Matrix Ordinary Differential Equations Research Articles

Pure Yang–Mills Solutions on dS4

We consider pure SU(2) Yang–Mills theory on four-dimensional de Sitter space dS4 and construct smooth and spatially homogeneous classical Yang–Mills fields. Slicing dS4 as $$\mathbb{R} \times {{S}^{3}},$$ via an SU(2)-equivariant ansatz we reduce the Yang–Mills equations to ordinary matrix differential equations and further to Newtonian dynamics in a particular three-dimensional potential. Its classical trajectories yield spatially homogeneous Yang–Mills solutions in a very simple explicit form, depending only on de Sitter time with an exponential decay in the past and future. These configurations have not only finite energy, but their action is also finite and bounded from below. We present explicit coordinate representations of the simplest examples (for the fundamental SU(2) representation). Instantons (Yang–Mills solutions on the Wick-rotated $${{S}^{4}}$$ ) and solutions on AdS4 are also briefly discussed.

Oscillatory and non oscillatory criteria for the systems of two linear first order two by two dimensional matrix ordinary differential equations

The Riccati equation method is used for study the oscillatory and non oscillatory behavior of solutions of systems of two first order linear two by two dimensional matrix differential equations. An integral and an interval oscillatory criteria are obtained. Two non oscillatory criteria are obtained as well. On an example, one of the obtained oscillatory criteria is compared with some well known results.

Solutions to Yang-Mills Equations on Four-Dimensional de Sitter Space.

We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter space dS_{4} and construct a smooth and spatially homogeneous magnetic solution to the Yang-Mills equations. Slicing dS_{4} as R×S^{3}, via an SU(2)-equivariant ansatz, we reduce the Yang-Mills equations to ordinary matrix differential equations and further to Newtonian dynamics in a double-well potential. Its local maximum yields a Yang-Mills solution whose color-magnetic field at time τ∈R is given by B[over ˜]_{a}=-1/2I_{a}/(R^{2}cosh^{2}τ), where I_{a} for a=1, 2, 3 are the SU(2) generators and R is the de Sitter radius. At any moment, this spatially homogeneous configuration has finite energy, but its action is also finite and of the value -1/2j(j+1)(2j+1)π^{3} in a spin-j representation. Similarly, the double-well bounce produces a family of homogeneous finite-action electric-magnetic solutions with the same energy. There is a continuum of other solutions whose energy and action extend down to zero.

A note on the use of step responses matrix and Lambert W function in the dynamics analysis of time delay systems

For investigation of the dynamics of linear time-delay system it is proposed to use the step responses matrix of the system and apply the Lambert W function method. The elements of this matrix in the first stage are being found on the interval (0 \tau) (here \tau is a fixed delay). For this it is sufficient to solve an ordinary matrix differential equation. After that, using obtained solution as initial vector-function, elements of the step responses matrix are being found on the interval (\tau +\infty) solving a linear homogeneous matrix delay differential equation. The well known Lambert W function method can be used for solution of this equation. The proposed method is illustrated on concrete time-delay systems.DOI: http://dx.doi.org/10.5755/j01.itc.46.2.16208

Resonant-state expansion of light propagation in nonuniform waveguides

A new rigorous approach for precise and efficient calculation of light propagation along non-uniform waveguides is presented. Resonant states of a uniform waveguide, which satisfy outgoing-wave boundary conditions, form a natural basis for expansion of the local electromagnetic field. Using such an expansion at fixed frequency, we convert the wave equation for light propagation in a non-uniform waveguide into an ordinary second-order matrix differential equation for the expansion coefficients depending on the coordinate along the waveguide. We illustrate the method on several examples of non-uniform planar waveguides and evaluate its efficiency compared to the aperiodic Fourier modal method and the finite element method, showing improvements of one to four orders of magnitude. A similar improvement can be expected also for applications in other fields of physics showing wave phenomena, such as acoustics and quantum mechanics.

Static analysis for magneto-electro-elastic plates based on the scaled boundary finite element method

The scaled boundary finite element method (SBFEM) and the precise integration algorithm (PIA) are utilized to analyze the extended displacement field in clamped or simple-supported magneto-electro-elastic plates produced by external transverse loadings. There are no limitation on boundary conditions and types of external forces. Only the in-plane dimensions are divided into 2D elements. By introducing a set of scaled boundary local coordinates, 3D governing partial differential equations are converted into the second order ordinary differential matrix equation. By means of the internal nodal force, a first order ordinary differential equation is obtained and its general solution is a matrix exponential. The PIA is introduced to calculate the matrix exponential and any desired accuracy can be obtained. Finally, several numerical examples are provided to validate the versatility of the proposed technique.

Thermal performance of stratified fluid-filled geomaterials with compressible constituents around a deep buried decaying heat source

This paper investigates the transient thermal performance of stratified fluid-filled geomaterials with compressible constituents due to a buried decaying heat source. Starting with the governing partial differential equations, the ordinary differential matrix equation is derived by means of Laplace transform and Hankel transform. Using the extended precise integration method, the ordinary differential matrix equation is solved in the transformed domain, and the actual solutions are acquired through the inversion of Laplace and Hankel transforms. Two numerical calculations are given to validate the accuracy and feasibility of the presented approach, and other calculations are performed to study the effects of the heat source’s half-life, compressible constituents, thermal expansion coefficients, and the stratified character on the transient thermal response of medium.

Elastic displacement fields of multi-layered transversely isotropic materials under rectangular loads

Based on the precise integration algorithm (PIA) and the technique of dual vector forms, analytical solutions of displacements for the complete elastic field induced by the vertical loadings uniformly distributed over a rectangular area are deduced. The rectangular area is assumed horizontally resting on the surface or embedded in the interior of a linearly elastic, homogeneous transversely isotropic continuum. The planes of transverse isotropy are chosen to be parallel to the horizontal boundary surface. Associated with the approach of dual vector, the classical integral transform method is adopted to convert the partial differential governing equations into the concise first-order ordinary differential matrix equation. As a highly accurate method to solve sets of the first-order ordinary differential equations, the procedure of PIA is introduced to evaluate the key matrix equation. As a result, any desired accuracy of the elastic solutions can be achieved. Additionally, dual vector forms of equations in the transformed domain make the assembly of the global stiffness matrix simple and convenient. Finally, some illustrative examples are analysed to verify the proposed solutions and elucidate the influences of the dimensions of the loading area, the type and degree of material anisotropy and the stratified characters on the load–displacement relationship in the transversely isotropic media.

A study on calculation method for mechanical impedance of air spring

This paper proposes an approximate analytic method of obtaining the mechanical impedance of air spring. The sound pressure distribution in cylindrical air spring is calculated based on the linear air wave theory. The influences of different boundary conditions on the acoustic pressure field distribution in cylindrical air spring are analysed. A 1-order ordinary differential matrix equation for the state vector of revolutionary shells under internal pressure is derived based on the non-moment theory of elastic thin shell. Referring to the transfer matrix method, a kind of expanded homogeneous capacity high precision integration method is introduced to solve the non-homogeneous matrix differential equation. Combined the solved stress field of shell with the calculated sound pressure field in air spring under the displacement harmonic excitation, the approximate analytical expression of the input and transfer mechanical impedance for the air spring can be achieved. The numerical simulation with the Comsol Multiphysics software verifies the correctness of theoretical analysis result.

Elastostatic solutions of a multilayered transversely isotropic piezoelectric system under axisymmetric loading

The precise integration algorithm (PIA) and the technique of dual vector formulation are extended to explore the axisymmetric response of multilayered transversely isotropic piezoelectric materials under the action of external forces. The planes of cross-anisotropy are assumed to be parallel to the horizontal surface. The mechanical and electrical loads distributed over a circular area can be applied either on the surface of the layered system or at the interface between neighboring layers with dissimilar materials. By means of the Hankel integral transformation and the approach of dual vector form, the governing partial differential equations are reduced to the standard ordinary differential matrix equation. The PIA is exploited to evaluate the first-order ordinary differential matrix equation in the transformed domain. As the PIA is a highly accurate algorithm, any desired accuracy of axisymmetric piezoelectric solutions can be accomplished. The whole calculation is based on standard matrix algebra, and computational effort can be saved to a great extent. Furthermore, dual vector forms of key equations can facilitate the combination of adjacent strata. Finally, several numerical examples are given to validate the accuracy of the proposed technique and to investigate the influence of loading positions, types of external forces, stratified characters, and weak and thin interlayer on the response of the layered piezoelectric medium.

Axisymmetric solutions for the multi-layered transversely isotropic piezoelectric medium

By introducing the precise integration algorithm (PIA) and the technique of the mixed variable formalism accounting for any number of layers, variations of components of the axisymmetric piezoelectric field including elastic displacement, vertical normal stress, electric potential and electric displacement along the axis of symmetry in the cylindrical coordinate system are explored. The piezoelectric field is induced by the vertical loadings of mechanical or electric types uniformly distributed over a circular region. With the aid of the mixed variable formulations, along with the Hankel integral transform and the corresponding matrix algebraic operations, the governing partial differential equations of equilibrium expressed in terms of displacements and electric potential are reduced to the first order ordinary differential matrix equations. As a highly accurate algorithm, the PIA is provided to evaluate the obtained ordinary differential matrix equations in the transformed domain. Both mechanical and electrical quantities in the physical domain can be acquired by taking the inversion of the Hankel integral transform. An example as benchmark is illustrated to examine the applicability and performance of the proposed technique compared with results in the literature. Numerical examples are used to demonstrate the influence of the degree of the material anisotropy and stratified parameters.

Elastodynamic response for the multi-layered transversely isotropic piezoelectric solid subjected to time-harmonic loads

The precise integration algorithm (PIA) and the approach of dual vector form are utilized to explore the elastodynamic axisymmetric response of the multi-layered transversely isotropic piezoelectric medium. The planes of transverse isotropy are assumed to be parallel to the horizontal surface of the layered system. The elastic solid is under the action of axisymmetric mechanical or electrical time-harmonic loads prescribed either at the external surface or in the interior of the stratified medium. There is no limit of the thickness and number of layers to be considered. Based on the dual vector form and the Hankel integral transform, the governing equations of the cross-anisotropic piezoelectric material are conveniently turned into the standard ordinary differential matrix equation. The PIA is introduced to evaluate the first-order ordinary differential matrix equation with specified two-end boundary conditions. As PIA is a highly accurate method, any desired accuracy of dynamic piezoelectric solutions can be accomplished. In addition, the dual vector form of motion equation facilitates the combination of two neighboring layers into a new one and computational efforts could be reduced to a great extent. The true piezoelectric field is obtained by taking inversion of the Hankel integral transform. Finally, a comparison with the numerical solutions for stratified piezoelectric half-space is made to confirm the accuracy of the proposed procedure. Meanwhile, some numerical examples are also included to discuss the influence of loading positions, types of external forces, stratified characters and weak and thin interlayer on the dynamic response of the layered piezoelectric medium.

Elastic solutions due to a time-harmonic point load in isotropic multi-layered media

A new analytical derivation of the elastodynamic point load solutions for an isotropic multi-layered half-space is presented by means of the precise integration method (PIM) and the approach of dual vector. The time-harmonic external load is prescribed either on the external boundary or in the interior of the solid medium. Starting with the axisymmetric governing motion equations in a cylindrical coordinate system, a second order ordinary differential matrix equation can be gained by making use of the Hankel integral transform. Employing the technique of dual vector, the second order ordinary differential matrix equation can be simplified into a first-order one. The approach of PIM is implemented to obtain the solutions of the ordinary differential matrix equation in the Hankel integral transform domain. The PIM is a highly accurate algorithm to solve sets of first-order ordinary differential equations and any desired accuracy of the dynamic point load solutions can be achieved. The numerical simulation is based on algebraic matrix operation. As a result, the computational effort is reduced to a great extent and the computation is unconditionally stable. Selected numerical trials are given to validate the accuracy and applicability of the proposed approach. More examples are discussed to portray the dependence of the load-displacement response on the isotropic parameters of the multi-layered media, the depth of external load and the frequency of excitation.

Consolidation analysis of transversely isotropic layered saturated soils in the Cartesian coordinate system by extended precise integration method

A general solution for three-dimensional consolidation of transversely isotropic layered saturated soils is presented as an extension to the extended precise integration solution for consolidation in the cylindrical coordinate system. Starting with the governing equations of Biot's consolidation of transversely isotropic saturated soils in the Cartesian coordinate system, an ordinary differential matrix equation is deduced with the aid of Laplace–Fourier transforms. Based on the precise integration method for two-point boundary value problems, an extended precise integration solution of multilayered systems subjected to internal loads or dislocations is presented and then used to solve the above ordinary differential matrix equation, the actual solution in the physical domain is obtained by taking a numerical inversion. The feasibility of the proposed method is proved by three examples, and the influence of transverse isotropy of the soil skeleton on the consolidation behavior is analyzed.

Axisymmetric coupled thermo-mechanical response of multilayered elastic medium

Most thermal stress analyses are based on the uncoupled theory of thermoelasticity, which considers that the determination of the temperature field is independent from the stress and displacement fields. This paper derives an extended precise integration solution for the coupled thermoelastic problems with the integral transformation techniques. From the axisymmetric fundamental governing equations, an ordinary differential matrix equation is achieved by means of a Laplace–Hankel transform. An extended precise integration method is introduced to solve the above matrix equation in the transformed domain, and the actual solution is acquired by a numerical inverse transformation. Finally, some numerical examples are given to demonstrate the accuracy of the proposed theory and to illustrate the influence of the coupling of thermoelasticity and layered characteristics on the thermo-mechanical response.

Response of multilayered transversely isotropic medium due to axisymmetric loads

Summary A novel procedure associated with the precise integration method (PIM) and the technique of dual vector is proposed to effectively calculate the magnitude and distribution of deformations in a homogeneous multilayered transversely isotropic medium. The planes of transverse isotropy are assumed to be parallel to the horizontal surface of the soil system. The linearly elastic medium is subjected to four types of vertically acting axisymmetric loads prescribed either at the external surface or in the interior of the soil medium. There are no limits for the thicknesses and number of soil layers to be considered. By virtue of the governing equations of motion and the constitutive equations of the transversely isotropic elastic body, and based on the Hankel integral transform and a dual vector formulation in a cylindrical coordinate system, the partial differential motion equations can be converted into first-order ordinary differential matrix equations. Applying the approach of PIM, it is convenient to obtain the solutions of ordinary differential matrix equations for the continuously homogeneous multilayered transversely isotropic elastic soil in the transformed domain. The PIM is a highly accurate algorithm to solve the sets of first-order ordinary differential equations, which can ensure to achieve any desired accuracy of the solutions. What is more, all calculations are based on the standard method with the corresponding algebraic operations. Computational efforts can be reduced to a great extent. Finally, numerical examples are provided to illustrate the accuracy and effectiveness of the proposed approach. Some more cases are analyzed to evaluate the influences of the elastic parameters of the transversely isotropic media on the load-displacement responses. Copyright © 2015 John Wiley & Sons, Ltd.

Axisymmetric dynamic response of the multi-layered transversely isotropic medium

By virtue of the precise integration method (PIM) and the technique of mixed variable formulations, solutions for the dynamic response of the multi-layered transversely isotropic medium subjected to the axisymmetric time-harmonic forces are presented. The planes of cross anisotropy are assumed to be parallel to the horizontal surface of the stratified media. Four kinds of vertically acting axisymmetric loads are prescribed either at the external surface or in the interior of the soil system. Thicknesses and number of the medium strata are not limited. Employing the Hankel integral transform in cylindrical coordinate, the axisymmetric governing equations in terms of displacements of the multi-layered media are uncoupled. Applying mixed variable formulations, more concise first-order ordinary differential matrix equations from the uncoupled motion equations can be obtained. Solutions of the ordinary differential matrix equations in the transformed domain are acquired by utilizing the approach of PIM. Since PIM is highly accurate to solve the sets of first-order ordinary differential equations, any desired accuracy of the solutions can be achieved. All calculations are based on the corresponding algebraic operations and computational efforts can be reduced to a great extent. Comparisons with the existing numerical solutions are made to confirm the accuracy of the present solutions proposed by this procedure. Several examples are illustrated to explore the influences of the type and degree of material anisotropy, the frequency of excitation and loading positions on the dynamic response of the stratified medium.

Extended precise integration method for axisymmetric thermo‐elastic problem in transversely isotropic material

SummaryThis paper presents a numerical solution for the analysis of the axisymmetric thermo‐elastic problem in transversely isotropic material due to a buried heat source by means of extended precise integral method. By virtue of the Laplace–Hankel transform applied into the basic governing equations, an ordinary differential matrix equation is achieved, which describes the relationship between the generalized stresses and displacements in transformed domain. An extended precise integration method is introduced to solve the aforementioned matrix equation, and the actual solution in the physical domain is acquired by inverting the Laplace–Hankel transform. Numerical examples are carried out to demonstrate the accuracy of the proposed method and elucidate the influence of the character of transverse isotropy, the anisotropy of linear expansion coefficient, the anisotropy of thermal diffusivity, and medium's stratification on the thermo‐elastic response. Copyright © 2015 John Wiley & Sons, Ltd.

Extended precise integration method for consolidation of transversely isotropic poroelastic layered media

A new numerical method for calculating the consolidation behavior of the stratified, transversely isotropic and poroelastic material is presented by combining the extended precise integration algorithm with the integral transformation techniques. Starting with the governing partial differential equations of a saturated medium with transversely isotropic skeleton and compressible fluid constituents, an ordinary differential matrix equation is deduced with the aid of a Laplace–Hankel transform. An extended precise integration method for internal loading situations is proposed to solve the ordinary differential matrix equation in the transformed domain, and the actual solution is recovered by a numerical inverse transformation. Numerical examples are also provided to prove the feasibility of this method.

First-order polynomial differential equations over matrix skew series

In this paper we prove that one can reduce the solution of first-order polynomial matrix ordinary differential equations to the integration of similar scalar equations, provided that equation parameters are triangular. We establish requirements to elements of the desired matrix in the case when its parameters are double diagonal matrices. We consider the Riccati equation over the set of third-order square matrices. The obtained results are formulated in terms of “skew series”, the notion of which was introduced by us earlier.