Second-order Linear Homogeneous Differential Equations Research Articles

Degenerate Canonical Forms of Ordinary Second-Order Linear Homogeneous Differential Equations

For each fundamental and widely used ordinary second-order linear homogeneous differential equation of mathematical physics, we derive a family of associated differential equations that share the same “degenerate” canonical form. These equations can be solved easily if the original equation is known to possess analytic solutions, otherwise their properties and the properties of their solutions are de facto known as they are comparable to those already deduced for the fundamental equation. We analyze several particular cases of new families related to some of the famous differential equations applied to physical problems, and the degenerate eigenstates of the radial Schrödinger equation for the hydrogen atom in N dimensions.

ANALYSIS OF EXAMPLES OF APPLYING DIFFERENTIAL EQUATIONS IN CHEMICAL AND FOOD TECHNOLOGIES

The article deals with applying mathematics in chemistry and chemistry-technology. Specifically, differential equations are extensively used in various fields of science and technology. That is why the theory of differential equations, as a separate topic in the course of higher mathematics, is of major importance in educational system of future mechanics, physicists, electrical engineers, chemists, mechanical engineers etc. A possibility of using differential equations in solving various chemical problems is demonstrated. Some chemical technology problems are exemplified whose general solution is reduced to separating variables equations, first-order linear differential equations, second-order linear homogeneous differential equations. It is noteworthy that in solving chemical technology problems we deal with all of these types of differential equations. First-order homogeneous differential equations are applied in solving the following problems: chemical compounds chlorination; chemical agent consumption with maximum end product yield in complex reactions. Second-order non-homogeneous differential equations with constant coefficients are used in solving problems of a system of reverse reactions running at constant volume; continuous hydrolysis of solid fat in a spray column. Second-order differential equations which allow reduction of order are utilized for problems such as liquid movement in capillaries. Second-order linear non-homogeneous differential equations with constant coefficients are applied to solve various problems, e.g. to find a law of motion of a particle that falls as a precipitate in a liquid having no initial velocity.

A powerful diagnostic tool of analytic solutions of ordinary second-order linear homogeneous differential equations

Very few of the ordinary second-order linear homogeneous (OSLH) differential equations are known to possess analytic, closed-form solutions and there is no general theory to predict whether such solutions exist in each particular case. In this work, we present a powerful new method for the discovery of closed-form solutions in the entire class of the OSLH differential equations. A sufficient condition for the existence of such solutions is that the equations can be transformed to forms with constant coefficients and we show that two predictors exist, a nonlinear first-order Bernoulli equation with index n=3/2 and a nonlinear second-order equation. There is no need to solve these equations, only to verify them based on the contents of a given OSLH equation. Because the predictors can be verified quite easily in all cases, we believe that this methodology is an important new diagnostic tool for studying all the equations of mathematical physics that belong to the OSLH class.

The intervals of oscillations in the solutions of the radial Schrödinger differential equation

We have previously formulated a program for deducing the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations. In this work, we demonstrate how the oscillation-detection program can be carried out for the radial Schrodinger equation when a Coulomb potential is used to describe the hydrogen atom. The method predicts that the oscillation intervals are finite in radius and their sizes are determined uniquely by the two quantum numbers n and l. Numerical integrations using physical boundary conditions at the origin confirm this oscillatory behavior of the radial Coulomb wavefunctions. Two related differential equations due to Kummer and Whittaker and other attractive electrostatic potentials are also discussed in the same context.

The intervals of oscillations in the solutions of the Legendre differential equations

We have previously formulated a program for deducing the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations. In this work, we demonstrate how the oscillation-detection program can be carried out around the regular singular points $x=\pm1$ of the Legendre differential equations. The solutions $y_{n}(x)$ of the Legendre equation are predicted to be oscillatory in $|x| < 1$ for $n\geq3$ and nonoscillatory outside of that interval for all values of n. In contrast, the solutions $y_{n}^{m}(x)$ of the associated Legendre equation are predicted to be oscillatory for $n\geq3$ and $m\leq n-2$ only in smaller subintervals $|x| < x_{*} < 1$ , the sizes of which are determined by n and m. Numerical integrations confirm that such subintervals are distinctly smaller than $(-1, +1)$ .

A program for predicting the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations

We derive a new criterion for deducing the intervals of oscillatory behavior in the solutions of ordinary second-order linear homogeneous differential equations from their coefficients. The validity of the method depends on one’s ability to transform a given differential equation to its simplest possible form, so a program must be executed that involves transformations of both variables before the criterion can be applied. The payoff of the program is the detection of oscillations precisely where they may occur in finite or infinite intervals of the independent variable. We demonstrate how the oscillation-detection program can be carried out for a variety of well-known differential equations from applied mathematics and mathematical physics.

The Similar Structure Method for Solving the Model of Fractal Dual-Porosity Reservoir

This paper proposes a similar structure method (SSM) to solve the boundary value problem of the extended modified Bessel equation. The method could efficiently solve a second-order linear homogeneous differential equation’s boundary value problem and obtain its solutions’ similar structure. A mathematics model is set up on the dual-porosity media, in which the influence of fractal dimension, spherical flow, wellbore storage, and skin factor is taken into cosideration. Researches in the model found that it was a special type of the extended modified Bessel equation in Laplace space. Then, the formation pressure and wellbore pressure under three types of outer boundaries (infinite, constant pressure, and closed) are obtained via SSM in Laplace space. Combining SSM with the Stehfest algorithm, we propose the similar structure method algorithm (SSMA) which can be used to calculate wellbore pressure and pressure derivative of reservoir seepage models clearly. Type curves of fractal dual-porosity spherical flow are plotted by SSMA. The presented algorithm promotes the development of well test analysis software.

二阶齐次线性微分方程解的相似结构&amp;lt;br&amp;gt;The Similar Structure of Solutions of Second-Order Li-near Homogeneous Differential Equation

二阶齐次线性微分方程解的相似结构&amp;lt;br&amp;gt;The Similar Structure of Solutions of Second-Order Li-near Homogeneous Differential Equation

An enhancement of transformation method

To solve Cauchy problem, for an ordinary second-order linear homogeneous differential equation, we use, so-called, transformation method. This numerical technique does not rely on expansion in Poincaré series, however, it requires integration of a nonlinear second-order ordinary differential equation. This formidable task substantially diminishes when instead of the nonlinear second-order equation we integrate a linear third-order equation.

On the Hyper-Order of Solutions of Some Second-Order Linear Differential Equations

In this paper, we investigate the growth of solutions of the second-order linear homogeneous differential equations with entire coefficients of the same order, and obtain precise estimates of the hyper-order of their solutions.

Accurate computation of the zeros of the generalized Bessel polynomials

A general method for approximating polynomial solutions of second-order linear homogeneous differential equations with polynomial coefficients is applied to the case of the families of differential equations defining the generalized Bessel polynomials, and an algorithm is derived for simultaneously finding their zeros. Then a comparison with several alternative algorithms is carried out. It shows that the computational problem of approximating the zeros of the generalized Bessel polynomials is not an easy matter at all and that the only algorithm able to give an accurate solution seems to be the one presented in this paper.

Two Generator Subgroups ofSL(2,C) and the Hypergeometric, Riemann, and Lamé Equations

For the purposes of constructing explicit solutions to second-order linear homogeneous differential equations on the Riemann sphere the Kovacic algorithm partitions the subgroups ofSL (2,C) into four classes and initially determines which class contains the differential Galois group of the input equation. We prove in the case of the hypergeometric and Riemann equations that the relevant class can be determined directly from the coefficients by elementary calculation. We also treat the (non-algebraic form of the) Lamé equation, to which the Kovacic algorithm is not directly applicable. In that instance we combine the Kovacic results with ours to produce an algorithm for determining the class of the associated group. From the group-theoretic viewpoint the problem solved herein is the following: given arbitrary S,T∈SL(2,C), determine which class contains the group 〈S, T〉 generated by S and T.

Gauge-invariant perturbations in a scalar field dominated universe

The authors propose a new covariant and gauge-invariant (GI) treatment of perturbations in a Robertson-Walker universe dominated by a classical scalar field phi . They first set up the formalism, based on the natural slicing of the problem by the surfaces phi =constant, and introduce a set of covariantly defined GI variables. In their approach the whole inhomogeneity of the matter field is incorporated in the GI spatial fluctuations of the momentum psi of phi ; then the GI density perturbations are simply proportional to the momentum perturbations. The inhomogeneity of the geometry is characterized by GI fluctuations of the 3-curvature scalar of the surfaces phi =constant. The time evolution of the matter and curvature perturbations are coupled by a pair of first-order linear differential equations. Correspondingly, each GI variable satisfies a second-order linear homogeneous differential equation. When the background curvature vanishes, k=0, the curvature variable is conserved for perturbation scales larger than the horizon, but this is no longer true in general if k not=0. They discuss simple examples, including the case when more than one scalar field is present, recovering standard results for inflationary universe models. They also demonstrate that in coasting solutions with k=-1, inhomogeneities are damped out on all scales.

On the Virial Coefficients for a Lennard-Jones Gas

Simpler methods for computing the second and third virial coefficients for a Lennard-Jones gas are given. The second virial coefficient is found to obey a second-order linear homogeneous differential equation in the temperature. The use of these methods is discussed.