scholarly journals An Attempt to Make Non-Elementary Functions That Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

2022 ◽  
Vol 10 (01) ◽  
pp. 56-67
Author(s):  
Magne Stensland
Keyword(s):  
Second Order ◽  
Elementary Functions

scholarly journals Second-order integrable Lagrangians and WDVV equations

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
E. V. Ferapontov ◽  
M. V. Pavlov ◽  
Lingling Xue
Keyword(s):  
Lagrangian Density ◽  
Theta Functions ◽  
Second Order ◽  
Elementary Functions ◽  
Lagrange Equations ◽  
Wdvv Equations ◽  
Integrability Conditions ◽  
Jacobi Theta Functions

AbstractWe investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ ∫ f ( u xx , u xy , u yy ) d x d y . By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

scholarly journals Integration of second order linear differential equations in elementary functions, example from quantum-mechanics

2018 ◽  
Vol 251 ◽  
pp. 04015
Author(s):  
Alexander Chichurin ◽  
Viktor Red’kov
Keyword(s):  
Differential Equation ◽  
Schwarzian Derivative ◽  
Second Order ◽  
Quantum Mechanical ◽  
Order Linear Differential Equation ◽  
Elementary Functions ◽  
Mechanical Problem ◽  
Order Linear ◽  
Special Procedure ◽  
Linear Differential

In the paper, special procedure for integrating a second order linear differential equation with six singular points, example of a quantum-mechanical problem for a spin zero particle with intrinsic Darwin—Cox structure, is considered. The method is based on the use of the Schwarzian derivative, conditions for parameters are derived at which solutions of the equation under consideration are constructed in terms of elementary functions. The graphs for several solutions are presented, their physical sense is discussed.

scholarly journals SECOND-ORDER QUASILINEAR PDEs AND CONFORMAL STRUCTURES IN PROJECTIVE SPACE

2010 ◽  
Vol 21 (06) ◽  
pp. 799-841 ◽  
Author(s):  
P. A. BUROVSKIY ◽  
E. V. FERAPONTOV ◽  
S. P. TSAREV
Keyword(s):  
Modular Forms ◽  
Sufficient Conditions ◽  
Elliptic Functions ◽  
Coefficient Matrix ◽  
Conformal Structure ◽  
Second Order ◽  
Elementary Functions ◽  
Integrable Equations ◽  
Integrability Conditions ◽  
Conformal Structures

We investigate second-order quasilinear equations of the form fijuxixj = 0, where u is a function of n independent variables x1, …, xn, and the coefficients fij depend on the first-order derivatives p1 = ux1, …, pn = uxn only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n + 1, R), which acts by projective transformations on the space Pn with coordinates p1, …, pn. The coefficient matrix fij defines on Pn a conformal structure fij(p)dpidpj. The necessary and sufficient conditions for the integrability of such equations by the method of hydrodynamic reductions are derived, implying that the moduli space of integrable equations is 20-dimensional. Any equation satisfying the integrability conditions is necessarily conservative, and possesses a dispersionless Lax pair. The integrability conditions imply that the conformal structure fij(p) dpidpj is conformally flat, and possesses infinitely many three-conjugate null coordinate systems parametrized by three arbitrary functions of one variable. Integrable equations provide examples of such conformal structures parametrized by elementary functions, elliptic functions and modular forms.

LAGRANGIAN FEYNMAN FORMULAS FOR SECOND-ORDER PARABOLIC EQUATIONS IN BOUNDED AND UNBOUNDED DOMAINS

2010 ◽  
Vol 13 (03) ◽  
pp. 377-392 ◽  
Author(s):  
YANA A. BUTKO ◽  
MARTIN GROTHAUS ◽  
OLEG G. SMOLYANOV
Keyword(s):  
Parabolic Equations ◽  
Diffusion Processes ◽  
Unbounded Domains ◽  
Variable Coefficients ◽  
Second Order ◽  
Cauchy Problems ◽  
Elementary Functions ◽  
Functional Integrals ◽  
Finite Dimensional ◽  
Variable Diffusion

In this note a class of second-order parabolic equations with variable coefficients, depending on coordinate, is considered in bounded and unbounded domains. Solutions of the Cauchy–Dirichlet and the Cauchy problems are represented in the form of a limit of finite-dimensional integrals of elementary functions (such representations are called Feynman formulas). Finite-dimensional integrals in the Feynman formulas give approximations for functional integrals in the corresponding Feynman–Kac formulas, representing solutions of these problems. Hence, these Feynman formulas give an effective tool to calculate functional integrals with respect to probability measures generated by diffusion processes with a variable diffusion coefficient and absorption on the boundary.

Stratified polymorphism and primitive recursion

1999 ◽  
Vol 9 (4) ◽  
pp. 507-522 ◽  
Author(s):  
NORMAN DANNER ◽  
DANIEL LEIVANT
Keyword(s):  
Programming Language ◽  
Lambda Calculus ◽  
Second Order ◽  
Type Inference ◽  
Language Theory ◽  
Inference Rules ◽  
Elementary Functions ◽  
Primitive Recursion ◽  
Primitive Recursive ◽  
Highly Uniform

Natural restrictions on the syntax of the second-order (i.e., polymorphic) lambda calculus are of interest for programming language theory. One of the authors showed in Leivant (1991) that when type abstraction in that calculus is stratified into levels, the definable numeric functions are precisely the super-elementary functions (level [Escr ]4 in the Grzegorczyk Hierarchy). We define here a second-order lambda calculus in which type abstraction is stratified to levels up to ωω, an ordinal that permits highly uniform (and finite) type inference rules. Referring to this system, we show that the numeric functions definable in the calculus using ranks < ω[lscr ] are precisely Grzegorczyk's class [Escr ][lscr ]+3 ([lscr ] [ges ] 1). This generalizes Leivant (1991), where this is proved for [lscr ] = 1. Thus, the numeric functions definable in our calculus are precisely the primitive recursive functions.

Disappearance Voltage Determinations Using a Convergent Beam

1971 ◽  
Vol 29 ◽  
pp. 184-185
Author(s):  
W. L. Bell
Keyword(s):  
Second Order ◽  
Objective Method ◽  
Uniform Thickness ◽  
Rocking Curve ◽  
Crystal Perfection ◽  
Kikuchi Line ◽  
Central Maximum ◽  
Diffraction Patterns ◽  
Convergent Beam ◽  
Beam Technique

Disappearance voltages for second order reflections can be determined experimentally in a variety of ways. The more subjective methods, such as Kikuchi line disappearance and bend contour imaging, involve comparing a series of diffraction patterns or micrographs taken at intervals throughout the disappearance range and selecting that voltage which gives the strongest disappearance effect. The estimated accuracies of these methods are both to within 10 kV, or about 2-4%, of the true disappearance voltage, which is quite sufficient for using these voltages in further calculations. However, it is the necessity of determining this information by comparisons of exposed plates rather than while operating the microscope that detracts from the immediate usefulness of these methods if there is reason to perform experiments at an unknown disappearance voltage.The convergent beam technique for determining the disappearance voltage has been found to be a highly objective method when it is applicable, i.e. when reasonable crystal perfection exists and an area of uniform thickness can be found. The criterion for determining this voltage is that the central maximum disappear from the rocking curve for the second order spot.

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