Comparing the Effect of Different Concentration of Sulfuric Acid Differentiation Solution on Acid Fast Dyeing

Objective: In the acid-fast staining experiment of pathological tissues, sulfuric acid and hydrochloric acid were used to evaluate the staining results, so as to get the best staining method. Methods: Using sulfuric acid differentiation solution and hydrochloric acid differentiation solution, the paraffin blocks of pathological tissues known to contain Mycobacterium tuberculosis were compared to evaluate the staining effect. Results: When 0.5% hydrochloric acid differentiation solution is used and the differentiation time is 6 s, the dyeing effect is better than that of sulfuric acid differentiation solution.

THE EXISTENCE OF PERIODIC SOLUTIONS FOR A CLASS OF FUNCTIONAL DIFFERENTIAL EQUATIONS AND THEIR APPLICATION

In this paper. four sufficiency theorems of existence of periodic solutions for aclass of retarded functional differential equations are given. The result of thesetheorems is better than the well-known Yoshizawa’s periodic solution theorem. Anexample of application is given at the end.

The Lanczos-Chebyshev Pseudospectral Method for Solution of Differential Equations

In this paper, we propose to replace the Chebyshev series used in pseudospectral methods with the equivalent Chebyshev economized power series that can be evaluated more rapidly. We keep the rest of the implementation the same as the spectral method so that there is no new mathematical principle involved. We show by numerical examples that the new approach works well and there is indeed no significant loss of solution accuracy. The advantages of using power series also include simplicity in its formulation and implementation such that it could be used for complex systems. We investigate the important issue of collocation point selection. Our numerical results indicate that there is a clear accuracy advantage of using collocation points corresponding to roots of the Chebyshev polynomial.

Extended tanh-function Method for Solving Traveling Wave Solutions of Nonlinear Kundu Equation

In this paper, we study the existence of multiple positive periodic solutions for the second order differential equation x′′（t） ＋ p（t）x′（t） ＋ q（t）x（t） = f（t, x（t））.By using Krasnoselskii fixed point theorem, we establish some criteria for the existence and multiple positive periodic solutions for this differential equation.

ASYMPTOTIC SOLUTION OF SINGULAR PERTURBATION PROBLEMS FOR THE FOURTH-ORDER ELLIPTIC DIFFERENTIAL EQUATIONS

In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.

DISCUSSION ON″THE BOUNDEDNESS AND ASYMPTOTIC BEHAVIOR OR SOLUTION DIFFERENTIAL SYSTEM OF SECOND-ORDER WITH VARIABLE COEFFICIENT" (App1ied Mathematics and Mechanics,Vo1.3,No.4,1982)

After reading the article "The Boundedness and Asymptotic Behavior of Solution of Differential System of Second-Order with Variable Coefficient" in "Applied Mathematics and Mechanics", Vol. 3, No. 4, 1982, we would like to put forward a few points to discuss with the author and the readers. Our opinions are presented as follows:

CONORMAL SINGULARITIES FOR SOLUTION OF SEMILINEAR PARTIAL DIFFERENTIAL EQUATIONS

In the studies of nonlinear partial differential equations, the influence, from the singularities of coefficients to the singularities of solution, is a field that has not been dealt with. In this paper, we discuss a simple case of semilinear equations under the frame of the space of conormal distributions. We prove the result that the solution has the same singularities on the hypersurface in which the coefficients have the conormal singularities.

THE CONTINUAL DIFFERENTIABLE PEAK-UNIMODAL SOLUTIONS OF FEIGENBAUM’S FUNCTIONAL EQUATIONS

For the famous Feigenbaum's equations, in this paper, we established its constructive theorem of the peak-unimodal, then we found out other paths to explore the peak-unimodal solutions. For example, we proceed on the direction to try the non-symmetrical continuous peak-unimodal solutions and C1 solutions.

NUMERICAL SOLUTION OF QUASILINEAR SINGULARLY PERTURBED ORDINARY DIFFERENTIAL EQUATION WITHOUT TURNING POINTS

In this paper we consider a quasilinear second order ordinary diferential equation with a small parameter Firstly an approximate problem is constructed. Then an iterative procedure is developed. Finally we give an algorithm whose accuracy is good for arbitrary e>0 .

Asymptotic Behavior of Solutions of Retarded Differential Equations

A sufficient condition is obtained for every solution of the nonlinear retarded differential equationx'(t) +f(t,x(t-τ)) =0to tend to zero as t→∞ , which extends and improves the corresponding results obtained by Ladas, Sficas and Gopalsamy.

A unique solution to a semilinear Black-Scholes partial differential equation for valuing multi-assets of American options

In this paper, by using the optimal stopping theory, the semilinear Black-Scholes partial differential equation （PDE） was invesigated in a fixed domain for valuing two assets of American （call-max/put-min） options. From the viscosity solution of a PDE, a unique viscosity solution was obtained for the semilinear Black-Scholes PDE.

DIFFERENTIABILITY FOR THE HIGH DIMENSIONAL POLYNOMIAL-LIKE ITERATIVE EQUATION

This paper studies the smoothness of solutions of the higher dimensional polynomial-like iterative equation. The methods given by Zhang Weinian([7]) and by Kulczvcki M, Tabor j.([3]) are improved by constructing a new operator for the structure of the equation in order to apply fixed point theorems. Existence, uniqueness and stability of continuously differentiable solutions are given.

On Skew Triangular Matrix Rings

In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of fractional differential equations. Our results are based on the fixed point theorems of increasing operator and the cone theory, some illustrative examples are also presented.

Weighted Pseudo Almost Periodic Solutions of N-th Order Neutral Differential Equations with Piecewise Constant Arguments

In this work, we present some existence theorems of weighted pseudo almost periodic solutions for N-th order neutral differential equations with piecewise constant argument by means of weighted pseudo almost periodic solutions of relevant difference equations.

Quasiperiodic Solutions for Nonlinear Differential Equations of Second Order with Symmetry

In this paper,we study the existence of quasi-periodic solutions and the bound- edness of solutions for a wide class nonlinear differential equations of second order.Using the KAM theorem of reversible systems and the theory of transformations,we obtain the existence of quasi-periodic solutions and the boundedness of solutions under some reasonable conditions.

Existence of Exactly n+1 Simple 4-Periodic Solutions of the Differential Delay Equation x(t)=-f(x(t-1))

By discussing the zeros of periodic.solutions we give in this paper a criterion for the existence of exactly n+1 simple 4-periodic solutions of the differential delay equation x(T)= -f(x(t-1)).

linear and nonlinear fractional differential equation modified Riemann–Liouville derivatives exact solutions fractional auxiliary sub-equation expansion method Mittag–Leffler function method

In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional Kd V equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the（3＋1）-spacetime fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag–Leffler function methods. The obtained results recover the well-know solutions when α = 1.

ON PERIODIC SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper,using Mawhin's continuation theorem in the theory of coincidence degree,we first prove the general existence theorem of periodic solutions for F.D.Es with infinite delay:dx(t)/dt=f(t,x_t),x(t)∈R^n,which is an extension of Mawhin's existence theorem of periodic solutions of F.D.Es with finite delay.Second,as an application of it,we obtain the existence theorem of positive periodic solutions of the Lotka-Volterra equations:dx(t)/dt=x(t)(a-kx(t)-by(t)),dy(t)/dt=-cy(t)+d integral from n=0 to +∞ x(t-s)y(t-s)dμ(s)+p(t).

THE NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE 0F PERIODIC SOLUTIONS OF A TYPE OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, using Fourier series, we study the problem of the existence of periodic solutionsof a type of periodic neutral differential difference system. Some necessary and sufficient conditionsfor the existence of periodic solutions of a type of neutral functional equation system are obtained,and at the same time, we present a method with formula shows how to find the periodicsolutions.