Homogeneous Linear Ordinary Differential Equations Research Articles

Riccati equations for holographic 2-point functions

Any second order homogeneous linear ordinary differential equation can be transformed into a first order non-linear Riccati equation. We argue that the Riccati form of the linearized fluctuation equations that determine the holographic 2-point functions simplifies considerably the numerical computation of such 2-point functions and of the corresponding transport coefficients by computing directly the response functions, eliminating the arbitrary source from the start. Moreover, it provides a neat criterion for the infrared regularity of the fluctuations. In particular, it is shown that the infrared regularity conditions for scalar and tensor fluctuations coincide, and hence they are either both regular or both singular. We demonstrate our numerical recipe based on the Riccati equations by computing the holographic 2-point functions for the stress tensor and a scalar operator in a number of asymptotically anti de Sitter backgrounds of bottom up scalar-gravity models. Analytical results are obtained for the 2-point function of the transverse traceless part of the stress tensor in two confining geometries, including a geometry that belongs to the class of IHQCD. We find that in this background the spin-2 spectrum is linear and, as expected, the position space 2-point function decays exponentially at large distances at a rate proportional to the confinement scale.

An analytic model for a [formula omitted] isotope separation process by cryogenic distillation

In this paper we present a structured approach to the modeling of an isotope separation plant that makes use of distillation principles for C13 isotope separation. In the first part of the paper, after a brief review of isotope separation processes with an accent on isotope separation by distillation, we define our initial-boundary-value problem, which is a partial differential equation. By applying the Laplace transform to the partial differential equation that governs the evolution of the desired isotope with respect to height and time, we obtain a linear homogeneous ordinary differential equation. After solving the obtained ordinary differential equation, we use Heaviside's expansion theorem to find the inverse Laplace transform, and thus, the analytic model of the isotope separation process follows. Using the analytic model it is possible to determine the concentration of the desired isotope at any height, at any moment of time, with respect to plant parameters and thus, expensive experiments can be avoided. In the second part of the paper we use the analytic model to simulate the isotope separation process, followed by the assessment of the results against experimental data.

Series solution of second-order linear homogeneous ordinary differential equations via complex integration

A method is presented, with standard examples, based on an elementary complex integral expression, for developing, in particular, series solutions for second-order linear homogeneous ordinary differential equations. Straightforward to apply, the method reduces the task of finding a series solution to the solution, instead, of a system of simple equations in a single variable. The method eliminates the need to manipulate power series and balance powers, which is a characteristic of the usual approach. The method originated with Herrera [3], but was applied to the solution of certain classes of nonlinear ordinary differential equations by him. Mathematics Subject Classification: 30B10, 30E20 34A25, 34A30

New complex soliton-like solutions of combined KdV equation with variable coefficients and forced term

The [G()]/[G()] expansion method is extensively studied to search for new infinite sequence of complex solutions to nonlinear evolution equations with variable coefficients. According to a function transformation, the solving of homogeneous linear ordinary differential equation with constant coefficients of second order can be changed into the solving of a one-unknown quadratic equation and the Riccati equation. Based on this, new infinite sequence complex solutions of homogeneous linear ordinary differential equation with constant coefficients of second order are obtained by the nonlinear superposition formula of the solutions to Riccati equation. By means of the new complex solutions, new infinite sequence complex soliton-like exact solutions to the combined KdV equation with variable coefficients and forced term are constructed with the help of symbolic computation system Mathematica.

On the procedure for the series solution of certain general-order homogeneous linear differential equations via the complex integration method

The theory of series solutions for two important classes of the general higherorder linear homogeneous ordinary differential equation is developed ab initio, using an elementary complex integral expression derived and applied in previous papers [10, 11], based on the original work of Herrera [5]. As well as producing general expressions for the recurrence relations for higher-order equations with analytic coefficients or the general-order Fuchs’ equation, the complex integral method is straight-forward to apply as an algorithm on its own. ‘Benchmark’ examples from the general mathematic literature, are presented and a brief discussion of ‘logarithmic’ solutions is included.

Solution of First Order Linear Non Homogeneous Ordinary Differential Equation in Fuzzy Environment Based on Lagrange Multiplier Method

In this paper the First Order Linear Non Homogeneous Ordinary Differential Equations (FOLNHODE) are described in fuzzy environment. Here coefficients and /or initial condition of FOLNHODE are taken as Generalized Triangular Fuzzy Numbers (GTFNs).The solution procedure of the FOLNHODE in fuzzy environment is developed by Lagrange multiplier Method. Finally a fish population problem is discussed in fuzzy environment.

New infinite sequence soliton-like solutions of (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation

This paper will study in detail homogeneous linear ordinary differential equation with constant coefficients of second order and draw new conclusion to construct new infinite sequence soliton-like solutions of high-dimensional nonlinear evolution equations. Step one: the solving of a homogeneous linear ordinary differential equation with constant coefficients of second order is changed into the solving of the quadratic equation with one unknown and the Riccati equation. Based on this, new infinite sequence solutions of homogeneous linear ordinary differential equation with constant coefficients of second order are found by using nonlinear superposition formula for the solutions to Riccati equation. Step two: new infinite sequence soliton-like solutions to (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation are constructed using the above conclusion and the symbolic computation system Mathematica.

First Order Linear Homogeneous Ordinary Differential Equation in Fuzzy Environment Based On Laplace Transform

In this paper the First Order Linear Ordinary Differential Equations (FOLODE) are described in fuzzy environment. Here coefficients and /or initial condition of FOLODE are taken as Generalized Triangular Fuzzy Numbers (GTFNs).The solution procedure of the FOLODE is developed by Laplace transform. It is illustrated by numerical examples. Finally imprecise bank account problem and concentration of drug in blood problem are described.

First Order Linear Homogeneous Fuzzy Ordinary Differential Equation Based on Lagrange Multiplier Method

In this paper the First Order Linear Fuzzy Ordinary Differential Equations are described. Here coefficients and /or initial condition of said differential equation are taken as the Generalized Triangular Fuzzy Numbers (GTFNs).The solution procedure of this Fuzzy Differential Equation is developed by Lagrange Multiplier Method. An imprecise barometric pressure problem is described.

Propagator for the general time-dependent harmonic oscillator with application to an ion trap

We present the simplest possible formula for the propagator of the general time-dependent quadratic Hamiltonian, including linear terms. The method is based on the use of a linear time-dependent invariant and requires only the solution of a linear homogeneous second-order ordinary differential equation corresponding to the classical quadratic Hamiltonian. We give an example for the case of the Paul trap.

Perturbation of solutions of ordinary linear homogeneous differential equations of the second order

In this paper we give a method for peturbation of solutions of linear homogeneous differential equation of the second order.

Analytical solution for wave-induced response of isotropic poro-elastic seabed

By use of separation of variables, the governing equations describing the Biot consolidation model is firstly transformed into a complex coefficient linear homogeneous ordinary differential equation, and the general solution of the horizontal displacement of seabed is constructed by employing a complex wave number, thus, all the explicit analytical solutions of the Biot consolidation model are determined. By comparing with the experimental results and analytical solution of Yamamoto etc. and the analytical solution of Hsu and Jeng, the validity and superiority of the suggested solution are verified. After investigating the influence of seabed depth on the wave-induced response of isotropic poro-elastic seabed based on the present theory, it can be concluded that the influence depth of wave-induced hydrodynamic pressure in the seabed is equal to the wave length.

On the optimal recovery of a solution of a system of linear homogeneous differential equations

We consider the optimal recovery problem for a solution of a system of linear homogeneous ordinary differential equations with constant self-adjoint coefficient matrix on the basis of inexactly given values of the solution at two time instants.

On continuous symmetries of second-order homogeneous linear ordinary differential equations

In this work, a method to extract continuous syrnmetries of general second-order linearl ordinary differential equation is presented. The formalism is illustrated by the examples.

Some classes of Chebyshev systems

Let L be a linear real space of solutions of a linear homogeneous ordinary differential equation with constant coefficients. This space is a Chebyshev space on any interval of length at most π/ω, where ω is maximal among the imaginary parts of the roots of characteristic polynomial. It turns out that this estimate for the length of the interval is exact in a certain sense. Bibliography: 2 titles.

Elasticity, shell theory and finite element results for the buckling of long sandwich cylindrical shells under external pressure

The buckling of a sandwich cylindrical shell under uniform external hydrostatic pressure is studied in three ways. The simplifying assumption of a long shell is made (or, equivalently, ‘ring’ assumption), in which the buckling modes are assumed to be two-dimensional, i.e. no axial component of the displacement field, and no axial dependence of the radial and hoop displacement components. All constituent phases of the sandwich structure, i.e. the facings and the core, are assumed to be orthotropic. First, the structure is considered a three-dimensional (3D) elastic body, the corresponding problem is formulated and the solution is derived by solving a set of two linear homogeneous ordinary differential equations of the second-order in r (the radial coordinate), i.e. an eigenvalue problem for differential equations, with the external pressure, p the parameter/eigenvalue. A complication in the sandwich construction is due to the fact that the displacement field is continuous but has a slope discontinuity at the face-sheet/core interfaces, which necessitates imposing ‘internal’ boundary conditions at the face-sheet/core interfaces, as opposed to the traditional two-end-point boundary value problems. Second, the structure is considered a shell and shell theory results are generated with and without accounting for the transverse shear effect. Two transverse shear correction approaches are employed, one based only on the core, and the other based on an effective shear modulus that includes the face-sheets. Third, finite element results are generated by use of the ABAQUS finite element code. In this part, two types of elements are used: a shear deformable shell element and a solid 3D (brick) element. The results from all these three different approaches are compared.

Resolvent Estimates for Plane Couette Flow

We discuss the problem of deriving estimates for the resolvent of the linear operator associated with three-dimensional perturbations of plane Couette flow and determining its dependence on the Reynolds number R.Depending on the values of the parameters involved, we derive estimates analytically. For the remaining values of the parameters, we prove that deriving estimates for the resolvent can be reduced to estimating the solutions of a 4th-order linear homogeneous ordinary differential equation with nonhomogeneous boundary conditions. We study these boundary value problems numerically. Our results indicate the L2 norm of the resolvent to be proportional to R2 .

Formal Solutions of Linear Ordinary Differential Equations Containing m-Hypergeometric Series

Let a linear homogeneous ordinary differential equation with polynomial coefficients over a field \mathbb{K} be given. For a singular point of the equation, the fundamental system of formal solutions that contain a finite number of power series with coefficients belonging to the algebraic extension of \mathbb{K} can be constructed by known algorithms. In this paper, an algorithm is suggested for construction of a space of formal solutions such that all series containing in these solutions have m-hypergeometric coefficients. The implementation of the algorithm in the computer algebra system Maple is discussed.

Influence functions for usual fourth-order homogeneous linear ordinary differential equations

Influence functions in various situations are derived. They are very useful for solving problems of static buckling, stability and vibration of elastic bodies. Some of them have been utilized to research vibrations of beams, thin-walled bars, shells and plates.

Inertia effects on non-parallel thermal instability of natural convection flow over horizontal and inclined plates in porous media

The inertia effect on the onset of thermal instability in natural convection flow over heated horizontal and inclined flat plates embedded in fluid-saturated porous media is analyzed. The linear non-parallel flow model is employed in the instability analysis, which takes into account the streamwise variation as well as the transverse variation of the disturbance amplitude functions. The set of partial differential equations for the disturbance amplitude functions are converted to a system of homogeneous linear ordinary differential equations with homogeneous boundary conditions by the local non-similarity method. The resulting eigenvalue problem is then solved by an implicit finite-difference method. Representative neutral stability curves and critical Rayleigh numbers are presented. It has been found that as the angle of inclination relative to the horizontal increases, the surface heat transfer rate increases, whereas the flow becomes more stable to the vortex mode of instability. Also, as the inertia effect, expressed in terms of Forchheimer number, Fr, increases, the heat transfer rate decreases, but the flow becomes more stable. It is demonstrated that the non-parallel flow model predicts a more stable flow than the parallel flow model.