REFINED DIFFERENTIAL EQUATIONS OF DEFLECTIONS IN AXIAL SYMMETRICAL BENDING PROBLEMS OF SPHERICAL SHELL AND THEIR SINGULAR PERTURBATION SOLUTIONS

This paper deals with the research of accuracy of differential equations of deflections. The basic idea is as follows. Firstly, considering the boundary effect the meridian midsurface displacement u=0, thus we derive the deflection differential equations; secondly we accurately prove that by use of the deflection differential equations or the original differential equations the same inner forces solutions are obtained; finally, we accurately prove that considering the boundary effect the meridian surface displacement u = 0 is an exact solution. In this paper we give the singular perturbation solution of the deflection differential equations. Finally we check the equilibrium condition and prove the inner forces solved by perturbation method and the outer load are fully equilibrated. It shows that perturbation solution is accurate. On the other hand, it shows again that the deflection differential equation is an exact equation.The features of the new differential equations are as follows:1. The accuracies of the new differential equations and the original differential e-quations are the same.2. The new differential equations can satisfy the boundary conditions simply.3. It is advantageous to use perturbation method with the new differential equations.4 We may obtain the deflection expression(w)and slope expression (dw/da) by using the new differential equations.The new differential equations greatly simplify the calculation of spherical shell. The notation adopted in this paper is the same as that in Ref. [1]

ON RELATIONS BETWEEN THE MODIFIED-ITERATIVE METHOD AND CHIEN’S PERTURBATION SOLUTION

In this paper, we gave analytical formulas of characteristic relation of circular plate in solving high-order solutions of modified-iterative method, which reduces the calculating quantities of the method. Having deduced the relations between the modified-iterative method and Chien's perturbation solution, we obtained the conclusion that the convergent regions of the two methods are the same.

SINGULAR PERTURBATION SOLUTIONS OF THE NONLINEAR STABILITY OF A TRUNCATED SHALLOW SHPERICAL SHELL UNDER LINEAR DISTRIBUTED LOADS ALONG THE INTERIOR EDGE

Using a singular perturbation method, the nonlinear stability of a truncated shallow, spherical shell without a nondeformable rigid body at the center under linear distributed loads along the interior edge is investigated in this paper. When the geometrical parameter k is large, the uniformly valid asymptotic solutions are obtained.

PERTURBATION SOLUTION FOR COMPARTMENT MODEL OF VARIABLE EXTRACELLULAR VOLUME

The two compartment model with variable extracellular volume is presented and solved by using both perturbation and analytical method. The computation for both creatinine and urea show that the perturbation solution is not only simple but also accurate enough and is a good substitute for the more exact analytical solution.

THE PERTURBATION SOLUTIONS OF THE FLOW IN A ROTATING CURVED ANNULAR PIPE

WT5”BZ]In this paper, the flow in a rotating curved annular pipe is examined by a perturbation method. A second order perturbation solution is presented. The characteristics of the secondary flow and the axial flow are studied in detail. The study indicates that the loops of the secondary flow are more complex than those in a curved annular pipe without rotation and its numbers depend on the ratio of the Coriolis force to centrifugal force F. As F≈-1, the secondary flow has eight loops and its intensity reaches the minimum value, and the distribution of the axial flow is like that of the Poiseuille flow. The position of the maximum axial velocity is pushed to either outer bend or inner bend, which is also determined by F. [WT5”HZ]

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.

ASYMPTOTIC SOLUTION FOR SINGULAR PERTURBATION PROBLEMS OF DIFFERENCE EQUATION

In this paper, we constructed a new asymptotic method for singular perturbation problems of difference equation with a small parameter.

AN APPLICATION OF THE EXPONENTIAL CUBIC SPLINES TO NUMERICAL SOLUTION OF A SELF-ADJOINT PERTURBATION PROBLEM

We present a class of the second order optimal splines difference schemes derived from ex- ponential cubic splines for self-adjoint singularly perturbed 2-point boundary value problem. We prove an optimal error estimate and give illustrative numerical example.

Approximate solutions of the Alekseevskii–Tate model of long-rod penetration

The Alekseevskii–Tate model is the most successful semi-hydrodynamic model applied to long-rod penetration into semi-infinite targets. However, due to the nonlinear nature of the equations, the rod(tail) velocity, penetration velocity, rod length, and penetration depth were obtained implicitly as a function of time and solved numerically By employing a linear approximation to the logarithmic relative rod length, we obtain two sets of explicit approximate algebraic solutions based on the implicit theoretica solution deduced from primitive equations. It is very convenient in the theoretical prediction of the Alekseevskii–Tate model to apply these simple algebraic solutions. In particular, approximate solution 1 shows good agreement with the theoretical(exact) solution, and the first-order perturbation solution obtained by Walters et al.(Int. J. Impac Eng. 33:837–846, 2006) can be deemed as a special form of approximate solution 1 in high-speed penetration. Meanwhile, with constant tail velocity and penetration velocity approximate solution 2 has very simple expressions, which is applicable for the qualitative analysis of long-rod penetration. Differences among these two approximate solutions and the theoretical(exact) solution and their respective scopes of application have been discussed, and the inferences with clear physical basis have been drawn. In addition, these two solutions and the first-order perturbation solution are applied to two cases with different initial impact velocity and different penetrator/target combinations to compare with the theoretical(exact) solution. Approximate solution 1 is much closer to the theoretical solution of the Alekseevskii–Tate model than the first-order perturbation solution in both cases, whilst approximate solution 2 brings us a more intuitive understanding of quasi-steady-state penetration.

SINGULAR PERTURBATION OF INITIAL BOUNDARY VALUE FROBLEMS OF REACTION DIFFUSlON EQUATION WITH DELAY

In this paper we consider the initial-boundary value problems for a class ofapplications, such as biomathematics and biochemistry.Applying the method ofcomposile expansion we construct the formally asymptotic solution of the problemdescribed. With the help of theory of upper and lower solutions we prove the uniformlyvalidity of the formal solution and the existence of solution of the original problem.

UNIQUENESS OF SOLUTIONS OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR THIRD QRDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].

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 .

NUMERICAL SOLUTION OF THE SINGULARLY PERTURBED PROBLEM FOR THE HYPERBOLIC EQUATION WITH INITIAL JUMP

In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constructed on a non-uniform grid. Finally, uniform convergence of the difference solution is proved in the sense of the discrete energy norm.

Variable viscosity in peristalsis of Sisko fluid

This paper aims to examine variable viscosity effects on peristalsis of Sisko fluids in a curved channel with compliant characteristics. Viscous dissipation in a heat transfer is studied. The resulting problems are solved using perturbation and numerical schemes to show qualitatively similar responses for velocity and temperature. A streamline phenomenon is also considered.

Peristaltic flow of Walter’s B fluid in endoscope

The peristaltic flow of a Walter＇s B fluid in an endoscope is studied. The problem is modeled in a cylindrical coordinate system. The main theme of the present analysis is to study the endoscopic effects on the peristaltic flow of the Waiter＇s B fluid. To the best of the authors＇ knowledge, no investigation has been made so far in the literatures to study the Walter＇s B fluid in an endoscope. Analytical solutions axe obtained using the regular perturbation method by taking 5 as a perturbation parameter. The approximate analytical solutions for the pressure rise and friction forces are evaluated using numerical integration. The effects of emerging parameters of the Waiter＇s B fluid are presented graphically.

Nonlinear transport of Bose-Einstein condensates in a double barrier potential

The stable nonlinear transport of the Bose-Einstein condensates through a double barrier potential in a waveguide is studied. By using the direct perturbation method we have obtained a perturbed solution of Cross-Pitaevskii equation. Theoretical analysis reveals that this perturbed solution is a stable periodic solution, which shows that the transport of Bose-Einstein condensed atoms in this system is a stable nonlinear transport. The corresponding numerical results are in good agreement with the theoretical analytical results.

Variations of water's local-structure induced by solvation of NaCl

The researches on the structure of water and its changes induced by solutes are of enduring interests. The changes of the local structure of liquid water induced by NaCl solute under ambient conditions are studied and presented quantitatively with some order parameters and visualized with 2-body and 3-body correlation functions. The results show that, after the NaCl are solvated, the translational order t of water is decreased for the suppression of the second hydration shells around H20 molecules; the tetrahedral order （q） of water is also decreased and its favorite distribution peak moves from 0.76 to 0.5. In addition, the orientational freedom k and the diffusion coefficient D of water molecules are reduced because of new formed hydrogen-bonding structures between water and solvated ions.

THE APPLICATION OF INTEGRAL EQUATIONS TO THE NUMERICAL SOLUTION OF NONLINEAR SINGULAR PERTURBATION PROBLEMS

The nonlinear singular perturbation problem is solved numerically on nonequidistant meshes which are dense in the boundary layers. The method presented is based on the numerical solution of integral equations [1]. The fourth order uniform accuracy of the scheme is proved. A numerical experiment demonstrates the effectiveness of the method.

A study on stochastic longitudinal wave equation in a magneto-electro-elastic annular bar to find the analytical solutions

In this paper,we set up dynamic solitary perturb solutions of a unidirectional stochastic longitudinal wave equation in a magneto-electro-elastic annular bar by a feasible,useful,and influential method named the dual(G’/G,1/G)-expansion method.Computer software,like Mathematica,is used to complete this discussion.The obtained solutions of the proposed equation are classified into trigonometric,hyperbolic,and rational types which play an important role in searching for numerous scientific events.The technique employed here is an extension of the(G’/G)-expansion technique for finding all previously discovered solutions.To illustrate our findings more clearly,we provide 2D and 3D charts of the various recovery methods.We then contrasted our findings with those of past solutions.The graphical illustrations of the acquired solutions are singular periodic solitons and kink solitons which are added at the end of this paper.

Series solution of unsteady peristaltic flow of a Carreau fluid in small intestines

The study of peristaltic flow of a Carreau fluid in a non-uniform tube under the con- sideration of long wavelength is presented. The flow is investigated in a wave frame of reference moving with velocity of the wave e. Numerical integration has been used to obtain the graphical results for pressure rise and frictional forces. The effects of various emerging parameters are investigated through graphs.