On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain

1 Introduction

Classical Laguerre polynomials [1,2,3,4,] are the main subject of a very extensive literature. If we denote by Ln(α)(x): n = 0,1,2,…}, the sequence of monic Laguerre polynomials, then their crucial property is the following orthogonality condition

where m, n ∈ ℕ and the parameter α satisfies the condition α > – 1 in order to assure the convergence of the integrals. For more details about classical Laguerre polynomials, please see [5].

In this framework, we define the modified generalized Laguerre polynomials Ln(α,β)(x), with the newly added parameter β, the existence of this parameter will affect on the convergence of Ln(α,β)(x), moreover, the classical Laguerre polynomials can be obtained as a direct special case of Ln(α,β)(x) by letting β = 1.

Many differential models in various research disciplines such as mathematics, fluid dynamics, chemistry, biology, viscoelasticity, engineering and physics have arisen in semi-infinite domains [6,7,8,9,10]. Consequently, a lot of researchers have utilized various transformations on classical orthogonal polynomials to map the bounded interval [–1,1] into [0,L] and [0,∞) maintaining their orthogonal property. This idea is acceptable, nevertheless, the rate of convergence as well as the stability of solutions will affected when ≫ 1 also when the number of retained modes is great. In this research we tame the proposed orthogonal polynomials Ln(α,β)(x) to handle some differential problems on a semi-infinite domain.

To be more precise we tackle three important problems over a semi-infinite domain, namely, the Lane-Emden equation, Bratu’s type equation and the nonlinear space-time Burger’s type equation.

The Lane-Emden equation [11,12,13] is one of the basic equations in the theory of stellar structure and has been the focus of many studies. This equation describes the temperature variation of a spherical gas cloud under the mutual attraction of its molecules and subject to the laws of classical thermodynamics. It also describes the variation of density as a function of the radial distance for a polytrope.

The initial value problems of Bratu’s type [14,15] was used to model a combustion problem in a numerical slab. Bratu’s problem is also used in a large variety of applications such as the fuel ignition model of the thermal combustion theory, the model of the thermal reaction process, the Chandrasekhar model [16] of the expansion of the universe, questions in geometry and relativity concerning the Chandrasekhar model, chemical reaction theory, radiative heat transfer and nanotechnology.

The one-dimensional Burgers equation first appeared in a paper by Bateman [17], who derived two of the essentially steady solutions. It is a special case of some mathematical models of turbulence introduced about twenty years ago by Burgers [18]. The distinctive feature of Burger’s equation is that it is the simplest mathematical formulation of the competition between convection and diffusion. It thus offers a relatively convenient means of studying not only turbulence but also the distortion caused by laminar transport of momentum in an otherwise symmetric disturbance and the decay of dissipation layers formed.

In this work, we attempt to solve differntial equations using the modified generalized Laguerre polynomials jointly with the spectral accurate tau and collocation methods. The Laguerre polynomials and the spectral collocation method for the numerical solution of differential equations have been extensively used in [19,20,21,22,23,24]. Guo et al. [25] applied the mixed generalized Laguerre spectral method for the exterior problems. In [26], the mixed Laguerre-Legendre spectral method and pseudospectral method are proposed for solving the fluid folw problems in an infinite channel. Rahmoune [27], presented a spectral collocation method using the scaled Laguerre functions for solving the Fredholm integral equations of the second kind on the half-line. In [28], a spectral collocation method based on the Laguerre polynomials has been proposed for solving the fractional pantograph equation.

In most of these applications, a formula is used that relates the expansion coefficients of derivatives appearing in the differential equation to those of the function itself. For the Galerkin and tau variants of the spectral methods, explicit expressions for the expansion coefficients for the solution are needed. Karageorghis [29], obtained an expression when the basis functions of expansion are shifted Chebyshev polynomials Tn∗(x),,x ∈ [0, 1]. A corresponding formula for Legendre polynomials P_n(x), x ∈ [–1, 1], is derived by Phillips [30]. Doha [31], has obtained a more general formula when the basis functions are the ultraspherical polynomials Cn(λ)(x), x ∈ [–1, 1], λ ∈ (–12, ∞); formulae for the first and second kinds of Chebyshev polynomials and Legendre polynomials T_n(x), U_n(x) and P_n(x) are given as special cases of Cn(λ)(x). A most general formula when the basis functions are the Jacobi polynomials Pn(ν,μ) (x), x ∈ [–1, 1], ν > –1, μ > –1, is given in Doha [32]. Another formula when the basis functions are the Hermite polynomials is obtained in Doha [33], two other formulae for the third and fourth kinds of Chebyshev polynomials V_n(x) and W_n(x) are given in Doha and Abd-Elhameed [34].

A more general situation which often arises in the numerical solution of differential equations with polynomial coefficients in spectral methods is the evaluation of the expansion coefficients of the moments of high-order derivatives of infinitely differentiable functions. A formula for the shifted Chebyshev coefficients of the moments of general-order derivatives of an infinitely differentiable function is given in Karageorghis [29]. Corresponding results for Chebyshev polynomials of the first and second kinds, Legendre, ultraspherical and Hermite polynomials are given in Doha [35], Doha and El-Soubhy [36], Doha [37] and Doha [38], respectively, another work on connection formulae were given in [39,40,41].

Up to now, and to the best of our knowledge, many formulae corresponding to those mentioned previously are not known and traceless in the literature for the modified generalized Laguerre expansions. This motivates our interest in such polynomials. Another motivation is that the theoretical and numerical analyses of numerous physical and mathematical problems very often require the expansion of an arbitrary polynomial or the expansion of an arbitrary function with its derivatives and moments into a set of orthogonal polynomials. This is in particular true (for modified generalized Laguerre polynomials) in quantum mechanical studies of physical systems, where the equation of motion or Schrp̈dinger equation is a second-order differential equation with polynomial coefficients. This is the case not only for the solution of the Schrödinger, Klein–Gordon and Dirac equations for the Coulomb field but also for many other potentials, as shown for example in Bagrov and Gitman [42] and Nikiforov and Uvarov [43].

The presentation of the paper is as follows, in Section 2, we present some important properties of the modified generalized Laguerre polynomials. In Section 3, we state and prove relations between the coefficients Section 3, we state and prove relations between the coefficients an(q) and a_n and the qth derivative of Ln(α,β)(x). In Section 4, we present explicit relation for the modified generalized Laguerre coefficients of the moments of one single modified Laguerre polynomial of any degree. In Section 5, we present the modified generalized Laguerre coefficients of a general-order derivative of an infinitely differentiable function in terms of its modified Laguerre coefficients. In Section 6, we implement an application to ordinary differential equations with variable coefficients. Finally, in Section 7, we present four applications for solving second-order differential equations with variable coefficients, namely, the Lane-Emden equation, Bratu’s type equation and the nonlinear space-time Burger’s type equation.

2 Some properties of modified generalized Laguerre polynomials

Let w_α,β(x) = x^α, e^{–β x} with α > – 1 and β > 0. Then the modified generalized Laguerre polynomials (MGLP) are a sequence of polynomials {Ln(α,β)(x) : n = 0,1,2,…}, each of degree n, satisfying the orthogonality relation

where δ_mn is the kronecker symbol. The set of (MGLP) is a complete Lwα,β(x)2(0, ∞)–orthogonal system. The modified generalized Laguerre polynomials may be generated by using Rodrigue’s formula

The following two recurrence relations are of fundamental importance in developing the present work. These are

with L−1(α,β)(x) = 0, and

where D ≡ ddx. Note that the recurrence relation (4) may be used to generate the (MGLP) starting from L0(α,β)(x) = 1, and L1(α,β)(x) = 1 + α – βx. If β = 1, then Ln(α,β)(x) is the standard modified generalized Laguerre polynomials.

Suppose now we are given a function f(x) which is infinitely differentiable in the interval [0, ∞), then we can write

and for the qth derivative of f(x),

Moreover, if f(x) satisfies, f(x) = O(e^αx) for x → ∞, and for some α < 12, then it can be shown (cf, [44]) that the modified generalized Laguerre expansion

converges faster than algebraically as the number of terms N → ∞.

3 Relations between the coefficients an(q) and a_n and the qth derivative of Ln(α,β) (x)

4 Modified generalized Laguerre coefficients of the moments of one single (MGLP) of any degree

For the evaluation of modified generalized Laguerre coefficients of the moments of higher-order derivatives of infinitely differentiable functions, the following theorem is needed.

It is worth noting here that, recalling the definition of Pochhammer’s symbol,

and the identity

formula (10) can be written in terms of a ₃F₂ hypergeometric function of unit argument.

5 Modified generalized Laguerre coefficients of a general-order derivative of an infinitely differentiable function

6 Application to ordinary differential equations with variable coefficients

1. Let y(x) be infinitely differentiable function defined on [0, ∞) and having the Laguerre expansion as in (5), and assume that it satisfies the linear nonhomogenous differential equation of order n > 0,

   ∑i=0npi(x)y(i)(x)=f(x),x∈[0,∞),(28)

   where p₀, p₁,…,p_n ≠ 0 are polynomials of x, and the coefficients of Laguerre series of f(x) are known; formula (7), (9) and (19) enable one to construct, in view of (28), the linear recurrence relation of order r, namely,

   ∑j=0rαj(k)ak+j=β(k),k≥0,(29)

   where α₀, α₁,…, α_r (α₀,α_r ≠ 0) are polynomials of the variable k. The interested reader is referred to [45] for a similar derivation of (29) when the basis of expansion is Jacobi polynomials.

2. Consider the linear ordinary differential equation of order n of the form

   ∑i=0nfi(x)y(i)(x)=g(x),x∈[0,∞),(30)

   where f_i(x) and g(x) are functions of x only. Suppose the equation to be solved in the interval [0, ∞) subject to n linear boundary conditions, and assume we approximate y(x) a truncated expansion of Laguerre polynomials

   ∑i=0nfi(x)∑j=0NajDjLj(α,β)(x)=g(x),(31)

   which may be written in the form

   ∑j=0Naj∑i=0nfi(x)ajDjLj(α,β)(x)=g(x).(32)

   The boundary conditions associated with (30) give rise to n equations connecting the coefficients a_j, and the remaining equations may be obtained in two ways:

1. We may equate the coefficients of the various Li(α,β)(x) after expanding the two sides of (33) in Laguerre series.

2. We may collocate at m = N – n selected points in (0, ∞).

The system of equations obtained from the collocation is of the form

where x_k are the collocation points, which are usually chosen at the zeros of Lm(α,β)(x), (see, for instance, [46]). Since the derivatives D^jLj(α,β)(x) are now expressible explicitly in terms of Lj(α,β)(x), then the problem of computing them solved by using the formula (8). Therefore, the resulting linear system obtained from (31) and the n linear boundary conditions can easily be solved using the standard direct solvers.

7 Applications and Numerical Results

Example 2

Consider the Lane-Emden type equation

xy″(x)+2y′(x)+xy(x)=0,y(0)=1,y′(0)=0.(40)

with the exact solution

y(x)=sin⁡xx.

If y(x) is expanded in the form

y(x)=∑i=0NaiLi(α,β)(x),(41)

then by virtue of formulae(18)and(19), Eq. (40), takes the form

bi2,1+2bi1,0+bi0,1=0,i≥0,(42)

wherebiq,lare as given in(19). In Table 1, we list the maximum absolute error of Example 2, for different values of α, βand N over the interval (0, 10). In Figures 1, 2, we depict the graph of approximate and absolute error of Example 2, respectively, for the case α = 1, β = 12and N = 40. In Figure 3, we show the convergence of the method when N increases.

Fig. 1

Approximate solution of Example 2

Fig. 2

Absolute Error of Example 2

Fig. 3

Log-Error of Example 2

Example 3

Consider the Bratu type equation

y″(x)−2ey(x)=0,y(0)=y′(0)=0.(43)

with the exact solution

y(x)=−logcos2⁡x.

If y(x) is expanded in the form

y(x)=∑i=0NaiLi(α,β)(xk),(44)

here, we apply the collocation technique. In Table 2, we list the maximum absolute error of Example 3, for different values α, β and N over the interval (0, 1.57). In Figures 4 and 5, we depict the graph of approximate and absolute error of Example 3, respectively, for the case α = 1, β = 14and N = 20. In Figure 6, we show the convergence of the method when N increases.

Table 2

Maximum absolute errors for Example 3

N	α =1, β = 2	α =1, β =1	α = 1, β =12	α =1, β =14
5	8.12 .10–1	2.27 . 10–2	6.29 . 10–2	3.27 . 10–2
10	9.76 . 10–2	3.86 .10–3	2.67 .10–3	4.44 .10–3
15	7.38 . 10–2	4.97 .10–4	2.81 .10–3	3.67 .10–4
20	5.38 .10–3	9.37 .10–4	3.94 .10–4	1.00 .10–5

Fig. 4

Approximate solution of Example 3

Fig. 5

Absolute Error of Example 3

Fig. 6

Log-Error of Example 3

Example 4

Consider the space-time nonlinear Burgers type equation

ut(x,t)−μuxx(x,t)+u(x,t)ux(x,t)=π2e−2π2μtsin⁡(2πx),(x,t)∈(−1,1)×(0,τ)(45)

subject to the homogenous boundary conditions

u(±1,t)=0,

and the initial condition

u(x,0)=−sin⁡(πx),

with the exact smooth solution

u(x,t)=−e−π2μtsin⁡(πx).

We write

u(x,t)≈uN(x,t)=∑i=0N∑j=0NcijTi(x)Lj(α,β)(t),

where N is even number and T_i(x) denotes the Chebyshev polynomials of degree i. The residual of Eq. (45)is given by

R(x,t)=utN(x,t)−μuxxN(x,t)+uN(x,t)uxN(x,t)−π2e−2π2μtsin⁡(2πx),(46)

we select the following collocation points

xi=cos(2i+1)π2N+2,tj=jτN+1,0≤i,j≤N+1,

where x_i are the distinct roots of the first kind Chebyshev polynomial T_N+1(x) and t_j are the equidistant Riemann points. Now we have the following system of nonlinear algebraic equations

R(xi,tj)=0,0≤i,j≤N−1(47)

u(±1,tj)=0,0≤j≤N2−1(48)

u(xi,0)=−sin⁡(πxi),0≤i≤N(49)

The dimension of this system is (N+1)², thanks to the Newton’s iterative method for solving this nonlinear system and finally we obtain the solution. We solve this system for the case corresponding to μ = 0.02, τ = 10, N = 20 and (α,β) = (32,12).In Figures 7, 8, we depict the approximate solution of Eq. (45)and the error E = u – u^N, respectively.

Fig. 7

Approximate solution of Example 4

Fig. 8

Error of Example 4

8 Concluding Remarks

A formula expressing the modified generalized Laguerre coefficients of a general-order derivative of an infinitely differentiable function in terms of its original coefficients is proved, and a formula expressing explicitly the derivatives of modified Laguerre polynomials of any degree and for any order as a linear combination of suitable modified Laguerre polynomials is deduced. A formula for the modified Laguerre coefficients of the moments of one single modified Laguerre polynomial of certain degree is given. Four applications were implemented to illustrate the effect of the new parameter β on the convergence and accuracy of the methods.