A Note on the Generalized and Universal Associated Legendre Equations (original) (raw)
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European Journal of Science and Technology
In this study, the Legendre operational matrix method based on collocation points is introduced to solve high order ordinary differential equations with some nonlinear terms arising in physics and mechanics. This technique transforms the nonlinear differential equation into a matrix equation with unknown Legendre coefficients via mixed conditions. This solution of this matrix equation yields the Legendre coefficients of the solution function. Thus, the approximate solution is obtained in terms of Legendre polynomials. Some test problems together with residual error estimation are given to show the usefulness and applicability of the method and the numerical results are compared.
Near-Legendre Differential Equations
Journal of Kufa for Mathematics and Computer, 2018
A differential equation of the form (()) integers is called a near-Legendre equation. We show that such an equation has infinitely many polynomial solutions corresponding to infinitely many λ. We list all of these equations for. We show, for , that these solutions are 'partially' orthogonal with respect to some weight functions and show how to expand functions using these polynomials. We give few applications to partial differential equations.
Legendre’s equation is key in various branches of physics. Its general solution is a linear function space, spanned by the Legendre functions of first and second kind. In physics however, commonly the only acceptable members of this set are the Legendre polynomials. Quantization of the eigenvalues of Legendre’s operator is a consequence of this. We present and explain a stand-alone, in-depth argument for rejecting all solutions of Legendre’s equation, but the polynomial ones, in physics. We show that the combination of the linearity, the mirror symmetry and the signature of the regular singular points of Legendre’s equation is quintessential to the argument. We demonstrate that the evenness or oddness of the Legendre polynomials is a consequence of the same ingredients.
Standard and non-standard associated Legendre equations and supersymmetric quantum mechanics
1997
A simple supersymmetric approach recently used by Dutt, Gangopadhyaya, and Sukhatme (hep-th/9611087, Am. J. Phys., to appear) for spherical harmonics is generalized to Gegenbauer and Jacobi equations. The coordinate transformation procedure is well known to the authors writing on supersymmetric quantum mechanics (see section 5 in [1]). Essentially, one starts with a one-dimensional Schrödinger equation and tries to obtain a new one by a coordinate transformation. In a recent work, Dutt, Gangopadhyaya and Sukhatme (DGS) [2] used a coordinate transformation to recast the associated Legendre equation in the Schrödinger form and then employed the concept of shape invariance to derive properties of spherical harmonics in a simple way. In the following, after presenting the case of spherical harmonics, we generalize the DGS scheme to Gegenbauer and Jacobi equations. 1. Spherical harmonics The equation for the associated Legendre polynomials dy dθ2 + cotθ dy dθ + [l(l + 1) − m 2 sin θ ]y =...
2020
The solution in hyperspherical coordinates for NNN dimensions is given for a general class of partial differential equations of mathematical physics including the Laplace, wave, heat and Helmholtz, Schrodinger, Klein-Gordon and telegraph equations and their combinations. The starting point is the Laplacian operator specified by the scale factors of hyperspherical coordinates. The general equation of mathematical physics is solved by separation of variables leading to the dependencies: (i) on time by the usual exponential function; (ii) on longitude by the usual sinusoidal function; (iii) on radius by Bessel functions of order generally distinct from cylindrical or spherical Bessel functions; (iv) on one latitude by associated Legendre functions; (v) on the remaining latitudes by an extension, namely the hyperspherical associated Legendre functions. The original associated Legendre functions are a particular case of the Gaussian hypergeometric functions, and the hyperspherical associ...
2004
In a recent paper, a new 3-parameter class of Abel type equations, so-called AIR, all of whose members can be mapped into Riccati equations, is shown. Most of the Abel equations with solution presented in the literature belong to the AIR class. Three canonical forms were shown to generate this class, according to the roots of a cubic. In this paper, a connection between those canonical forms and the differential equations for the hypergeometric functions 2F1, 1F1 and 0F1 is unveiled. This connection provides a closed form pFq solution for all Abel equations of the AIR class.
The Study of Triple Integral Equations with Generalized Legendre Functions
Abstract and Applied Analysis, 2008
A method is developed for solutions of two sets of triple integral equations involving associated Legendre functions of imaginary arguments. The solution of each set of triple integral equations involving associated Legendre functions is reduced to a Fredholm integral equation of the second kind which can be solved numerically.
Legendre Polynomials: a Simple Methodology
Journal of Physics: Conference Series, 2019
Legendre polynomials are obtained through well-known linear algebra methods based on Sturm-Liouville theory. A matrix corresponding to the Legendre differential operator is found and its eigenvalues are obtained. The elements of the eigenvectors obtained correspond to the Legendre polynomials. This method contrast in simplicity with standard methods based on solving Legendre differential equation by power series, using the Legendre generating function, using the Rodriguez formula for Legendre polynomials, or by a contour integral.
Hypergeometric-type differential equations: second kind solutions and related integrals
Journal of Computational and Applied Mathematics, 2003
A Rodrigues-type representation for the second kind solutions of a second-order di erential equation of hypergeometric type is given. This representation contains some integrals related with relevant special functions. For these integrals, a general recurrence relation, which only involves the coe cients of the di erential equation, is also presented. Finally, an extension of the Rodrigues type representation for the second solution of a second-order di erence equation of hypergeometric type is indicated.
Polynomial solutions of the classical equations of Hermite, Legendre, and Chebyshev
International Journal of Mathematical Education in Science and Technology, 2003
The classical differential equations of Hermite, Legendre, and Chebyshev are well-known for their polynomial solutions. These polynomials occur in the solutions to numerous problems in applied mathematics, physics, and engineering. However, since these equations are of second order, they also have second linearly independent solutions that are not polynomials. These solutions usually cannot be expressed in terms of elementary functions alone. In this paper, the classical differential equations of Hermite, Legendre, and Chebyshev are studied when they have a forcing term x M on the right hand side. It will be shown that for each equation, choosing a certain initial condition is a necessary and sufficient condition for ensuring a polynomial solution. Once this initial condition is determined, the exact form of the polynomial solution is presented.