Separation Of Variables Technique Research Articles

Analytical solution for an interfacial crack subjected to dynamic anti-plane shear loading

A closed form solution for an interfacial crack problem is obtained. The crack lies on the interface of two bonded dissimilar orthotropic strips. Its surfaces are subjected to a dynamic anti-plane shear traction. Separation of variables technique is employed, to reduce the problem to a singular system of triple series equations, then to a singular integral equation. That is solved exactly, such that the asymptotic stress field distribution and the stress intensity factor are obtained in closed form expressions. The validity of the solution is proved. Further, a parametric study is introduced to investigate the effects of elastic and geometric characteristics of the composition on the values of the dynamic stress intensity factor (DSIF).

Electromagnetic Wave Scattering by Elliptic Chiral Cylinder

An exact boundary value solution for the scattering of transverse magnetic (TM) wave by an elliptic chiral cylinder is developed and presented in this paper. The solution is based on the separation of variable technique in the elliptic cylinder coordinates system and expressed in terms of Mathieu and modified Mathieu functions. The incident, transmitted and scattered electromagnetic waves are expressed in terms of infinite series of wave functions. The matrix form of the expansion coefficients is found by applying the boundary conditions and orthogonality of the Mathieu functions. Numerical results of the forward and back scattered echo widths for co- and cross-polarized waves with TM and TE polarized incident fields, for various cases are presented and discussed.

Transient responses of a magneto-electro-elastic hollow sphere for fully coupled spherically symmetric problem

By introducing a dependent variable and a special function satisfying the inhomogeneous mechanical boundary conditions, the governing equation for a new variable with homogeneous mechanical boundary conditions is derived. Then by means of the separation of variables technique and the electric and magnetic boundary conditions, the dynamic problem of a magneto-electro-elastic hollow sphere under spherically symmetric deformation is transformed to two Volterra integral equations of the second kind about two functions of time. Cubic Hermite polynomials are adopted to approximate the two undetermined functions at each time subinterval and the recursive formula is obtained to solve the integral equations successfully. The transient responses of displacements, stresses, electric and magnetic potentials are completely determined at the end. Numerical results are presented.

The transient responses of a special non-homogeneous magneto-electro-elastic hollow cylinder for axisymmetric plane strain problem

By virtue of the introduction of new dependent variable and the separation of variables technique, the transient responses of a special non-homogeneous magneto-electro-elastic hollow cylinder are transformed to two Volterra integral equations of the second kind of about two functions with respect to time. These integral equations can be solved successfully by means of the interpolation method. Then, the complete solutions of displacements, stresses, electric potential, electric displacements, magnetic potential and magnetic inductions are obtained. The present method is suitable for a magneto-electro-elastic hollow cylinder with an arbitrary thickness subjected to arbitrary axisymmetric mechanical and electromagnetic loads. Numerical results are finally presented.

Effect of the thermal conductivity of building materials on the steady-state thermal behaviour of underground building envelopes

Using a mathematical model of heat transfer of basement, the calculation area is divided into eight rectangles according to the interzone temperature profile estimation (ITPE) technology, and the solution obtained for all the parts by the separation of variables technique. During the solution, the Fourier coefficients are determined by the continuity of the heat flux and boundary conditions, as a result, a system of linear equation group including 14 N equations has been obtained. The effect of building materials on the temperature and heat flux for the building envelope has been evaluated by calculating the temperature and heat flux for three buildings built with different building materials. The results show that the highest relative error in heat flux among the three envelopes built with various non-insulation material is 27.8%, which indicates the thermal conductivity of the building materials is an important factor of the heat transfer of the envelope.

Propagation and Radiation Characteristics of Dielectric-Wall Conical Horn Antenna

Propagation behaviour, aperture field distribution, radiation pattern and gain of dielectric wall conical horn antenna have been studied theoretically and/or experimentally. The solution of Maxwell's equations is obtained for the hybrid (HE/EH) modes using separation of variables technique. The characteristic equation, which has been obtained by applying continuity of electric and magnetic fields at the dielectric-air boundaries, has been solved for the propagation constant for different hybrid modes. The dependence of the root of the characteristic equation related to radial propagation constant for different hybrid modes, on cone length, flare angle and dielectric constant of the horn dielectric, εr2 is also studied. The estimated aperture field distributions obtained for HE11 mode of the horns of different dimensions but each with the same value of the dielectric constant of the horn dielectric, εr2 = 2.56 in φ = 0° and φ = 90° planes are compared with corresponding experimental ones at 9.4 GHz in X-band. The radiation characteristics of these horns are also measured.

On the analytical separation of variables solution for a class of partial integro-differential equations

The present work deals with the derivation of analytical solutions for a particular class of partial intregro-differential equations, by employing the separation of variables technique. The equation under consideration consists of parabolic and hyperbolic terms, so that the character of the equation is determined by their relative weight. The key feature for the analysis presented here is that the eigenfunctions of the problem, which can be found in a closed form, do not comprise an orthogonal set of functions. The convergence of the new series solution is getting faster as the parabolic term of the integro-differential equation dominates over the hyperbolic (integral) one.

Threshold Voltage Model for Mesa-Isolated Small Geometry Fully Depleted SOI MOSFETs Based on Analytical Solution of 3-D Poisson's Equation

A threshold voltage model for mesa-isolated fully depleted silicon-on-insulator (FDSOI) MOSFETs, based on the analytical solution of three-dimensional (3-D) Poisson's equation is presented for the first time in this paper. The separation of variables technique is used to solve the 3-D Poisson's equation analytically with appropriate boundary conditions. Simple and accurate analytical expressions for the threshold voltage of the front and the back gate are derived. The model is able to predict short channel as well as narrow width effects in mesa-isolated FDSOI MOSFETs. The model is validated by comparing with the experimental results as well as with the numerical results available in the literature.

Joule heating effects in electroosmotically driven microchannel flows

Analytical solutions for temperature distributions, heat transfer coefficients, and Nusselt numbers of steady electroosmotic flows are obtained for two-dimensional straight microchannels. This analysis is based on an infinitesimal electric double layer in which flow velocity becomes “plug-like”, except very close to the wall. Both constant surface temperature and constant surface heat flux conditions are considered in this study. Separation of variables technique is applied to obtain analytical solutions of temperature distributions from the energy equation of electroosmotically driven flows. The thermal analysis considers interaction among inertial, diffusive and Joule heating terms in order to obtain thermally developing behavior of electroosmotic flows. Heat transfer characteristics are presented for low Reynolds number microflows where the viscous and electric field terms are very dominant. For the parameter range studied here ( Re⩽0.7), the Nusselt number is independent of the thermal Peclet number, except in the thermally developing region. Analytical results for cases with and without Joule heating are also compared with the existing heat transfer results, and an excellent agreement is obtained between them.

Analytical solution of transient heat conduction in a two-layer anisotropic cylindrical slab excited superficially by a short laser pulse

This paper presents an exact analytical solution of 2D transient temperature, caused by a non-periodical pulse heating of a cylindrical two-layer slab. The corresponding partial differential equations are solved step-by-step, using the separation of variables technique. Analytical/computational procedure for eigenvalue derivation is considered separately. General initial assumptions, such as existence of thermal contact resistance, thermal conduction and diffusion anisotropy, radiative heat losses, non-uniform heating, and finite absorption depth are accounted in the solution. Numerical simulations of temperature distribution throughout the sample are given at the end of the paper.

Analytical solution for the electroelastic dynamics of a nonhomogeneous spherically isotropic piezoelectric hollow sphere

By introduction of a special dependent variable and separation of variables technique, the electroelastic dynamic problem of a nonhomogeneous, spherically isotropic hollow sphere is transformed to a Volterra integral equation of the second kind about a function of time. The equation can be solved by means of the interpolation method, and the solutions for displacements, stresses, electric displacements and electric potential are obtained. The present method is suitable for a piezoelectric hollow sphere with an arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are presented at the end.

Dynamic responses of a functionally graded pyroelectric hollow sphere for spherically symmetric problems

The spherically symmetric thermoelastic problem of a functionally graded pyroelectric hollow sphere is considered. The functionally graded material is restricted to the variations of density and other material constants as a certain power of the radial distance. By virtue of the introduction of a dependent variable as well as the separation of variables technique, a Volterra integral equation of the second kind related to a function with respect to time is derived, which is solved successfully by using an interpolation polynomial to approximate the unknown function. The corresponding recursive formulae are derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently and is suitable for a functionally graded pyroelectric hollow sphere with an arbitrary thickness and subjected to arbitrary spherically symmetric thermal loads. Numerical results are finally presented.

Analytical solution of a special non-homogeneous pyroelectric hollow cylinder for piezothermoelastic axisymmetric plane strain dynamic problems

By virtue of the introduction of a dependant variable and the separation of variables technique, the piezothermoelastic axisymmetric plane strain dynamic problem of a special non-homogeneous pyroelectric hollow cylinder is transformed to a Volterra integral equation of the second kind about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solutions of displacements, stresses, electric displacements and electric potential are obtained. The present method is suitable for a non-homogeneous pyroelectric hollow cylinder with an arbitrary thickness subjected to arbitrary axisymmetric thermal loads. Numerical results are presented graphically.

Transient Responses in a Piezoelectric Spherically Isotropic Hollow Sphere for Symmetric Problems

By virtue of the separation of variables technique, the spherically symmetric electroelastic dynamic problem of a spherically isotropic hollow sphere is transformed to an integral equation about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solution of displacements, stresses, electric displacements, and electric potential are obtained. The present method is suitable for a piezoelectric hollow sphere with an arbitrary thickness subjected to spherically symmetric electric potential and radial mechanical loads, that both can be arbitrary functions about the time variable, at the internal and external surfaces.

Analytical solution for the axisymmetric plane strain electroelastic dynamics of a special non-homogeneous piezoelectric hollow cylinder

By virtue of the introduction of a dependent variable and the separation of variables technique, the axisymmetric plane strain electroelastic dynamic problem of a special non-homogeneous piezoelectric hollow cylinder is transformed to a Volterra integral equation of the second kind about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solutions of displacements, stresses, electric displacements and electric potential are obtained. The present method is suitable for a piezoelectric hollow cylinder with an arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are finally presented.

On Stress-Focusing Effect in a Uniformly Heated Solid Sphere

By using the separation of variables technique, the dynamic thermal stress responses in an isotropic solid sphere subjected to uniform temperature rise all over the sphere and a sudden constant pressure at the external surface are performed successfully. The analytical solutions of the radial and hoop dynamic stresses at the center are also obtained. By means of the present method, integral transform can be avoided. Numerical results denote that a very high dynamic stress peak appears periodically at the center of the isotropic solid sphere subjected to uniform temperature rise all over the sphere and a sudden constant pressure at the external surface.

Propagation of EM waves in composite bianisotropic cylindrical structures

Propagation of electromagnetic waves in a bianisotropic cylinder embedded in an unbounded bianisotropic space and enclosing an array of parallel bianisotropic circular rods is studied. Based on a separation of variables technique which is facilitated by the use of suitable translation-addition relations, the analysis ends up with an infinite homogeneous system of linear algebraic equations. All matrix elements are given by pole-free, single-term, closed-form expressions. Numerical results are presented for several cases along with comparisons with previously published data. These results reveal the possibility to dynamically control the dispersion characteristics of the structure via changes in the constitutive parameters of the materials involved.

ANALYTICAL SOLUTION OF A PYROELECTRIC HOLLOW CYLINDER FOR PIEZOTHERMOELASTIC AXISYMMETRIC DYNAMIC PROBLEMS

By virtue of the separation of variables technique, the piezothermoelastic axisymmetric dynamic problem of a pyroelectric hollow cylinder is transformed to a second kind of Volterra integral equation about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solution of displacements, stresses, electric displacements, and electric potential are obtained. The present method is suitable for a pyroelectric hollow cylinder with an arbitrary thickness subjected to arbitrary thermal loads. Numerical results are presented.

ELECTROMAGNETIC SCATTERING BY A CONDUCTING CYLINDER COATED WITH METAMATERIALS - ABSTRACT

The electromagnetic scattering from a conducting cylinder coated with metamaterials, which have both negative permittivity and permeability, is derived rigorously by using the classic separation of variables technique. It is found that a conducting cylinder coated with metamaterials has anomalous scattering cross section compared to that coated with conventional dielectric materials. Numerical results are presented and discussed for the scattering cross section of a conducting cylinder coated with metamaterials.

The transient responses of piezoelectric hollow cylinders for axisymmetric plane strain problems

By virtue of the separation of variables technique, the axisymmetric plane strain electroelastic dynamic problem of hollow cylinder is transferred to an integral equation about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solution of the displacements, stresses, electric displacements and electric potentials are finally obtained. The present method is suitable for the hollow cylinder with arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are also presented.