scholarly journals Ring-Shaped Potential and a Class of Relevant Integrals Involved Universal Associated Legendre Polynomials with Complicated Arguments

2017 ◽  
Vol 2017 ◽  
pp. 1-4 ◽  
Author(s):  
Wei Li ◽  
Chang-Yuan Chen ◽  
Shi-Hai Dong
Keyword(s):  
Differential Equation ◽  
Generating Function ◽  
Legendre Polynomials ◽  
Present Method ◽  
Special Functions ◽  
Analytical Result

We find that the solution of the polar angular differential equation can be written as the universal associated Legendre polynomials. Its generating function is applied to obtain an analytical result for a class of interesting integrals involving complicated argument, that is,∫-11Pl′m′xt-1/1+t2-2xtPk′m′(x)/(1+t2-2tx)(l′+1)/2dx, wheret∈(0,1). The present method can in principle be generalizable to the integrals involving other special functions. As an illustration we also study a typical Bessel integral with a complicated argument∫0∞Jn(αx2+z2)/(x2+z2)nx2m+1dx.

The number of meetings in free Poisson traffic

1974 ◽  
Vol 11 (2) ◽  
pp. 320-331
Author(s):  
Hans D. Unkelbach ◽  
Helmut Wegmann
Keyword(s):  
Differential Equation ◽  
Generating Function ◽  
Practical Interest ◽  
Time Interval ◽  
Integro Differential Equation

Using Rényi's model of free Poisson traffic the distribution of the number of meetings of vehicles on a highway section during a given time interval is investigated. An integro-differential equation for the generating function of that variable is deduced and the first moments are calculated. The generating function is given explicitly in simple cases and approximately in cases of practical interest.

LEFM Analysis of V-Notched Aluminum Plates Using Higher-Order Layerwise Model

2014 ◽  
Vol 898 ◽  
pp. 355-358
Author(s):  
Kwang Sung Woo ◽  
Yoo Mi Kwon ◽  
Dong Woo Lee ◽  
Hee Joong Kim
Keyword(s):  
Legendre Polynomials ◽  
Present Method ◽  
Higher Order ◽  
Virtual Crack Closure Technique ◽  
Shape Functions ◽  
Intensity Factors ◽  
Displacement Fields ◽  
Closure Technique ◽  
Aluminum Plates ◽  
Convergent Approach

Higher-order layerwise model is proposed to determine stress intensity factors using virtual crack closure technique for V-notched plates. Present method is based on p-convergent approach and adopts the concept of subparametric element. In assumed displacement field, strain-displacement relations and 3-D constitutive equations of a layer are obtained by combination of 2-D and 1-D higher-order shape functions. Thus, it allows independent implementation of p-refinement for in-plane and transversal displacements. In the proposed elements, the integrals of Legendre polynomials and Gauss-Lobatto technique are employed to interpolate displacement fields and to implement numerical quadrature, respectively.

A carrier-borne epidemic with multiple stages of infection

1991 ◽  
Vol 28 (01) ◽  
pp. 1-8 ◽  
Author(s):  
J. Gani ◽  
Gy. Michaletzky
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Markov Chain ◽  
Generating Function ◽  
Continuous Time ◽  
Probability Generating Function ◽  
Death Process ◽  
Partial Differential ◽  
Multiple Stages

This paper considers a carrier-borne epidemic in continuous time with m + 1 > 2 stages of infection. The carriers U(t) follow a pure death process, mixing homogeneously with susceptibles X 0(t), and infectives Xi (t) in stages 1≦i≦m of infection. The infectives progress through consecutive stages of infection after each contact with the carriers. It is shown that under certain conditions {X 0(t), X 1(t), · ··, Xm (t) U(t); t≧0} is an (m + 2)-variate Markov chain, and the partial differential equation for its probability generating function derived. This can be solved after a transfomation of variables, and the probability of survivors at the end of the epidemic found.

scholarly journals Moments of the weighted Cantor measures

2019 ◽  
Vol 52 (1) ◽  
pp. 256-273
Author(s):  
Steven N. Harding ◽  
Alexander W. N. Riasanovsky
Keyword(s):  
Probability Measure ◽  
Generating Function ◽  
Exponential Decay ◽  
Legendre Polynomials ◽  
Borel Probability Measure ◽  
Moment Generating Function ◽  
Unit Interval ◽  
Uniform Error ◽  
Natural Case ◽  
Borel Probability

AbstractBased on the seminal work of Hutchinson, we investigate properties of α-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, α is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure μα is the unique Borel probability measure on [0, 1] satisfying {\mu ^\alpha }(E) = \sum\nolimits_{n = 0}^{N - 1} {{\alpha _n}{\mu ^\alpha }(\varphi _n^{ - 1}(E))} where ϕn : x ↦ (x + n)/N. In Sections 1 and 2 we examine several general properties of the measure μα and the associated Legendre polynomials in L_{{\mu ^\alpha }}^2 [0, 1]. In Section 3, we (1) compute the Laplacian and moment generating function of μα, (2) characterize precisely when the moments Im = ∫[0,1]xm dμα exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the first m moments within uniform error ε in O((log log(1/ε)) · m log m). We also state analogous results in the natural case where α is palindromic for the measure να attained by shifting μα to [−1/2, 1/2].

scholarly journals Spherical curves and quadratic relationships for special functions

1985 ◽  
Vol 27 (1) ◽  
pp. 111-124
Author(s):  
Even Mehlum ◽  
Jet Wimp
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Special Functions ◽  
Position Vector ◽  
Third Order ◽  
Generalized Hypergeometric Functions ◽  
Transcendental Functions ◽  
Horn Functions ◽  
Vector Differential Equation

AbstractWe show that the position vector of any 3-space curve lying on a sphere satisfies a third-order linear (vector) differential equation whose coefficients involve a single arbitrary function A(s). By making various identifications of A(s), we are led to nonlinear identities for a number of higher transcendental functions: Bessel functions, Horn functions, generalized hypergeometric functions, etc. These can be considered natural geometrical generalizations of sin2t + cos2t = 1. We conclude with some applications to the theory of splines.

Studies in multiplicative solutions to linear differential equations

1987 ◽  
Vol 106 (3-4) ◽  
pp. 277-305 ◽  
Author(s):  
F. M. Arscott
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Floquet Theory ◽  
Special Functions ◽  
Linear Differential Equations ◽  
Ordinary Linear Differential Equation ◽  
Closed Path ◽  
Starting Point ◽  
Linear Differential

SynopsisGiven an ordinary linear differential equation whose singularities are isolated, a solution is called multiplicative for a closed path C if, when continued analytically along C, it returns to its starting-point merely multiplied by a constant. This paper first classifies such paths into three types, then investigates combinations of two such paths, in which a number of qualitatively different situations can arise. A key result is also given relating to a three-path combination. There are applications to special functions and Floquet theory for periodic equations.

The Variational-Iteration Method to Solve the Nonlinear Boltzmann Equation

2008 ◽  
Vol 63 (3-4) ◽  
pp. 131-139 ◽  
Author(s):  
Essam M. Abulwafa ◽  
Mohammed A. Abdou ◽  
Aber H. Mahmoud
Keyword(s):  
Differential Equation ◽  
Distribution Function ◽  
Boltzmann Equation ◽  
Generating Function ◽  
Iteration Method ◽  
Nonlinear Partial Differential Equation ◽  
Variational Iteration Method ◽  
Inhomogeneous Media ◽  
Nonlinear Boltzmann Equation ◽  
Variational Iteration

The time-dependent nonlinear Boltzmann equation, which describes the time evolution of a single-particle distribution in a dilute gas of particles interacting only through binary collisions, is considered for spatially homogeneous and inhomogeneous media without external force and energy source. The nonlinear Boltzmann equation is converted to a nonlinear partial differential equation for the generating function of the moments of the distribution function. The variational-iteration method derived by He is used to solve the nonlinear differential equation of the generating function. The moments for both homogeneous and inhomogeneous media are calculated and represented graphically as functions of space and time. The distribution function is calculated from its moments using the cosine Fourier transformation. The distribution functions for the homogeneous and inhomogeneous media are represented graphically as functions of position and time.

scholarly journals Weighted Average Based Differential Quadrature Method for One-Dimensional Homogeneous First Order Nonlinear Parabolic Partial Differential Equation

2021 ◽  
Vol 1 (1) ◽  
pp. 15-28
Author(s):  
Kedir Aliyi Koroche
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Present Method ◽  
Differential Quadrature Method ◽  
Grid Point ◽  
Weighted Average ◽  
Absolute Error ◽  
Quadrature Method ◽  
Maximum Absolute Error ◽  
Error Norm

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

scholarly journals Legendre-Gould Hopper-Based Sheffer Polynomials and Operational Methods

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2051
Author(s):  
Nabiullah Khan ◽  
Mohd Aman ◽  
Talha Usman ◽  
Junesang Choi
Keyword(s):  
Generating Function ◽  
Legendre Polynomials ◽  
Sheffer Polynomials ◽  
Operational Methods

A remarkably large of number of polynomials have been presented and studied. Among several important polynomials, Legendre polynomials, Gould-Hopper polynomials, and Sheffer polynomials have been intensively investigated. In this paper, we aim to incorporate the above-referred three polynomials to introduce the Legendre-Gould Hopper-based Sheffer polynomials by modifying the classical generating function of the Sheffer polynomials. In addition, we investigate diverse properties and formulas for these newly introduced polynomials.

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