Indefinite integrals of products of special functions

A method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals of products of Gauss hypergeometric functions are also presented, which seem to be the first integrals of this type. All results presented have been numerically checked with Mathematica.     

Indefinite integrals of Lommel functions from an inhomogeneous Euler–Lagrange method

A method given recently for deriving indefinite integrals of special functions which satisfy homogeneous second-order linear differential equations has been extended to include functions which obey inhomogeneous equations. The extended method has been applied to derive indefinite integrals for the Lommel functions, which obey an inhomogeneous Bessel equation. The method allows integrals to be derived for the inhomogeneous equation in a manner which closely parallels the homogeneous case, and a number of new Lommel integrals are derived which have well-known Bessel analogues. Results will be presented separately for other special functions which obey inhomogeneous second-order linear equations.     

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.     

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.     

Indefinite integrals of special functions from inhomogeneous differential equations

A method is presented for deriving integrals of special functions which obey inhomogeneous second-order linear differential equations. Inhomogeneous equations are readily derived for functions satisfying second-order homogeneous equations. Sample results are derived for Bessel functions, parabolic cylinder functions, Gauss hypergeometric functions and the six classical orthogonal polynomials. For the orthogonal polynomials the method gives indefinite integrals which reduce to the usual orthogonality conditions on the usual orthogonality intervals. These indefinite integrals for the orthogonal polynomials appear to be new. All results have been checked with Mathematica.     

Numerical solution of some partial differential equations by means of a deterministic method of approximate functional integration

A numerical method of solution of some partial differential equations is presented. The method is based on representation of Green functions of the equations in the form of functional integrals and subsequent approximate calculation of the integrals with the help of a deterministic approach. In this case the solution of the equations is reduced to evaluation of usual (Riemann) integrals of relatively low multiplicity. A procedure allowing one to increase accuracy of the solutions is suggested. The features of the method are investigated on examples of numerical solution of the Schrödinger equation and related diffusion equation.     

New insight into classical equilibrium solutions of kinetic equations

Summary. The equilibrium solutions to kinetic equations of weak turbulence (weakly nonlinear wave systems) are analyzed in a systematic manner. The study is performed for kinetic equations involving any number of interacting waves of an arbitrary dimension. Conditions for the equilibrium solutions are reduced to generalized Cauchy functional equations defined at specific hypersurfaces. The analysis proves that, among differentiable functions, the formal equilibrium solutions correspond to equipartition of a linear combination of the energy, the components of momentum, and the number of particles. The structure of the set of equilibrium solutions does not depend on the dimension of wave systems, nor does it explicitly depend on the particular dispersion relation. It depends only on the number of wave components in resonance sets, whereas the number of particles does not enter into the equilibrium distribution in interactions of an odd number of harmonics. The most important consequence of this difference is that wave systems with four-wave resonance conserve the number of particles (wave action), whereas systems with three-wave or five-wave resonance violate this law. All the formal solutions, notwithstanding their physical realizability, are proved to be linearly stable with respect to small disturbances. Received October 5, 2000; accepted August 17, 2001 Online publication November 5, 2001     

构造Birkhoff函数(组)的参数调节法

根据偏微分方程的Cauchy-Kovalevski可积性定理,将欠定的Birkhoff方程组转化为以Birkhoff函数组为未知变量的完备的偏微分方程组,提出了构造Birkhoff动力学函数的参数调节法．通过调节补偿方程中的两类可调的函数参数就能得到不同的Birkhoff函数组．并把构造Birkhoff函数组的参数调节法与Santilli构造方法进行了比较,例如研究了利用动力学系统独立的第一积分构造Birkhoff函数组的Hojman方法与参数调节法之间的关系．最后,给出应用实例验证了参数调节法的实用性及其与Santilli 3种构造方法的关系     

Indefinite integrals involving the Jacobi Zeta and Heuman Lambda functions

Jacobian elliptic functions are used to obtain formulas for deriving indefinite integrals for the Jacobi Zeta function and Heuman's Lambda function. Only sample results are presented, mostly obtained from powers of the twelve Glaisher elliptic functions. However, this sample includes all such integrals in the literature, together with many new integrals. The method used is based on the differential equations obeyed by these functions when the independent variable is the argument u of elliptic function theory. The same method was used recently, in a companion paper, to derive similar integrals for the three canonical incomplete elliptic integrals.     

Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions

Integration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the independent variable u is the argument of Jacobian elliptic function theory. This allows a limitless number of indefinite integrals with respect to the amplitude to be derived for these three elliptic integrals. Sample results are given, including the integrals derived from powers of the 12 Glaisher elliptic functions. New recurrence relations and integrals are also given for the 12 Glaisher elliptic functions.     

Generalized self-adjustment method for statistical mechanics of composite materials

A new method is developed for the statistical mechanics of composite materials — the generalized selfadjustment method — which makes it possible to reduce the problem of predicting effective elastic properties of composites with random structures to the solution of two simpler “averaged” problems of an inclusion with transitional layers in a medium with the desired effective elastic properties. The inhomogeneous elastic properties and dimensions of the transitional layers take into account both the “approximate” order of mutual positioning, and also the variation in the dimensions and elastics properties of inclusions through appropriate special averaged indicator functions of the random structure of the composite. A numerical calculation of averaged indicator functions and effective elastic characteristics is performed by the generalized self-adjustment method for a unidirectional fiberglass on the basis of various models of actual random structures in the plane of isotropy.     

Fluid-particle flow: a symmetric formulation

In this Note, we present a numerical method to simulate the motion of solid particles in a moving viscous fluid. The fluid is supposed to be Newtonian and incompressible. The Arbitrary Lagrangian Eulerian formulation of the Navier-Stokes equations is discretized at the first order in time, as are the equations for the solid bodies. The advection term is taken into account by a method of characteristics. The variational formulation of the coupled problem is then established, and the boundary integrals expressing the hydrodynamical forces are eliminated. By introduction of an appropriate Finite Element approximation, a symmetric linear system is obtained. This system is solved by an inexact Uzawa algorithm, preconditionned by a Laplace operator with Neumann boundary conditions on the pressure. Numerical results are presented, for 2 and 100 particles: The Reynolds number in both cases is of the order of 100.     

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.     

Analytical formulation of meshless local integral equation method

In this paper, the exact forms of integrals in the meshless local boundary integral equation method are derived and implemented for elastostatic problems. A weak form for a set of governing equations with a unit test function or polynomial test functions is transformed into local integral equations. Each node has its own support domain and is surrounded by a local integral domain with different shapes of boundaries. The meshless approximation based on the radial basis function (RBF) is employed for the implementation of displacements. A completed set of closed forms of the local boundary integrals are obtained. As there are no numerical integrations to be carried out the computational time is significantly reduced. Three examples are presented to demonstrate the application of this approach in solid mechanics.     

Computation of Autocorrelations of Interdeparture Times by Numerical Transform Inversion

The generating function of the autocorrelations of the interdeparture times in stationary M/G/1 and GI/M/1 systems involves the probability generating function of the number of customers served in a busy period. The latter function is implicitly determined as a solution to a functional equation. Standard methods for the numerical inversion of generating functions require the values of these functions at many complex arguments. A recently discovered substitution method for contour integrals allows the numerical inversion of implicitly determined generating functions without the numerical solution of the functional equations.     

Generalised functions and divergent integrals

A method is given for the interpretation of a class of divergent integrals in terms of a sum of function evaluations over an arbitrary partition of the integration interval. The class of integrands considered includes functions continuous on the integration interval, except at a finite number of algebraic or algebraico-logarithmic singularities, and the delta function and related generalised functions, or products of these. The interpretation assigned to such integrals coincides with that of generalised function theory. Possible applications of the method to the computation of functions are discussed.     

Modular equations and distortion functions

Modular equations occur in number theory, but it is less known that such equations also occur in the study of deformation properties of quasiconformal mappings. The authors study two important plane quasiconformal distortion functions, obtaining monotonicity and convexity properties, and finding sharp bounds for them. Applications are provided that relate to the quasiconformal Schwarz Lemma and to Schottky’s Theorem. These results also yield new bounds for singular values of complete elliptic integrals.      

The macroscopic system of Einstein-Maxwell equations for a system of interacting particles

We derive the macroscopic Einstein—Maxwell equations up to the second-order terms, in the interaction for systems with dominating electromagnetic interactions between particles (e.g., radiation-dominated cosmological plasma in the expanding Universe before the recombination moment). The ensemble averaging of the microscopic Einstein and Maxwell equations and of the Liouville equations for the random functions of each type of particle leads to a closed system of equations consisting of the macroscopic Einstein and Maxwell equations and the kinetic equations for one-particle distribution functions for each type of particle. The macroscopic Einstein equations for a system of electromagnetically and gravitationally interacting particles differ from the classical Einstein equations in having additional terms in the lefthand side due to the interaction. These terms are given by a symmetric rank-two traceless tensor with zero divergence. Explicitly, these terms are represented as momentum-space integrals of the expressions containing one-particle distribution functions for each type of particle and have much in common with similar terms in the left-hand side of the macroscopic Einstein equations previously obtained for a system of self-gravitating particles. The macroscopic Maxwell equations for a system of electromagnetically and gravitationally interacting particles also differ from the classical Maxwell equations in having additional terms in the left-hand side due to simultaneous effects described by general relativity and the interaction effects. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 107–131, October, 2000.     

Representation of a multiple integral of special form by a series

Multiple integrals generalizing the iterated kernels of linear integral equations are expressed by a series each of whose terms is proportional to the product of two orthogonal functions in the case of a similar representation of the kernel. Besides integral equations, these integrals have applications in the theory of Markov processes. The results obtained are illustrated by several examples.