusion equations with time delays are considered. Under suitable conditions, using theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied.

A CLASS OF NONLINEAR SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

A class of nonlinear singularly perturbed problems for reaction diffusion equations are considered. Under suitable conditions, by using the theory of differential inequalities, the asymptotic behavior of solutions for the initial boundary value problems are studied, reduced problems of which possess two intersecting solutions.

THE SINGULARLY PERTURBED NONLOCAL REACTION DIFFUSION SYSTEM

In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Unsing the iteration method and the comparison theorem, the existence, uniqueness and its asymptotic behavior of solution for the problem are studied.

THE NONLINEAR NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed, finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation is discussed.

ASYMPTOTIC METHOD OF TRAVELLING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR REACTION DIFFUSION EQUATION

In this article the travelling wave solution for a class of nonlinear reaction diffusion problems are considered. Using the homotopic method and the theory of travelling wave transform, the approximate solution for the corresponding problem is obtained.

Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation

In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.

NONLINEAR SINGULARLY PERTURBED PREDATOR-PREY REACTION DIFFUSION SYSTEMS

A class of nonlinear predator prey reaction diffusion systems for singularly pe rturbed problems are considered.Under suitable conditions, by using theory of di fferential inequalities the existence and asymptotic behavior of solution for in itial boundary value problems are studied.

A CLASS OF SINGULARLY PERTURBED REACTION DIFFUSION SYSTEMS

A class of singularly perturbed reaction diffusion systems are considered. Under suitable conditions, using the comparison theorem the asymptotic behavior of solution for the initial boundary value problems is studied.

Nonlinear predator-prey singularly perturbed Robin Problems for reaction diffusion systems

The nonlinear predator-prey reaction diffusion systems for singularly perturbed Robin Problems are considered. Under suitable conditions, the theory of differential inequalities can be used to study the asymptotic behavior of the solution for initial boundary value problems.

THE NONLINEAR NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH A BOUNDARY PERTURBATION

The nonlinear nonlocal singularly perturbed initial boundary value problems for reaction diffusion equations with a boundary perturbation is considered. Under suitable conditions, the outer solution of the original problem is obtained. Using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. And then using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems is studied. Finally the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.

A Class of Nonlinear Nonlocal Singularly Perturbed Reaction Diffusion Systems

The singularly perturbed initial boundary value problems for the reaction diffusion system are raised. Firstly, under suitable conditions, using a iteration technique, the differential inequalities theorem is constructed and introducing two auxiliary functions the existence and uniqueness theorem of solution for the basic reaction diffusion system is proved. Using the singularly perturbed method the formal asymptotic expressions of the solution are constructed with power series theory. By using the comparison theorem the existence and its asymptotic behavior of solution for the original problem are studied. Finally, using method of estimate inequalities, the structure of solutions for the problem is discussed thoroughly in three cases and asymptotic solution of the original problem is given. The asymptotic behavior of solution in the three cases is proved.

Asymptotic solution for a class of weakly nonlinear singularly perturbed reaction diffusion problem

Under appropriate conditions, with the perturbation method and the theory of differential inequalities, a class of weakly nonlinear singularly perturbed reaction diffusion problem is considered. The existence of solution of the original problem is proved by constructing the auxiliary functions. The uniformly valid asymptotic expansions of the solution for arbitrary mth order approximation are obtained through constructing the formal solutions of the original problem, expanding the nonlinear terms to the power in small parameter ε and comparing the coefficient for the same powers of ε. Finally, an example is provided, resulting in the error of 0（ε^2）.

Singularly perturbed reaction diffusion equations with time delay

A class of initial boundary value problems of differential-difference equations for reaction diffusion with a small time delay is considered. Under suitable conditions and by using the stretched variable method, a formal asymptotic solution is constructed. Then, by use of the theory of differential inequalities, the uniform validity of the solution is proved.