A SPECIAL METHOD OF FOURIER SERIES WHICH IS EQUAL TO THE METHOD OF SEPARATION OF VARIABLES ON BOUNDARY VALUE PROBLEM

By the separation of singularity, a special Fourier series solution of the boundary value problem for plane is obtained, which can satisfy all boundary conditions and converges rapidly. II is proved that the solution is equal to the result of separation of variables. As a result, the non-linear characteristic equations resulting from the method of separation of variables are transformed into polynomial equations that can provide a foundation for approximate computation and asymptotic analysis.

On the feasibility of variable separation method based on Hamiltonian system for plane magnetoelectroelastic solids

The eigenvalue problem of an infinite-dimensional Hamiltonian operator appearing in the isotropic plane magnetoelectroelastic solids is studied. First, all the eigenvalues and their eigenfunctions in a rectangular domain are solved directly. Then the completeness of the eigenfunction system is proved, which offers a theoretic guarantee of the feasibility of variable separation method based on a Hamiltonian system for isotropic plane magnetoelectroelastic solids. Finally, the general solution for the equation in the rectangular domain is obtained by using the symplectic Fourier expansion method.

Extended Group Foliation Method and Functional Separation of Variables to Nonlinear Wave Equations

Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.

Numerical Solutions of Finite Well in Two Dimensions Using the Finite Difference Time Domain Method

The higher excited states for two dimensional finite rectangular well potential are calculated numerically,by solving the Schrödinger equation using the finite difference time domain method.Although,this method is suitable to calculate the ground state of the quantum systems,it has been improved to calculate the higher excited states directly.The improvement is based on modifying the iterative process involved in this method to include two procedures.The first is known as cooling steps and the second is known as a heating step.By determining the required length of the cooling iteration steps using suitable excitation energy estimate,and repeating these two procedures using suitable initial guess function for sufficient times.This modified iteration will lead automatically to the desired excited state.In the two dimensional finite rectangular well potential problem both of the suitable excitation energy and the suitable initial guess wave function are calculated analytically using the separation of variables technique.

Prediction and Analysis of the Force and Shape Parameters in Variable Gauge Rolling

Variable gauge rolling is a new process to obtain a plate for which the thickness changes continuously by continuously and dynamically adjusting the roll gap upward and downward in the rolling process.This technology is an efective method for producing lightweight,low-cost,and economical plates.However,variable gauge rolling is an unsteady process,and the changes in the force and deformation parameters are complex.In this research,based on the minimum energy theory of the variational principle and considering the characteristics of the roll movement and workpiece deformation comprehensively,the internal plastic deformation,friction,shear and tension powers,and the minimum result of the total power functional in upward and downward rolling are obtained with the frst integral and then with a variation of adopting the specifc plastic power and strain rate vector inner product.The analytical results of the deformation and force parameters are also established using the variational method.Then the precision of this model is certifed using the measured values in a medium plate hot rolling plant and the experimental data for Tailor Rolled Blank rolling.Good agreement is found.Additionally,the variation rule of bite angle,neutral angle,and location neutral points are shown,and the change mechanism of the friction parameter on the stress state efect coefcient is given in variable gauge rolling.This research proposes a new mathematical model for rolling process control that provides a scientifc basis and technical support for obtaining an accurate section shape in variable gauge rolling production.

Hermiticity of Hamiltonian Matrix using the Fourier Basis Sets in Bond-Bond-Angle and Radau Coordinates

In quantum calculations a transformed Hamiltonian is often used to avoid singularities in a certain basis set or to reduce computation time. We demonstrate for the Fourier basis set that the Hamiltonian can not be arbitrarily transformed. Otherwise, the Hamiltonian matrix becomes non-hermitian, which may lead to numerical problems. Methods for cor- rectly constructing the Hamiltonian operators are discussed. Specific examples involving the Fourier basis functions for a triatomic molecular Hamiltonian （J=0） in bond-bond angle and Radau coordinates are presented. For illustration, absorption spectra are calculated for the OC10 molecule using the time-dependent wavepacket method. Numerical results indicate that the non-hermiticity of the Hamiltonian matrix may also result from integration errors. The conclusion drawn here is generally useful for quantum calculation using basis expansion method using quadrature scheme.

CFD-DEM simulation of fluid-solid flow of a tapered column separation bed

Research on recycling waste Printed Circuit Boards(PCB) is at the forefront of preventing environmental pollution and finding ways to recycle resources.The Tapered Column Separation Bed(TCSB) is invented aiming at disposing the problem that fine particles of waste printed circuit boards cannot be separated efficiently so as to obtain further insight about the underlying mechanisms and demonstrate the separation feasibility in the tapered column separation bed.In this work,a Computational Fluid Dynamics(CFD) coupled with Discrete Element Method(DEM) model for two-phase flow has been extended to simulate the fluid-solid flow in the tapered column separation bed.Its validity is demonstrated by its successful capturing the key features of particles' flow pattern,velocity,the pressure distribution,the axial position with time and axial force for particles with different densities.Simulation results show that the plastic particles and resin particles become overflow,while copper particles,iron particles and aluminum particles successively become underflow,with a discharge water flow rate of 1 m^3/h,an obliquity of 30°.The simulated results agree reasonably well with the experimental observation.Using this equipment to separate waste PCBs is feasible,theoretically.

Solutions of novel soliton molecules and their interactions of(2+1)-dimensional potential Boiti-Leon-Manna-Pempinelli equation

The method of variable separation has always been regarded as a crucial method for solving nonlinear evolution equations.In this paper,we use a new form of variable separation to study novel soliton molecules and their interactions in(2+1)-dimensional potential Boiti–Leon-Manna–Pempinelli equation.Dromion molecules,ring molecules,lump molecules,multi-instantaneous molecules,and their interactions are obtained.Then we draw corresponding images with maple software to study their dynamic behavior.

Analytical solution for the stress field of hierarchical defects:multiscale framework and applications

Hierarchical defects are defined as adjacent defects at different length scales.Involved are the two scales where the stress field distribution is interrelated.Based on the complex variable method and conformal mapping,a multiscale framework for solving the problems of hierarchical defects is formulated.The separated representations of mapping function,the governing equations of potentials,and the stress field are subsequently obtained.The proposed multiscale framework can be used to solve a variety of simplified engineering problems.The case in point is the analytical solution of a macroscopic elliptic hole with a microscopic circular edge defect.The results indicate that the microscopic defect aggregates the stress concentration on the macroscopic defect and likely leads to global propagation and rupture.Multiple micro-defects have interactive effects on the distribution of the stress field.The level of stress concentration may be reduced by the coalescence of micro-defects.This work provides a unified method to analytically investigate the influence of edge micro-defects within the scope of multiscale hierarchy.The formulated multiscale approach can also be potentially applied to materials with hierarchical defects,such as additive manufacturing and bio-inspired materials.

Theoretical solution of a spherically isotropic hollow sphere for dynamic thermoelastic problems

The separation of variables method was successfully used to resolve the spherically symmetric dynamic thermoelastic problem for a spherically isotropic elastic hollow sphere. Use of the integral transform can be avoided by means of this method, which is also appropriate for an arbitrary thickness hollow sphere subjected to arbitrary thermal and mechanical loads. Numerical results are presented to show the dynamic stress responses in the uniformly heated hollow spheres.

New Families of Rational Form Variable Separation Solutions to(2+1)-Dimensional Dispersive Long Wave Equations

With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the （2＋1）-dimensional dispersive long wave equations are constructed by means of a function transformation, improved mapping approach, and variable separation approach, among which there are rational solitary wave solutions, periodic wave solutions and rational wave solutions.

Analytical Solution for Thermal Flutter of Laminates in Supersonic Speeds

As a basic component of engineering fields such as aeronautics, astronautics and shipbuilding, panel structure has been widely used in engineering and scientific research. It is of great theoretical and practical significance to study the vibration of panels. The panel flutter problem has caused widely concerned by researchers at home and abroad during to the emergence of high-speed aircrafts. With regard to the eigenvalue problem of rectangular panels, it is generally believed that it is difficult to obtain a closed form eigen solution in the case of an adjacent boundaries clamped-supported or a free boundary that cannot be decoupled. Aiming at the problem, this paper studies the two-dimensional symmetric orthogonal laminated plate structure in the hypersonic flow in the thermal environment, and combines the first-order piston aerodynamic theory to study a high-precision separation variable method. Through this method, analytical solution to the closed form of the thermal flutter problem of rectangular panels can be obtained under any homogeneous boundary conditions.

THE ACTIVATION METHOD FOR DISCRETIZED CONSERVATIVE NONLINEAR STABILITY PROBLEMS WITH MULTIPLE PARAMETER AND STATE VARIABLES

For nonlinear stability problems of discretized conservative systems with multiple parameter variables and multiple state variables, the activation method is put forward, by which activated potential functions and activated equilibrium equations are derived. The activation method is the improvement and enhancement of Liapunov-Schmidt method in elastic stability theory. It is more generalized and more normalized than conventional perturbation methods. The activated potential functions may be transformed into normalized catastrophe potential functions. The activated equilibrium equations may be treated as bifurcation equations. The researches in this paper will motivate the combination of elastic stability theory with catastrophe theory and bifurcation theory

THE FEM ANALYSIS OF SHAFT UNDER THENON-AXIAL SYMMETRICAL LOAD

The changes of stress and strain around the shaft under the non-axial symmetrical loads during the excavation are analyzed with the finite element semi-analytical method based on the separated variable method. The change laws of deformation and stress of surrounding rocks are obtaied. Moreover, an optimum method of the design and construction of the shaft lining is de veloped. which presents a new train of thought of the design and costruction of the shaft and has important theoretical value and extensive application prospects.

Analysis of steady heat conduction for 3D axisymmetric functionally graded circular plate

The thermal conduction behavior of the three-dimensional axisymmetric functionally graded circular plate was studied under thermal loads on its top and bottom surfaces. Material properties were taken to be arbitrary distribution functions of the thickness. A temperature function that satisfies thermal boundary conditions at the edges and the variable separation method were used to reduce equation governing the steady state heat conduction to an ordinary differential equation （ODE） in the thickness coordinate which was solved analytically. Next, resulting variable coefficients ODE due to arbitrary distribution of material properties along thickness coordinate was also solved by the Peano-Baker series. Some numerical examples were given to demonstrate the accuracy, efficiency of the present model, mad to investigate the influence of different distributions of material properties on the temperature field. The numerical results confirm that the influence of different material distributions, gradient indices and thickness of plate to temperature field in plate can not be ignored.

Variable Separation and Exact Solutions for the Kadomtsev-Petviashvili Equation

In the paper, we will discuss the Kadomtsev-Petviashvili Equation which is used to model shallow-water waves with weakly non-linear restoring forces and is also used to model waves in ferromagnetic media by employing the method of variable separation. Abundant exact solutions including global smooth solutions and local blow up solutions are obtained. These solutions would contribute to studying the behavior and blow up properties of the solution of the Kadomtsev-Petviashvili Equation.

Functional Separable Solutions of Nonlinear Heat Equations in Non-Newtonian Fluids

We study the functional separation of variables to the nonlinear heat equation： ut = （A（x）D（u）ux^n）x＋ B（x）Q（u）, Ax≠0. Such equation arises from non-Newtonian fluids. Its functional separation of variables is studied by using the group foliation method. A classification of the equation which admits the functional separable solutions is performed. As a consequence, some solutions to the resulting equations are obtained.

Restrained Torsion of Thin-walled Box Beam with Honeycomb Core

Restrained torsion of thin-walled box beam with honeycomb core is analyzed on the basis of rigid profile assumption. The method of variable separation is applied and two ordinary differential governing equations are established and solved. The boundary conditions are satisfied rigorously and the solutions are expressed by means of eigen function expansions. The diagram of torque is formulated by trigonometric series and used to determine the coefficients in above expansions. The results of computation provide the chord-wise and span-wise distributions of normal and shear stress in the face plate along with shear stress in the honeycomb core.

Restrained Bending of Thin-Walled Box Beam with Honeycomb Core

Restrained bending of thin-walled box beam with honeycomb core is analyzed on the basis of rigid profile assumption. The method of variable separation is applied and two ordinary differential governing equations are established and solved. The boundary conditions are satisfied rigorously and the solutions are expressed by means of eigen function expansions. The diagram of shearing force is formulated by trigonometric series and used to determine the coefficients in above expansions. The computational resuits give the chord and span wise distributions of nomal and shear stress in the cover plate and the honeycomb core. At the same time, the attenuation of additional stress from fixed end to free end along the length of beam is shown clearly.

Explicit Analytical Solutions of Radial Permeable Power Rate Flow

The research of different kinds of permeable non-Newtonian fluid flow is increasing day by day owing to the development of science,technology and production modes.It is most common to use power rate equation to describe such flows.However,this equation is nonlinear and very difficult to derive explicit exact analytical solutions.Generally,people can only derive approximate solutions with numerical methods.Recently,an advanced separating variables method which can derive exact analytical solutions easier is developed by Academician CAI Ruixian（the method of separating variables with addition）.It is assumed that the unknown variable may be indicated as the sum of one-dimensional functions rather than the product in the common method of separating variables.Such method is used to solve the radial permeable power rate flow unsteady nonlinear equations on account of making the process simple.Four concise（no special functions and infinite series） exact analytical solutions is derived with the new method about this flow to develop the theory of non-Newtonian permeable fluid,which are exponential solution,two-dimensional function with time and radius,logarithmic solution,and double logarithmic solution,respectively.In addition,the method of separating variables with addition is developed and applied instead of the conventional multiplication one.It is proven to be promising and encouraging by the deducing.The solutions yielded will be valuable to the theory of the permeable power rate flow and can be used as standard solutions to check numerical methods and their differencing schemes,grid generation ways,etc.They also can be used to verify the accuracy,convergency and stability of the numerical solutions and to develop the numerical computational approaches.