Derivation of the integration formula is provided in. Note that in the special case lambda0, the associated laguerre differential equation is of the form. Based on the analysis of the boundary value problem of laguerre equation, this paper studies the similar structure of its solution expression. Differential equations for generalized laguerre and jacobi.
Just as for the hermite equation, solutions exist for noninteger. Research article a modified generalized laguerre spectral. Laguerre waveletgalerkin method for the numerical solution of one dimensional partial di erential equations nite element methods have good spatial localization, but poor spectral localization. Moreover, the formulae of the order and type of the solutions are established. Polynomials laguerreln,lambda,z differential equations 15 formulas. Gaussian, hermitegaussian, and laguerregaussian beams. Note that if, then the solution to the associated laguerre differential equation is of the form 3 and the solution can be found using an integrating factor 4 so 5. Generalized laguerre functions and differential recursion.
Wavelet bases execute to combine the advantages of both spectral and nite element bases. We refer the reader to the literature for a more indepth look at the laguerre xop families. For certain special arguments, laguerrel automatically evaluates to exact values. The con volution of two laguerrre polynomials in z is a simple sum 2. Laguerre polynomials, are found in many important physical problems. The first ode is the following 5parameter laguerre type differential equation. A modified generalized laguerre spectral method for. They have circular symmetry and can be written in terms of the laguerre polynomials, where is the radial index and is the azimuthal index. A laguerre galerkin method is proposed and analyzed for the burgers equation and benjaminbonamahony bbm equation on a semiinfinite interval. Find the generalized laguerre polynomials where input arguments n and x are matrices. The boundary conditions quantize the energy of the hydrogen atom. A laguerregalerkin method is proposed and analyzed for the burgers equation and benjaminbonamahony bbm equation on a semiinfinite interval.
Using the generating function for the legendre polynomial show that. Adm based on laguerre polynomials to solve differential equation by the adomian decomposition method, for an arbitrary integer number, gx can be expressed in the taylor series and laguerre series, that is pointed by g x, kx and g j, kx, respectively, where g x. Kiwne, application of combine natural transform and adomian decomposition method in volterra integrodifferential equations, 2016. In terms of the normal laguerre polynomials, 6 the rodrigues formula for the associated laguerre polynomials is 7 8 and the generating function is 9. Solution to laguerre differential equation using generating. Laguerres differential equation article about laguerre. Using the plane wave representation of the fundamental gaus. Follow these steps to show that solutions to kummers equation 7. Request pdf differential equations for deformed laguerre polynomials the distribution function for the first eigenvalue spacing in the laguerre unitary ensemble of finite size may be expressed.
On a differential equation for koornwinders generalized laguerre polynomials. Generalizations of laguerre polynomials sciencedirect. The usual textbook approach for deriving these modes is to solve the helmoltz electromagnetic wave equation within the paraxial approximation. By reformulating these equations with suitable functional transforms, it is shown that the laguerregalerkin approximations are convergent on a semiinfinite interval with spectral accuracy. The rodrigues formula for the laguerre polynomials is. Does anybody know what the use of the laguerre differential equation would be.
Taha 5 department of mathematics and computer sciences, faculty of arts and sciences, cankaya university, eskisehir yolu. In particular, a spectral study of the x 1laguerre polynomials 1. Adomian decomposition method with laguerre polynomials. In this paper, we propose two integration processes for ordinary di.
Polynomials laguerreln,z differential equations 10 formulas ordinary linear differential equations and wronskians 10 formulas. This equation has nonsingular solutions only if n is a nonnegative integer sometimes the name laguerre polynomials is used for solutions of. Pdf laguerre polynomial solutions of a class of delay partial. For example, functions here take the role of vectors there, and linear. Laguerre waveletgalerkin method for the numerical solution. In this paper, we rigorously show the wellposedness of the models in sabzikar et al. In this case we only have one endpoint of the interval of orthogonality of the classical orthogonal polynomials. Laguerre polynomials a differential equation that arises in the study of the hydrogen atom is the laguerre equation. The term confluent refers to the merging of singular points of families of differential equations. Laguerre and jacobi polynomials ftp server of the gwdg. The origination of the type iii laguerre along with a comprehensive look at all three types of laguerre xop systems may be found in 19. But there exists at least one solution to the above equation, of the form y z. This is an exercise in modern quantum mechanics by sakurai and napolitano.
The solutions of the laguerre equation are called the laguerre polynomials, and together with the solutions of other differential equations, form the functions that describe. These polynomials, termed exceptional laguerre and jacobi polynomials, have two. Laguerre functions and differential recursion relations p. Laguerregalerkin method for nonlinear partial differential. Finding differential equations for symmetric generalized ultraspherical polynomials by using inversion methods. Multiplier sequences for laguerre bases diva portal. Series solutions of laguerre and jacobitype differential equations. The laguerre equation has coefficients that are not analytic at xx00 and yet. We can look forward to numerical methods based on wavelet bases to. Laguerre differential equation from wolfram mathworld. The laguerre polynomials are orthogonal with weight function. Dn ncomplex matrix of independent standard complex gaussian. Integration processes of ordinary differential equations based on laguerreradau interpolations benyu guo, zhongqing wang, hongjiong tian, and lilian wang abstract. Plug the above result into the radial equation to recover the associated laguerre equation for l.
Section 5, is devoted to approximations by the generalized laguerre polynomials and a useful theorem. The method uses properties of laguerre polynomials to convert the integral to a sum, while the remaining differential equation is solved analytically using an evolution operator approach to. We study a homogeneous partial differential equation and get its entire solutions represented in convergent series of laguerre polynomials. Differential equations 15 formulas ordinary linear differential equations and wronskians 14 formulas integral equations whose solutions contain the direct function 1 formula,, differential equations 15 formulas laguerrel.
Solutions to the laguerre differential equation with are called laguerre polynomials. For with the laguerre differential equation has a solution for which prove that the laplace transform of is my lecturer gave me a hint saying to use the bessel function. While the similar kernel function is constructed by both the function of guide solution and coefficients of right. Keywords tempered fractional differential equations singularity laguerre functions generalized laguerre functions weighted sobolev spaces approximation results spectral accuracy this work is supported in part by nsfc grants 171298, 1142111, 91630204 and 516615011.
A differential equation in 4 we found a second order differential equation for our polynomials in the case jv 1. When n is not a nonnegative integer, laguerrel returns the laguerre function. An application of the method for a higher order linear differential equation is presented in section 7. This operational matrix is applied together with the modified generalized laguerre tau method for solving general linear multiterm fractional differential equations fdes. Power series solutions of laguerre differential equation. We refer to it as laguerre type because the factors multiplying the first and second order derivatives have the same structure as that of the laguerre differential equation. So we study generalizations of the classical laguerre polynomials which are also further generalizations of the laguerre type and koornwinders generalized laguerre polynomials. Orthogonal polynomials with respect to the laguerre measure. Moment representations of type i x exceptional laguerre.
Consider for example the principle of argument that counts the. The term confluent refers to the merging of singular points of families of differential. I read earlier from differential equations by lomen and mark that a power series about xx0 solution of an ode exists when all polynomial coefficients are analytic at xx0. Find a solution of the differential equation of th. A new modified generalized laguerre operational matrix of.
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. Polynomials laguerreln,z differential equations 10 formulas ordinary linear differential equations and wronskians. Laguerres equation is equivalent to the statement that l m x is the eigenfunction with eigenvalue m of the second order di. The same method can be used in the general case, but in 3 j. Series solutions about an ordinary point if z z0 is an ordinary point of eq. A modified generalized laguerre spectral method for fractional differential equations on the half line d. A more interesting example of the application of newtons ideas has to do with. This form motivates the expansion of pz and fz, t in terms of laguerre polynomials, whose orthogonality condition has the exponential kernel. Series solutions of laguerre and jacobitype differential. In this paper, we derived a new operational matrix of fractional integration of arbitrary order for modified generalized laguerre polynomials.
Tempered fractional diffusion equations tfdes involving tempered fractional derivatives on the whole space were first introduced in sabzikar et al. Differential equations for deformed laguerre polynomials article in journal of approximation theory 1624. Differential equations 10 formulas ordinary linear differential equations and wronskians 10 formulas,,, differential equations 10 formulas laguerrel. Generalized laguerre function and laguerre polynomials.
B 195 1982, 237 is used to solve integrodifferential equations found in high energy scattering processes and solid state physics. Solutions to the associated laguerre differential equation with are called associated laguerre polynomials. Equation is a special case of the more general associated laguerre differential equation, defined by 2 where and are real numbers iyanaga and kawada 1980, p. These sets are less common in mathematical physics than the legendre and bessel functions of chapters 11 and 12, but hermite polynomials occur in solutions of the simple harmonic oscillator of quantum mechanics and laguerre polynomials in wave functions of the hydrogen atom. Laguerre differential equation and its solutions, i. This paper deals with modified generalized laguerre spectral tau and collocation methods for solving linear and nonlinear multiterm fractional differential equations fdes on the half line. Series solutions for the laguerre and laneemden fractional. Proceedings of the american mathematical society 112, 1991, 10451054. Lastly, we find a differential equation for the multiple orthogonal polynomials by the generating function, which is a reconstruction of the works by aptekarev et al. The hermite and laguerre polynomial bases are investigated in detail. Laguerregaussian modes of paraxial wave equation wolfram. This paper is found that its solution can be obtained by combining similar kernel function with coefficients of left boundary condition. The laguerre method for solving integrodifferential equations.
Research article laguerre collocation method for solving. Jul 18, 2017 tempered fractional diffusion equations tfdes involving tempered fractional derivatives on the whole space were first introduced in sabzikar et al. In mathematics, the laguerre polynomials, named after edmond laguerre 18341886, are solutions of laguerres equation. This paper discusses two new integration processes based on modi. Laguerrel can be evaluated to arbitrary numerical precision. I am having a hard time finding what areas of physics this diff. The fractional integration is described in the riemannliouville sense.
Laguerregaussian modes are solutions of the paraxial wave equation. In this case the highest derivative occurring is the second order, so this is called a differential equation of second order. Over time, an extensive list of differential equations and their solutions. Global solutions of some firstorder partial differential equations or system were studied by berenstein and li, hu and yang, hu and li, li, li and saleeby, and so on. We reach the laguerre polynomial solution ux, t in eq. By reformulating these equations with suitable functional transforms, it is shown that the laguerre galerkin approximations are convergent on a semiinfinite interval with spectral accuracy. Differential equations for deformed laguerre polynomials. In this paper, we concentrate on the following partial differential equation pde for a real. Pdf in this study, we develop a novel matrix collocation method. A linear homogeneous partial differential equation with. In particular, a spectral study of the x 1 laguerre polynomials 1. The laguerre polynomials are illustrated above for and, 2. A new formula expressing the caputo fractional derivatives of modified generalized laguerre polynomials of any degree and for any fractional order in terms of the modified generalized laguerre polynomials. This differential equation is important in quantum mechanics because it is one of several equations that appear in the quantum mechanical description of the hydrogen atom.
Laplace transform laguerre differential equation the. A method for solving a class of boundary value problems of. Find out information about laguerres differential equation. The laguerre method proposed by furmanski and petronzio nucl. Legendre polynomials are usually derived from differential equations of the following form. The generalized laguerre matrix method or solving linear. Laguerre functions and their applications to tempered. Koekoek gave a simple proof of the differential equation. Status offline join date apr 20 posts 3,635 thanks 3,519 times thanked 1,095 time awards. Laplaces method for ordinary differential equations. In section 6, the proposed method is applied for three numerical experiments. This equation has nonsingular solutions only if n is a nonnegative integer.
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