SOLUTIONS OF MAGNETOHYDRODYNAMICS EQUATIONS-THE THEORY OF FUNCTIONS OF A COMPLEX VARIABLE UNDER DIRAC-PAULI REPRESENTATION AND ITS APPLICATION IN FLUID DYNAMICS (Ⅳ)

This work is the continuation of the discussion of Refs.[1 3].(A) We turn the Magnetohydrodynamics equaitons of isentropic compresible andnon-dissipative magneto-flow into the form of the ideal hydrodynamics equations in thispaper;we can obtain the general Chaplygin equation from Ref.[3],and the generalsduction of this equation.(B)We apply the theory of functions of a complex rariable under Dirac-paulirepresentation,turn the general Magnetohydrodyamics equations of incompressiblemageto-flow into two nonlinear equaitons for flow function and"magneto-flow function",and obtain the exact stable solution of incompressible magnetohydrodynamics equationsunder the condition of stable magnetic field(i.e.under conditon of equality for kinematicalviscid coefficient or viscid diffusion coefficient with magnetic diffusion coefficient).     

ON THE GENERAL EQUATION AND THE GENERAL SOLUTION IN PROBLEMS FOR PLASTODYNAMICS WITH RIGID-PLASTIC MATERIAL

This work is the continuation of the discussion of refs.[1-2].We discuss thedynamics problems of ideal rigid—plastic material in the flow theory of plasticity in thispaper.From introduction of the theory of functions of complex variable under Dirac-Paulirepresentation we can obtain a group of the so-called“general equations”(i.e.have twoscalar equations)expressed by the stream function and the theoretical ratio.In this paperwe also testify that the equation of evolution for time in plastodynamics problema is neitherdissipative nor disperive,and the eigen-equation in plastodynamics problems is a stationarySchr(?)dinger equation,in which we take partial tensor of stress-increment as eigenfunctionsand take theoretical ratio as eigenvalues.Thus,we turn nonlinear plastodynamics problemsinto the solution of linear stationary Schr(?)dinger equation,and from this we can obtain thegeneral solution of plastodynamics problems with rigid-plastic material.     

THE SCHRDINGER EQUATION OF THIN SHELL THEORIES

This work is the continuation of the discussions of[50]and[51].In this paper:(A)The Love-Kirchhoff equation of small deflection problem for elastic thin shellwith constant curvature are classified as the same several solutions of Schr(?)dingerequation,and we show clearly that its form in axisymmetric problem;(B)For example for the small deflection problem,we extract the general solution ofthe vibration problem of thin spherical shell with equal thickness by the force in centralsurface and axisymmetric external field,that this is distinct from ref.[50]in variable.Today the variable is a space-place,and is not time;(C)The von Kármán-Vlasov equation of large deflection problem for shallow shellare classified as the solutions of AKNS equations and in it the one-dimensional problem isclassified as the solution of simple Schr(?)dinger equation for eigenvalues problem,and wetransform the large deflection of shallow shell from nonlinear problem into soluble linearproblem.     

CHAPLYGIN EQUATION IN THREE-DIMENSIONAL NON-CONSTANT ISENTROPIC FLOW-THE THEORY OF FUNCTIONS OF A COMPLEX VARIABLE UNDER DIRAC-PAULI REPRESENTATION AND ITS APPLICATION IN FLUID DYNAMICS(Ⅲ)

This work is the continuation of the discussion of Ref.[1].In this Paper we resoive theequations of isentropic gas dynamics into two problems:the three-dimensional non-constant irrotational flow (thus the isentropic flow,too),and the three-dimensional non-constant indivergent flow (i.e.the in compressible isentropic flow).We apply the theory offunctions of a complex variable under Dirac-Pauli representation and the Legendretransformation,transform these equations of two problems from physical space intovelocity space,and obtain two general Chaplygin equations in this paper.The generalChaplygin equation is a linear difference equation,and its general solution can be expressedat most by the hypergeometric functions.Thus we can obtain the general solution of generalproblems for the three-dimensional non-constant isentropic flow of gas dynamics.     

NEW METHOD OF SOLVING LAME-HELMHOLTZ EQUATION AND ELLIPSOIDAL WAVE FUNCTIONS

Despite the great significance of equations with doubly-periodic coefficients in the methods of mathematical physics, the problem of solving Lamé-Helmholtz equation still remains to be tackled. Arscott and M(?)glich method of double-series expansion as well as Malurkar nonlinear integral equation are incapable of reaching the final explicit solution. Our main result consists in obtaining analytic expressions for ellipsoidal wave functions of four species including the well known Lame functions E_(ci)(sna),E_(si)(sna) as special cases.This is effected by deriving two integro-differential equations with variable coefficients and solving them by integral transform. Generalizing Riemann's idea of P function, we introduce D function to express their transformation properties o     

SOLUTION FOR FREE VIBRATION PROBLEM OF MEMBRANE WITH UNEQUAL TENSION IN TWO DIRECTIONS

In this paper,we obtain the analytic solution of free vibra-tion frequency and mode shapes of rectangle,circle and ellip-tic membranes.The approximate solution of membrane with arbi-trary boundary is also obtained.All of these membranes areacted on by unequal tension in two directions.For the rectangle membrane,in this paper we transform itsvibration equation into one of usual membranes by trnasformingthe coordinate,thus it is easy to get the solution.For thecircle membrane,first we transform the coordinate in the sameway we deal with the rectangle membrane.Next we transform thevibration equation into the Mathieu equation,then we get a for-mula of frequeney of that membrane with some Mathieu function’sproperty.In the solution the elliptic membrane is similar tothat of the circle membrane.Finally,some examples are given.     

FURTHER STUDY OF THE RELATION OF VON KF8B6RMN EQUATION FOR ELASTIC LARGE DEFLECTION PROBLEM AND SCHRDINGER EQUATION FOR QUANTUM EIGENVALUES PROBLEM

This work is the continuation and improvement of the discussion of Ref.[1]. We alsoimprove the discussion of Refs.[2-3] on the elastic large deflection problem by results ofthis paper.We again simplify the von Kármán equation for elastic large deflection problem,and finally turn it into the nonlinear Schr(?)dinger equation in this paper.Secondly,weexpand the AKNS equation to still more symmetrical degree under many dimensionalconditions in this paper.Owing to connection between the nonlinear Schr(?)dinger equationand the integrability condition for the AKNS equation or the Dirac equation,we can obtainthe exact solution for elastic large deflection problem by inverse scattering method.In otherwords,the elastic large deflection problem wholly becomes a quantum eigenvalues problem.The large deflection problem with orthorhombic anisotropy is also deduced in thispaper.     

Large time asymptotics for solutions to a nonhomogeneous Burgers equation

In this article, the solutions to a nonhomogeneous Burgers equation subject to bounded and compactly supported initial profiles are constructed. In an interesting study, Kloosterziel （Journal of Engineering Mathematics 24, 213-236 （1990）） represented a solution to an initial value problem （IVP） for the heat equation, with an initial data in a class of rapidly decaying functions, as a series of self-similar solutions to the heat equation. This approach quickly revealed the large time behaviour for the solution to the IVP. Inspired by Kloosterziel＇s approach, the solution to the nonhomogeneous Burgers equation is expressed in terms of the self-similar solutions to the heat equation. The large time behaviour of the solutions to the nonhomogeneous Burgers equation is obtained.     

THE ANTIPLANE PROBLEM OF DOUBLE PERIOD NON-UNIFORM DISTRIBUTION CRACK FIELD

Originating from Robert Hooke’s law and equilibrious equations,and using the theoryof the complex variable function,this paper transfers the rectaugle region in which thecracks emerge into upper-half part of ξplane by means of conformal mapping.Thenaccording to the theory of ref.[1].We find the solution in closed form,thereby the stressintensity factors are derived.     

DISSIPATION MECHANICS AND EXACT SOLUTIONS FOR NONLINEAR EQUATIONS OF DISSIPATIVE TYPE—PRINCIPLE AND APPLICATION OF DISSIPATION MECHANICS(Ⅰ)

This work is the continuation and the distillation of the discussion of Refs. [1-4].(A)From complementarity principle we build up dissipation mechanics in this paper.It is a dissipative theory of correspondence with the quantum mechanics.From this theorywe can unitedly handle problems of macroscopic non-equilibrium thermodynamics andviscous hydrodynamics. and handle each dissipative and irreversible problems in quantummechanics.We prove the basic equations of dissipation mechanics to eigenvalues equationsof correspondence with the Schr(?)dinger equation or Dirac equation in this paper.(B)We unitedly merge the basic nonlinear equations of dissipative type, especially theNavier-Stokes equation as a basic equation for macroscopic non-equilibrium ther-modynamics and viscous hydrodynamics into integrability condition of basic equation ofdissipation mechanics. And we can obtain their exact solutions by the inverse scatteringmethod in this paper.     

NUMERICAL-PERTURBATION ANALYSIS OF EDGE EFFECT IN BENDING LAMINATED PLATE

In this paper,we consider a bending laminated plate.At first,the dimensionlessvariables are used to transform the equilibrium equations of any layer to perturbationdifferential equations.Secondly,the composite expansion is used and the solution domain isdivided into interior and boundary layer regions and the mathematical models for the outersolution and the inner solution are yielded respectively.Then,the inner solution is expressedwith the boundary intergral equation.     

Multiresolution method for bending of plates with complex shapes

A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains. To realize this method, we design a new wavelet basis function, by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains. In the solution of differential equations, various derivatives of the unknown function are denoted as new functions. Then, the integral relations ...     

THE THEORY OF FUNCTIONS OF A COMPLEX VARIABLE UNDER DIRAC—PAULI REPRESENTATION AND ITS APPLICATION IN FLUID DYNAMICS(Ⅰ)

In this paper:(A) We cast aside the traditional quaternion theory and build up the theory offunctions of a complex variable under Dirac-Pauli representation. Thus the multivariateand multidimensional problems become rather simple problems.(B) We simplify the Navier-Stokes equation of incompressible viscous fluid dynamicsand the equations group of isentropic aerodynamics by theory of functions of a complexvariable under Dirac-Pauli representation. And the above-equations, as central problems offluid dynamics, are classified as the nonlinear equation with only one complex unknownfunction.So the changes are the Great Pole, and give birth to the Two Bearings; the TwoBearings give birth to the Four Quadrants, and the Four Quadrants give birth to the EightDiagrams.——Commentaries on Changes,Copula(Ⅰ)     

Solution for free vibration problem of membrane with unequal tension in two directions

In this paper, we obtain the analytic solution of free vibration frequency and mode shapes of rectangle, circle and elliptic membranes. The approximate solution of membrane with arbitrary boundary "is also obtained. All of these membranes are acted on by unequal tension in two directions.For the rectangle membrane, in this paper we transform its vibration equation into one of usual membranes by trnasforming the coordinate, thus it is easy to get the solution. For the circle membrane, first we transform the coordinate in the same way we deal with the rectangle membrane. Next we transform the vibration equation into the Mathieu equation, then we get a formula of frequency of that membrane with some Mathieu function’s property. In the solution the elliptic membrane is similar to that of the circle membrane.Finally, some examples are given.     

ON GENERAL FORM OF NAVIER-STOKES EQUATIONS AND IMPLICIT FACTORED SCHEME

A general weak conservative form of Navier-Stokes equations expressed with respect to non-orthogonal curvilinear coordinates and with primitive variables was obtained by using tensor analysis technique, where the contravariant and covariant velocity components were employed. Compared with the current coordinate transformation method, the established equations are concise and forthright, and they are more convenient to be used for solving problems in body-fitted curvitinear coordinate system. An implicit factored scheme for solving the equations is presented with detailed discussions in this paper. For n-dimensional flow the algorithm requires n-steps and for each step only a block tridiagonal matrix equation needs to be solved. It avoids inverting the matrix for large systems of equations and enhances the speed of arithmetic. In this study, the Beam-Warming's implicit factored schceme is extended and developed in non-orthogonal curvilinear coordinate system.     

HOMOTOPY PERTURBATION SOLUTION AND PERIODICITY ANALYSIS OF NONLINEAR VIBRATION OF THIN RECTANGULAR FUNCTIONALLY GRADED PLATES

In this paper nonlinear analysis of a thin rectangular functionally graded piate is formulated in terms of von-Karman＇s dynamic equations. Functionaily Graded Material （FGM） properties vary through the constant thickness of the plate at ambient temperature. By expansion of the solution as a series of mode functions, we reduce the governing equations of motion to a Duffing＇s equation. The homotopy perturbation solution of generated Duffing＇s equation is also obtained and compared with numerical solutions. The sufficient conditions for the existence of periodic oscillatory behavior of the plate are established by using Green＇s function and Schauder＇s fixed point theorem.     

THE RELATION OF VON KRMN EQUATION FOR ELASTIC LARGE DEFLECTION PROBLEM AND SCHRDINGER EQUATION FOR QUANTUM EIGENALUES PROBLEM

In this paper the solutions of von Karman for elastic large deflection problem areclassified as the several solutions of Schr(?)dinger equation for quantum eigenvaluesproblem,and we present the transfrom for elastic large deflection problem from non-linearequation into linear equation.Thus,we create favourable conditions of the adoption ofconverse scattering methnd and B(?)cklund transformation.We also discuss the largedeflection problem of long and narrow plate.We can study the non-linear transition of elastic thin plate by furnished method fromthis paper.     

THE FINITE DEFLECTION EQUATIONS OF ANISOTROPIC LAMINATED SHALLOW SHELLS

In this paper,basing on ref.[1]we improved andextended that Which is concerned with a view of inves-tigating the finite deflection equations of anisotro-pic laminated shallow shells subjected to static loads,dynamic loads and thermal loads.We have consideredthe most general bending-stretching couplings and theshear deformations in the thickness direction.and de-rived the equilibrium equations,boundary conditionsand initial conditions.The differential equationsexpressed in terms of generalized displacements u_o,υ_o,φ_x, φ_y and w are obtained.From them,we could solvethe problems of stress analysis,deformation,stabilityand vibration.For some commonly encountered cases,we derived the simplified equations and methods.     

NEW THEORY FOR EQUATIONS OF NON-FUCHSIAN TYPE——REPRESENTATION THEOREM OF TREE SERIES SOLUTION (Ⅱ)

Our main result consists in proving the representation theorem. Irregular integral is a new type of analytic function, represented by a compound Taylor-Fourier tree series, in which each coefficient of the Fourier series is a Taylor series, while the Taylor coefficients are tree series in terms of equations parameters, higher order correction terms to each coefficient having tree structure with inexhaustible proliferation. The solution obtained is proved to be convergent absolutely and uniformly in the region defined by coefficient functions of the original equation, provided the structure parameter is less than unity. Direct substitution shows that our tree series solution satisfies the equation explicity generation by generation. As compared with classical theory our method not only furnishes explicit expression of irregular integral, leading to the solution of Poincaré problem, but also provides possibility of extending the scope of investigation for analytic theory to equations with various kinds o     

CORRESPONDENCE FUNCTION D(z)AND GENERALIZED IRREGULAR EQUATIONS

Extending Riemann’s idea of P function(using equation’s parameters to represent thefunction defined by the equation).we introduce correspondence functions (?)(z) to describeregular and irregular integrals in a unifying way.By explicit solution discuss monodromy group of non-Fuchsian equations.The explicitexpressions of exponent and expansion coefficients for Floquet solution are obtained.Method of correspondence functions permits us to obtain systematically the solutionsof generalized irregular equations. having regular,irregular poles,essential.algebraic,transcendental.logarithmic singularities as well as singular line.The representation of basic set of solutions by (?) function makes it possible to enlargethe scope of investgation of analytic theory.The significance of Poincaré’s conjecture is discussed,as(?)functions are automorphic.