Chebyshev Equations Research Articles

ON SOLUTIONS OF NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS BASED ON ELASTIC TRANSFORMATION METHODS

We consider difficult problems in solving nonlinear ordinary differential equations with variable coefficients. We use elastic transformation methods (elastic upgrading transformation and elastic reduced transformation) to transform the first and third order equations into the associated Chebyshev equation. Then, according to the general solution of the associated Chebyshev equation, we obtain the general solutions of the first-order and third-order nonlinear ordinary differential equations with variable coefficients, giving curves of general solution. The elastic transformation method provides a new idea and expands the solvable classes for solving nonlinear ordinary differential equation with variable coefficients.

Elastic transformation method for solving the initial value problem of variable coefficient nonlinear ordinary differential equations

<abstract><p>Aiming at the initial value problems of variable coefficient nonlinear ordinary differential equations, this paper introduces the elastic transformation method into the process of solving the initial value problems of nonlinear ordinary differential equations with variable coefficients. A class of first-order and a class of third-order nonlinear ordinary differential equations with variable coefficients can be transformed into Chebyshev equations through elastic upgrading transformation and elastic reduction transformation respectively. According to the properties of Chebyshev polynomials and the initial conditions, the solutions to the initial value problems of the original first-order and third- order differential equations can be obtained through the elastic inverse transformation, and then the curves of the solutions can be drawn. The introduction of the elastic transformation method not only provides a new idea for solving the initial value problems of nonlinear differential equations, but also expands the solvable classes of ordinary differential equations.</p></abstract>

Uyumlu Kesir Mertebeden Chebyshev Diferensiyel Denklemleri ve Kesirsel Chebyshev Polinomları

Uyumlu Kesir Mertebeden Chebyshev Diferensiyel Denklemleri ve Kesirsel Chebyshev Polinomları

Floquet theory for second order linear homogeneous difference equations

In this paper we provide a version of the Floquet’s theorem to be applied to any second order difference equations with quasi-periodic coefficients. To do this we extend to second order linear difference equations with quasi-periodic coefficients, the known equivalence between the Chebyshev equations and the second order linear difference equations with constant coefficients. So, any second order linear difference equations with quasi-periodic coefficients is essentially equivalent to a Chebyshev equation, whose parameter only depends on the values of the quasi-periodic coefficients and can be determined by a non-linear recurrence. Moreover, we solve this recurrence and obtaining a closed expression for this parameter. As a by-product we also obtain a Floquet’s type result; that is, the necessary and sufficient condition for the equation has quasi-periodic solutions.

Computational geometric and boundary value properties of Oblate Spheroidal Quaternionic Wave Functions

This paper introduces the Oblate Spheroidal Quaternionic Wave Functions (OSQWFs), which extend the oblate spheroidal wave functions introduced in the late 1950s by C. Flammer. We show that the theory of the OSQWFs is determined by the Moisil–Teodorescu type operator with quaternionic variable coefficients. We show the connections between the solutions of the radial and angular equations and of the Chebyshev equation, on one hand, and the quaternionic hyperholomorphic and anti-hyperholomorphic functions on the other. We proceed the paper establishing an analogue of the Cauchy’s integral formula as well as analogues of the boundary value properties such as Sokhotski–Plemelj formulas, the Dk-hyperholomorphic extension of a given Hölder function and on the square of the singular integral operator for this version of quaternionic function theory. To progress in this direction, we show how the D0-hyperholomorphic OSQWFs (with a bandwidth parameter c=0) of any order look like, without belabor them. With the help of these functions, we construct a complete orthogonal system for the L2-space consisting of D0-hyperholomorphic OSQWFs. A big breakthrough is that the orthogonality of the basis elements does not depend on the shape of the oblate spheroids, but only on the location of the foci of the ellipse generating the spheroid. As an application, we prove an explicit formula of the quaternionic D0-hyperholomorphic Bergman kernel function over oblate spheroids in R3. In addition, we provide the reader with some plot examples that demonstrate the effectiveness of our approach.

Statistical Average of Spin Operators for Calculation of Three-ComponentMagnetization (II): Solution of Equation

In this paper, the solution of Chebyshev equation with its argument beinggreater than 1 is obtained. The initial value of the derivative of thesolution is the expression of magnetization, which is valid for any spinquantum number S. The Chebyshev equation is transformed from an ordinarydifferential equation obtained when we dealt with Heisenberg model, in orderto calculate all three components of magnetization, by many-body Green'sfunction under random phase approximation. The Chebyshev functions withargument being greater than 1 are discussed. This paper shows that theChebyshev polynomials with their argument being greater than 1 have theirphysical application.

The Hamilton-Jacobi method for arbitrary rheo-linear dynamical systems

The Hamilton-Jacobi method is briefly summarized and then applied to arbitrary rheo-linear systems with a single degree of freedom. Various means of finding a complete solution of the Hamilton-Jacobi equation and applying the Jacobi theorem to solve the canonical differential equations are discussed. Basic to the procedure is the separation of variables in the Hamilton-Jacobi equation which leads to a Riccati equation which must be solved for the particular rheonomic differential equation. Five different cases of complete solution are illustrated by a simple rheonomic example. As a direct application of the method, a number of canonical systems corresponding to many “named” equations are solved. They are the: Airy, Bessel, Chebyshev, Error function, Euler, Gegenbauer, Hermite, Hypergeometric, Kelvin, Kummer, Laguerre, Legendre, Jacobi, Mathieu, Spherical Bessel, Weber-Hermite and Whittaker equations. Finally, the conservation law is given for a forced and damped oscillator.

SOLVING SECOND ORDER DIFFERENTIAL EQUATIONS IN QUANTUM MECHANICS BY ORDER REDUCTION

Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable differential equation of first order, a simple common solution is provided to cover all the existing standard solutions of these named equations. It is easier than the method of generating functions and more powerful than the Frobenius method of power series.

Taylor polynomial solutions of linear differential equations

A matrix method, which is called Taylor-matrix method, for the approximate solution of linear differential equations with specified associated conditions in terms of Taylor polynomials about any point. The method is based on, first, taking truncated Taylor series of the functions in equation and then substituting their matrix forms into the given equation. Thereby the equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. To illustrate the method, it is applied to certain linear differential equations and the generalized Hermite, Laguerre, Legendre and Chebyshev equations given by Costa and Levine [Int. J. Math. Edu. Sci. Technol. 20 (1989) 1].

Solutions of a certain class of fractional differintegral equations

Recently, several authors demonstrated the usefulness of fractional calculus in obtaining particular solutions of a number of such familiar second-order differential equations as those associated with Gauss, Legendre, Jacobi, Chebyshev, Coulomb, Whittaker, Euler, Hermite, and Weber equations. The main object of this paper is to show how some of the latest contributions on the subject by Tu et al. [1], involving the associated Legendre, Euler, and Hermite equations, can be presented in a unified manner by suitably appealing to a general theorem on particular solutions of a certain class of fractional differintegral equations.

On a Class of Differential Equations That Contains the Equations of Euler and Chebyshev

Linear differential equations with variable coefficients are usually solved by formal power series methods or by constructing a suitable integral transform that relates well with some differential operator. The typical w x w x w x example of such construction is the Mellin transform. See 1 , 2 , and 7 . In this paper we consider a class of linear differential equations with variable coefficients that contains the Euler and Chebyshev equations. We apply a general algebraic method, based on generating functions, to find explicit solutions of homogeneous and inhomogeneous equations with w x w x given initial conditions. Our method was introduced in 3 and 6 . Ž . Ž . For a given differential operator of the form L s c t D, where c t is some suitable function, we construct a vector space G that contains all the Ž . solutions of homogeneous equations u L g s 0, where u is a polynomial. We also construct in a algebraic way a convolution product on G that can Ž . be used to obtain a one-sided inverse for the operator u L . We find an integral representation for the convolution that allows us to solve inhomoŽ . geneous equations u L g s f for forcing functions f in a fairly general class. In Section 2 we present the basic definitions and the main results of our general method. In Sections 3 and 4 we apply the general results to the class of equations mentioned above, and we give some particular examples that illustrate our results in Section 5.

A method for the approximate solution of the second‐order linear differential equations in terms of Taylor polynomials

A matrix method is introduced for the approximate solution of the second‐order linear differential equation with specified associated conditions in terms of Taylor polynomials about any point. Examples are presented which illustrate the pertinent features of the method. Also it is applied to the generalized Hermite, Laguerre, Legendre and Chebyshev equations given by Costa and Levine.

Polynomial solutions of certain classes of ordinary differential equations

Several classes of ordinary differential equations which have polynomial solutions are studied. In particular, generalizations of the Hermite, Laguerre, Legendre, and Chebyshev equations are given for which such solutions exist. These solutions turn out to be generalizations of well‐known polynomials and enjoy similar properties.

Thermodynamic properties of organic oxygen compounds L. The vapour pressures of 1,2-ethanediol (ethylene glycol) and bis(2-hydroxyethyl) ether (diethylene glycol)

The vapour pressures of 1,2-ethanediol (ethylene glycol) (from 2.2 to 201 kPa 374 to 496 K) and of bis(2-hydroxyethyl) ether (diethylene glycol) (from 2.2 to 101 kPa, 411 to 520 K) were measured by comparative ebulliometry, and the results were fitted by third-order Chebyshev equations. Antoine equations were also fitted over restricted ranges including the normal boiling temperatures, but this equation will not reproduce the measured values satisfactorily over more extended ranges. The Chebyshev equations were extrapolated to ambient temperatures, and Antoine equations are given to represent these extrapolated values between 270 and 310 K. Measurements of the vapour pressure of ethylene glycol made subsequently confirm the correctness of the extrapolation for this compound and support the view that vapour-pressure curves may be extrapolated downwards reliably if they are based on sufficiently accurate measurements.

Comparison of the Antoine equation and a 3-term Chebyshev equation for correlation of vapor pressures

The closeness of fit of the Antione vapor pressure equation has been compared with that of a 3-term Chebysher polynomial, in order to select a simple equation for correlating the data from an automated vapor pressure measuring apparatus. Literature data for 13 compounds were used, covering the range of vapor pressure from a few kilopascals to the critical pressure. The compounds were water, acetone, diethyl ether, butanone, methanol, ethanol, 1-propanol, 2-propanol, 1-butanol, 2-methyl-2-propanol, benzene, toluene and 1,2-dimethylbenxene. In each case the Antoine equation gave the closer fit, with average deviations of 0.2–0.9% and maximum deviations of 0.5–2.3%, and was therefore the preferred correlating equation. The Chebyshev equation produced deviations about twice as large. The constants of the two equations are given for each compound, with an outline of the program for obtaining the constants by least squares using a programmable desk calculator.

Thermodynamic properties of organic oxygen compounds XLIII. Vapour pressures of some ethers

Vapour pressures of methyl propyl, isopropyl methyl, butyl methyl, ethyl propyl, t-butyl methyl, dipropyl, di-isopropyl, di- t-butyl, and decyl methyl ethers were measured at pressures up to 205 kPa. The measured values were fitted by Antoine and by Chebyshev equations, values already published from this laboratory for three aromatic ethers were recomputed uniformly with the present results, and published values for four additional compounds were incorporated in a scheme for correlation of the vapour pressures of ethers. Estimates were made of the vapour pressures of 10 other ethers. Between 5 and 200 kPa the vapour pressures of ethers may be represented by a single equation in which carbon number or an effective carbon number is a parameter. Chebyshev equations are given for interpolation between the upper bounds of the measurements and the critical pressures of 11 ethers for which this property has been previously determined.

Thermodynamic properties of organic oxygen compounds XXXVIII. Vapour pressures of some aliphatic ketones

Vapour pressures were measured of 3-methylbutan-2-one, hexan-2-one, 3,3-dimethylbutan-2-one, heptan-2-one, nonan-5-one, undecan-2-one, undecan-6-one, tridecan-2-one, and pentadecan-8-one below 210 kPa. The measured values were fitted by Antoine and by Chebyshev equations, values already published for some C 4 and C 5 ketones were recomputed uniformly with the present results, and estimates were made of the vapour pressures of fourteen other ketones. Between 5 and 200 kPa the vapour pressures of ketones may be represented by a single equation in which carbon number or an effective carbon number is a parameter. The vapour pressure of butanone between 190 kPa and the critical pressure was measured and the results were fitted by a Chebyshev equation; equations of this type are also given for interpolation in this high range for several other ketones of which the critical temperatures and pressures are known, and for representation of the scheme proposed by Pitzer et al. for correlation of vapour pressures by means of the acentric factor.

Thermodynamic properties of fluorine compounds. Part 15.—Vapour pressures of the three tetrafluorobenzenes and 1,3,5-trichloro-2,4,6-trifluorobenzene

The vapour pressures were measured of 1,2,4,5-tetrafluorobenzene from 5 kPa to the critical pressure, of 1,2,3,5-tetrafluorobenzene from 5 kPa to 470 kPa, of 1,2,3,4-tetrafluorobenzene from 7 kPa to 202 kPa, and of 1,3,5-trichloro-2,4,6-trifluorobenzene from 3 kPa to 180 kPa. The results were fitted by Antoine equations (applicable at pressures up to 200 kPa) and by Chebyshev equations. For each compound one of the latter equations is applicable over the whole range up to the critical temperature, and it is shown that the variation with temperature of ΔH/ΔZ calculated from this equation is similar to that found for other compounds.

The vapour pressure of mercury

The vapour pressure of mercury has been determined ebulliometrically over the range 0.05 to 800 kPa (380 to 770 K): above 3 kPa (480 K) by the comparative method with water as the standard, and below 5 kPa by direct measurement of the pressure with a mercury manometer. The normal boiling temperature obtained from the measurements is in agreement with the value quoted as a secondary reference point in the International Practical Temperature Scale of 1968; variation in the boiling temperature with change in atmospheric pressure is expressed by means of an Antoine equation. For the wider range 0.15 to 255 kPa (400 to 685 K) the results are well represented by a Cragoe equation or its equivalent, a third-order Chebyshev equation. For the full experimental range a higher-order Chebyshev equation is required and the one given, of fifth order, is applicable from 400 K to the critical temperature. The values now reported for the vapour pressure below 3 kPa will allow use of mercury as a standard supplementary to water in comparative ebulliometric measurements.