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.

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.

Exactly solving some typical Riemann-Liouville fractional models by a general method of separation of variables

Finding exact solutions for Riemann–Liouville(RL)fractional equations is very difficult.We propose a general method of separation of variables to study the problem.We obtain several general results and,as applications,we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation.In particular,we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation.In addition,we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions.

Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation

In this paper,using the fractional Fourier law,we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system.The method of variable separation is used to solve the timefractional heat conduction equation.The Caputo fractional derivative of the order 0 〈 α≤ 1 is used.The solution is presented in terms of the Mittag-Leffler functions.Numerical results are illustrated graphically for various values of fractional derivative.

CHARACTERISTIC EQUATIONS AND CLOSED-FORM SOLUTIONS FOR FREE VIBRATIONS OF RECTANGULAR MINDLIN PLATES

The direct separation of variables is used to obtain the closed-form solutions for the free vibrations of rectangular Mindlin plates. Three different characteristic equations are derived by using three different methods. It is found that the deflection can be expressed by means of the four characteristic roots and the two rotations should be expressed by all the six characteristic roots,which is the particularity of Mindlin plate theory. And the closed-form solutions,which satisfy two of the three governing equations and all boundary conditions and are accurate for rectangular plates with moderate thickness,are derived for any combinations of simply supported and clamped edges. The free edges can also be dealt with if the other pair of opposite edges is simply supported. The present results agree well with results published previously by other methods for different aspect ratios and relative thickness.

Bending of simply supported incompressible saturated poroelastic beams with axial diffusion

Based on the mathematical model of the bending of the incompressible saturated poroelastic beam with axial diffusion, the qUasi-static bendings of the simply supported poroelastic beam subjected to a suddenly applied constant load were investigated, and the analytical solutions were obtained for different diffusion conditions of the pore fluid at the beam ends. The deflections, the bending moments of the solid skeleton and the equivalent couples of the pore pressures were presented in figures. It is also shown that the behavior of the saturated poroelastic beams depends closely on the diffusion conditions at the beam ends, especially for the equivalent couples of the pore pressures. It is found that the Mandel-Cryer effect also exists in the bending of the saturated poroelastic beams under specific diffusion conditions at the beam ends.

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.

Necessary and Sufficient Conditions for the Separability and the R-Separability of the Irrotational Stokes Equation and Applications

In the present manuscript, we formulate and prove rigorously, necessary and sufficient conditions for all kinds of separation of variables that a solution of the irrotational Stokes equation may exhibit, in any orthogonal axisymmetric system, namely: simple separation and R-separation. These conditions may serve as a road map for obtaining the corresponding solution space of the irrotational Stokes equation, in any orthogonal axisymmetric coordinate system. Additionally, we investigate how the inversion of the coordinate system, with respect to a sphere, affects the type of separation. Specifically, we prove that if the irrotational Stokes equation separates variables in an axisymmetric coordinate system, then it R-separates variables in the corresponding inverted coordinate system. This is a quite useful outcome since it allows the derivation of solutions for a problem, from the knowledge of the solution of the same problem in the inverted geometry and vice-versa. Furthermore, as an illustration, we derive the eigenfunctions of the irrotational Stokes equation governing the flow past oblate spheroid particles and inverted oblate spheroidal particles.

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.

SYMPLECTIC DUALITY SYSTEM ON PLANE MAGNETOELECTROELASTIC SOLIDS

By means of the generalized variable principle of magnetoelectroelastic solids, the plane magnetoelectroelastic solids problem was derived to Hamiltonian system. In symplectic geometry space, which consists of original variables, displacements, electric potential and magnetic potential, and their duality variables, lengthways stress, electric displacement and magnetic induction, the effective methods of separation of variables and symplectic eigenfunction expansion were applied to solve the problem. Then all the eigen-solutions and the eigen-solutions in Jordan form on eigenvalue zero can be given, and their specific physical significations were shown clearly. At last, the special solutions were presented with uniform loader, constant electric displacement and constant magnetic induction on two sides of the rectangle domain.

On superintegrable systems with a position-dependent mass in polar-like coordinates

For a superintegrable system defined in plane polar-like coordinates introduced by Szumiński et al. and studied by Fordy, we show that the system with a position-dependent mass is separable in three distinct coordinate systems. The corresponding separation equations and additional integrals of motion are derived explicitly. The closure algebra of integrals is deduced. We also make a generalization of this system by employing the classical Jacobi method. Lastly a sufficient condition which ensures flatness of the underlying space is derived via explicit calculation.

Achieving thermal magnification by using effective thermal conductivity

thermal magnification device is proposed by using effective thermal conductivity. Different fromtransformation optics method, the magnification design is realized analytically by enforcingequality of effective thermal conductivity on the magnification device and the reference case inspecified domains. The validity of theoretical analysis is checked by numerical simulation results,which demonstrates the magnifying effects of the proposed design. The device only needsisotropic and homogeneous materials that are easy to obtain in nature. It is also shown that theobtained magnifying conditions are the same as those derived by separation of variables. But theproposed method proves more flexible for multilayered materials and simpler for non-sphericalobjects under non-uniform thermal fields. It can also be extended to other fields and applicationsgoverned by Laplace equation.

A New 2 + 1-Dimensional Integrable Variable Coefficient Toda Equation

In this paper, a new integrable variable coefficient Toda equation is proposed by utilizing a generalized version of the dressing method. At the same time, we derive the Lax pair of the new integrable variable coefficient Toda equation. The compatibility condition is given, which insures that the new Toda equation is integrable. To further analyze the character of the Toda equation, we derive one soliton solution of the obtained Toda equation by using separation of variables.

G-Function Solutions for Schrodinger Equation in Cylindrical Coordinates System

In this paper, the Schr?dinger equation is solved by Modified separation of variables (MSV) method suggested by Pishkoo and Darus. Using this method, Meijer’s G-function solutions are derived in cylindrical coordinate system for quantum particle in cylindrical can. All elementary functions and most of the special functions which are the solution of extensive problems in physics and engineering are special cases of Meijer’s G-functions.

HYDRAULIC CHARACTERISTICS OF VERTICAL VORTEX AT HYDRAULIC INTAKES

The trace of vertical vortex flow at hydraulic intakes is of the shape of spiral lines, which was observed in the presented experiments with the tracer technique. It represents the fluid particles flow spirally from the water surface to the underwater and rotate around the vortex-axis multi-cycle. This process is similar to the movement of screw. To describe the multi-circle spiral characteristics under the axisymmetric condition, the vertical vortex would change not only in the radial direction but also in the axial direction. The improved formulae for three velocity components for the vertical vortex flow were deduced by using the method of separation of variables in this article. In the improved formulae, the velocity components are the functions of the radial and axial coordinates, so the multi-circle spiral flow of vertical vortex could be simulated. The calculated and measured results for the vertical vortex flow were compared and the causes of errors were analyzed.

THE GENERAL SCHEME FOR HIGHER－ORDER DECOMPOSITIONS OF ZERO－CURVATURE EQUATIONS ASSOCIATED WITH■（2）

Within the framework of zero-curvature representation theory, the decompositions of eachequation in a hierarchy of zero-curvature equations associated with loop algebra 81(2) by meansof higher-order constraints on potential are given a unified treatment, and the general schemeand uniform formulas for the decompositions are proposed. This provides a method of separationof variables to solve a hierarchy of (1+1)-dimensional integrable systems. TO illustrate the general scheme, new higher-order decompositions of two hierarchies of zero-curvature equations arepresented.

A refined invariant subspace method and applications to evolution equations

The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. A two-component nonlinear system of dissipative equations is analyzed to shed light oi1 the resulting theory, and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differentii equations and their corresponding exact solutions with generalized separated variables.

Jacobi Inversion Problem for the Modified Jaulent-Miodek Hierarchy

The r\|matrices and classical Poisson structures are constructed for x\| and t n\|constrained flows of the modified Jaulent\|Miodek (MJM) hierarchy.The Lax matrix is used to study the separation of variables method for these constrained flows. The Jacobi inversion problem for the MJM equation is obtained through the factorization of the MJM equation and the separability of the constrained flows. This is analogous to separation of variables for solving the MJM equation.

SOLVING AKNS EQUATIONS VIA JACOBI INVERSION PROBLEM

By using Lax representation, we study the separation of variables for x- and tn-finitedimensional integrable Hamiltonian system (FDIHS) obtained from the factorization of AKNShierarchy. Then the separability of X- and tn-FDIHS and the factorization of AKNS hierarchy give rise to the Jacobi inversion problem for soliton equations in AKNS hierarchy. By a standard Jacobi inversion technique, the soliton equations can be solved in terms of Riemann theta function.