Indefinite integrals involving complete elliptic integrals of the third kind

A method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete elliptic integrals. This gives a total of 27 integrals for the complete integral of the third kind, including the single integral given in the literature. Some typographical errors in a previous related paper are corrected.     

基于第二类椭圆积分的椭圆弧长公式变换与应用

以第二类椭圆积分为理论基础,通过推导,将椭圆弧长公式变换为以椭圆离心角、极角等常用角度参数为自变量的第二类椭圆积分的标准形式,建立起椭圆弧长公式与第二类椭圆积分标准形式之间的关系,并分析了椭圆上的弧微分变化规律及椭圆周长与离心率的变化关系.公式反映了椭圆弧长的本质问题即为第二类椭圆积分问题.因此,各类涉及椭圆弧长计算的应用问题,均可化为第二类椭圆的计算问题,应用时直接调用各类编程软件的函数库中的第二类椭圆积分函数,无需复杂编程即可实现椭圆弧长的高精度计算.文章以GPS采用的WGS-84椭球子午线弧长为例进行计算分析,验证了给出的公式及相关分析的正确性及应用价值.     

Indefinite integrals involving the incomplete elliptic integral of the third kind

A substantial number of new indefinite integrals involving the incomplete elliptic integral of the third kind are presented, together with a few integrals for the other two kinds of incomplete elliptic integral. These have been derived using a Lagrangian method which is based on the differential equations which these functions satisfy. Techniques for obtaining new integrals are discussed, together with transformations of the governing differential equations. Integrals involving products combining elliptic integrals of different kinds are also presented.     

Boundary value problems for two types of degenerate elliptic systems in R~4

Firstly, the Riemann boundary value problem for a kind of degenerate elliptic system of the first order equations in R~4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system's solution, the boundary value problem as stated above is transformed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R~4 are derived.     

Boundary value problems for two types of degenerate elliptic systems in <Emphasis Type="Italic">R</Emphasis><Superscript>4</Superscript>

Firstly, the Riemann boundary value problem for a kind of degenerate elliptic system of the first order equations in R ⁴ is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system’s solution, the boundary value problem as stated above is transformed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R ⁴ are derived.     

Indefinite integrals involving the incomplete elliptic integrals of the first and second kinds

A substantial number of indefinite integrals are presented for the incomplete elliptic integrals of the first and second kinds. The number of new results presented is about three times the total number to be found in the current literature. These integrals were obtained with a Lagrangian method based on the differential equations which these functions obey. All results have been checked numerically with Mathematica. Similar results for the incomplete elliptic integral of the third kind will be presented separately.     

On the curious series related to the elliptic integrals

By using the theory of the elliptic integrals, a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect to the elliptic modulus and integral representations of several of the series in terms of the inverse Mellin transforms related to the Riemann zeta function. The relation with the corresponding case of the Voronoi summation formula is exhibited. The involved series are expressed in closed form in terms of complete elliptic integrals of the first and second kind, and some special cases are calculated in terms of particular values of the Euler gamma function.     

The analyticity of solutions to a class of degenerate elliptic equations

In the present paper, the analyticity of solutions to a class of degenerate elliptic equations is obtained. A kind of weighted norms are introduced and under such norms some degenerate elliptic operators are of weak coerciveness.     

Bounds for symmetric elliptic integrals

Lower and upper bounds for the four standard incomplete symmetric elliptic integrals are obtained. The bounding functions are expressed in terms of the elementary transcendental functions. Sharp bounds for the ratio of the complete elliptic integrals of the second kind and the first kind are also derived. These results can be used to obtain bounds for the product of these integrals. It is shown that an iterative numerical algorithm for computing the ratios and products of complete integrals has the second order of convergence.     

On the stable difference scheme for source identification nonlocal elliptic problem

Approximation of source identification problem for elliptic equation with integral-type nonlocal condition is discussed. The first order of accuracy difference scheme for elliptic nonlocal identification problem is studied. By using spectral resolution of a self-adjoint operator, we establish stability inequalities for solution of constructed scheme. Subsequently, the difference scheme for approximate solution of multidimensional boundary value problem with integral-type nonlocal and first kind boundary conditions is investigated on stability. Numerical test examples are presented.     

一类各向异性外问题的重叠型区域分解算法

本文以椭圆外调和问题的自然边界归化为基础，提出了求解各向异性常系数椭圆方程的一种重叠型区域分解算法，并分析了算法的收敛性及收敛速度．理论分析及数值实验表明，该方法对于求解各向异性外问题非常有效．     

A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation

Based on the homogeneous balance method,the Jacobi elliptic expansion method and the auxiliary equation method,the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations.New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney (BBM) equations are obtained with the aid of computer algebraic system Maple.The method is also valid for other (1+1)-dimensional and higher dimensional systems.     

(2+1)维modified Zakharov-Kuznetsov方程的复合型新解

给出函数变换,变量分离形式解与第一种椭圆方程相结合的方法,构造了(2+1)维modified Zakharov-Kuznetsov(m ZK)方程的多种复合型新解.步骤一,给出两种函数变换,将(2+1)维m ZK方程转化为能够获得变量分离解的非线性发展方程.步骤二,给出非线性发展方程的变量分离形式解,通过第一种椭圆方程及其相关结论,构造了(2+1)维m ZK方程的双孤子解和双周期解等复合型新解.     

Periodic solutions of a derivative nonlinear Schrödinger equation: Elliptic integrals of the third kind

The nonlinear Schrödinger equation (NLSE) is an important model for wave packet dynamics in hydrodynamics, optics, plasma physics and many other physical disciplines. The ‘derivative’ NLSE family usually arises when further nonlinear effects must be incorporated. The periodic solutions of one such member, the Chen-Lee-Liu equation, are studied. More precisely, the complex envelope is separated into the absolute value and the phase. The absolute value is solved in terms of a polynomial in elliptic functions while the phase is expressed in terms of elliptic integrals of the third kind. The exact periodicity condition will imply that only a countable set of elliptic function moduli is allowed. This feature contrasts sharply with other periodic solutions of envelope equations, where a continuous range of elliptic function moduli is permitted.     

Exact solution of the cubic-quintic Duffing oscillator

In this paper, we derive the exact solution of the cubic-quintic Duffing oscillator based on the use of Jacobi elliptic functions. We also showed that the exact angular frequency of this cubic-quintic Duffing equation is given in terms of the complete elliptic integral of the first kind.     

A class of motions of a top in the Goryachev-Chaplygin case

A solution of the Euler-Poisson equations is studied for the Goryachev-Chaplygin case /1/ under the condition when the ultraelliptic integrals degenerate to elliptic /2/. A solution is constructed for a class of motions in which both quantities, u and v, brought in by Chaplygin, vary with time, but one of them tends asymptotically to a constant when the time increases without limit. The dependence of the Euler-Poisson variables on time is expressed in terms of elliptic functions and an elliptic integral of the third kind. Fairly simple approximate formulas are given for determining all six variables sought.     

A two-parameter generalization of the complete elliptic integral of second kind

A two-parameter generalization of the complete elliptic integral of second kind is expressed in terms of the Appell function F ₄. This function is further reduced to a quite simple bilinear form in the complete elliptic integrals K and E. Some physical applications are briefly mentioned.     

Convexity of the complete elliptic integrals of the first kind with respect to Hölder means

In this paper, we establish a necessary and sufficient condition for the convexity of the complete elliptic integrals of the first kind with respect to Hölder means.     

A family of potentials for elliptic equations with one singular coefficient and their applications

Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with one degeneration line are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in the form of an integral equation. By using some properties of the Gaussian hypergeometric function, we first prove limiting theorems and derive integral equations concerning the densities of the double- and simple-layer potentials. The obtained results are then applied in order to find an explicit solution of the Holmgren problem for the multidimensional singular elliptic equation in the half of the ball.     

椭圆边界上的自然积分算子及各向异性外问题的耦合算法

1.引 言为求解微分方程的外边值问题常需要引进人工边界（见[1-4]）,对人工边界外部区域作自然边界归化得到的自然积分方程即Dirichlet-Neumann映射,正是人工边界上的准确的边界条件（见[2-6]）,这是一类非局部边界条件.自然积分算子即Dirichlet-Neumann算子,