Cheng Xu,Xuhong Yu and Zhongqing Wang

School of Science,University of Shanghai for Science and Technology,Shanghai 200093,China.

Abstract.Laguerre dual-Petrov-Galerkin spectral methods and Hermite Galerkin spectral methods for solving odd-order differential equations in unbounded domains are proposed.Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems.Numerical results demonstrate the effectiveness of the suggested approaches.

Key words:Dual-Petrov-Galerkin spectral methods,Laguerre functions,Hermite functions,Sobolev bi-orthogonal functions,odd-order differential equations.

1 Introduction

Spectral methods possess high-order accuracy and play an important role in scientific and engineering computations,see[2–4,6,20,21]and the references therein.For problems set in unbounded domains,such as fluid flows in an infinite strip,nonlinear wave equations in quantum mechanics and so on,the direct and commonly used spectral approaches are based on orthogonal systems on infinite intervals,i.e.,the Hermite and Laguerre orthogonal polynomials/functions.There exist a number of investigations on Laguerre and Hermite spectral methods for second and higher even-order equations,see[1,7,8,11,12,18,25,26,28,29].However,some physically interesting equations,e.g.,the Korteweg-de Vries equation,are odd-order equations.It is noteworthy that relatively few studies have focused on odd-order equations.This is partly due to the facts that the usual spectral Galerkin or collocation methods for odd-order problems lead to the condition number of algebraic systems growing too fast,and often exhibit unstable modes,cf.[10,17].

Since the main differential operators in odd-order differential equations are not symmetric,it is reasonable to use the Petrov-Galerkin spectral method to solve this kind of problems.Recently,Ma and Sun[14,15]developed a stable Legendre-Petrov-Galerkin and Chebyshev collocation method for the third-order differential equations in bounded domains.Shen[19]proposed an efficient Legendre dual-Petrov-Galerkin spectral method for the third and higher odd-order equations in bounded domains.Shen and Wang[23]also presented Legendre and Chebyshev dual-Petrov-Galerkin spectral methods for the first-order hyperbolic equations in bounded domains.For semi-infinite interval,Shen and Wang[22]considered a Laguerre dual-Petrov-Galerkin spectral method for the Korteweg-de Vries equation,which led to a strongly coercive and easily invertible linear system.It is pointed out that,the resulting linear systems mentioned above is sparse or compactly sparse.However,in most cases,people always hope to get a completely diagonalized algebraic system.

As it’s known that,the Fourier system{exp(ik·)}k∈Zis the most desirable basis owing to the facts:(i)the availability of Fast Fourier Transform;and(ii)the Sobolev orthogonality,which makes the corresponding algebraic system completely diagonal.However,the Fourier spectral method is only available for periodic problems.For non-periodic problems,the usual spectral methods merely get a sparse rather than diagonal system.Recently,Liuetal.[11–13]considered the diagonalized Laguerre and Hermite spectral methods for even-order problems in unbounded domains.Motivated by the works[11–13,19,24]and by those on Sobolev orthogonal basis functions[5,16],the main purpose of this paper is to construct Sobolev bi-orthogonal Laguerre and Hermite basis functions,and propose efficient Laguerre dual-Petrov-Galerkin and Hermite Galerkin spectral methods for oddorder problems.

The main advantages of the suggested algorithms include:

·the approximate solutions can be represented as truncated Fourier-like series;

·the resulting linear systems are diagonal and the condition numbers are equal to one;

·the computational cost is much lower than that of the classical Laguerre/Hermite spectral methods.

The remainder of this paper is organized as follows.In Section 2,we introduce the generalized Laguerre and Hermite functions.In Section 3,we construct two kinds of Sobolev bi-orthogonal generalized Laguerre functions corresponding to the third and fifth order differential equations on the half line,and propose the diagonalized Laguerre dual-Petrov-Galerkin spectral methods.In Section 4,we construct two kinds of Sobolev bi-orthogonal generalized Hermite functions corresponding to the third and fifth order differential equations on the whole line,and propose the diagonalized Hermite Galerkin spectral methods.Some numerical results are also presented in Sections 3 and 4 to demonstrate the effectiveness and accuracy of our approaches.The final section is for some concluding remarks.

2 Notations and preliminaries

LetΛbe a certain interval andω(x)be a generic weight function in the usual sense.For integerr≥0,we define the weighted Sobolev space(Λ)as usual,with the inner product(u,v)r,ω,the semi-norm|v|r,ωand the norm‖v‖r,ω,respectively.We omit the subscriptrorωwheneverr=0 orω(x)≡1.For simplicity,we denoteand

2.1 Generalized Laguerre functions

2.2 Generalized Hermite functions.

3 Diagonalized Laguerre dual-Petrov-Galerkin spectral methods

In this section,we propose diagonalized Laguerre dual-Petrov-Galerkin spectral methods for solving odd-order equations.The main idea is to find Sobolev bi-orthogonal basis functions with respect to the coercive bilinear form,such that the approximate solution can be expressed explicitly.

3.1 Third-order equation

Let us consider the following third-order equation

whereμ,ν,λare given constants.Since the third-order operator is not symmetric,it is natural to use a Petrov-Galerkin method,in which the trial and test function spaces are different.It is shown in[19,22]that for third and higher odd-order equation,it is advantageous to choose the trail and test function spaces satisfying the dual boundary conditions.

Define the dual approximation spaces by

3.2 Fifth-order equation

3.3 Numerical results

In this subsection,we examine the effectiveness and accuracy of the diagonalized Laguerre dual-Petrov-Galerkin method for solving odd-order elliptic equations.

We first examine the third-order problem on the half line.We takeμ=ν=λ=1 in(3.1)and consider the following two cases:

·u(x)=e-xsin(2x),which decays exponentially at infinity with oscillation.In Figure 1,we plot the log10of the discreteL2-errors vs.N.The near straight lines indicate geometric convergence rates(i.e.,e-cNfor certainc＞0).

·u(x)=x(1+x2)-3,which decays algebraically at infinity.In Figure 2,we plot the log10of the discreteL2-errors vs..Clearly,sub-geometric convergence rates of orderwithc＞0 are observed.

We next examine the fifth-order problem on the half line.We takeμ=ν=1 in(3.13)and consider the following two cases.

·u(x)=xsin(2x)e-x,which decays exponentially at infinity with oscillation.In Figure 3,we plot the log10of the discreteL2-errors vs.N.The near straight lines indicate geometric convergence rates.

Figure 1:Errors of scheme(3.2)with solutions of exponential decay.

Figure 2:Errors of scheme(3.2)with solutions of algebraic decay.

·u(x)=xsin(x)(1+x2)-3,which decays algebraically at infinity with oscillation.In Figure 4,we plot the log10of the discreteL2-errors vs..Clearly,sub-geometric convergence rates are observed in this case.

Figure 3:Errors of scheme(3.14)with solutions of exponential decay.

Figure 4:Errors of scheme(3.14)with solutions of algebraic decay.

4 Diagonalized Hermite Galerkin spectral methods

In this section,we propose the diagonalized Hermite Galerkin spectral methods for solving odd-order differential equations on the whole line.

4.1 Third-order equation.

Consider the third-order equation:

4.2 Fifth-order equation.

Consider the fifth-order equation:

4.3 Numerical results

In this subsection,we examine the effectiveness and accuracy of the diagonalized Hermite Galerkin spectral method for solving odd-order elliptic equations.

We first consider the third-order equation on the whole line.Takeμ=ν=λ=1 in(4.1)and consider the following two cases:

·u(x)=e-x2sin(2x),which decays exponentially at infinity with oscillation.In Figure 5,we plot the log10of the discreteL2-errors vs.N.The near straight lines indicate a geometric convergence rate(i.e.,e-cNfor certainc＞0).

·u(x)=sin(x)(1+x2)-3,which decays algebraically at infinity with oscillation.In Figure 6,we plot the log10of the discreteL2-errors vs..Clearly,sub-geometric convergence of orderwithc＞0 are observed.

We next examine the fifth-order equation on the whole line.Takeμ=ν=1 in(4.14)and consider the following two cases:

·u(x)=e-x2sin(2x),which decays exponentially at infinity with oscillation.In Figure 7,we plot the log10of the discreteL2-errors vs.N.The near straight lines indicate a geometric convergence rate.

·u(x)=sin(2x)(x2+1)-5,which decays algebraically at infinity with oscillation.In Figure 8,we plot the log10of the discreteL2-errors vs..Clearly,sub-geometric convergence rates are observed.

To demonstrate the essential superiority of our diagonalized Hermite Galerkin spectral method to the classical Hermite spectral method,we examine the issues on the 2-norm condition numbers for the resulting algebraic systems and the computational cost.

Since the basis functions in the diagonalized Galerkin spectral method are Sobolev biorthogonal,and hence the condition numbers are always equal to 1.For the classical Hermite Galerkin spectral method,the basis functions are chosen asfor odd-order problems on the whole line.The corresponding total stiff matrices have off-diagonal entries.In Table 1,we list the condition numbers of the total stiff matrices of the classical Hermite Galerkin spectral method for(4.1)and(4.14).We notice that the condition numbers increase asymptotically as O(N2)for the third-order problem(4.1)withμ=ν=λ=1 andβ=2,and as O(N3)for the fifth-order problem(4.14)withμ=ν=1 andβ=2.

To compare the computational cost between the diagonalized Hermite spectral method and the classical Hermite spectral method,we consider the problem(4.14)withμ=ν=1andβ=2.In Table 2,we show the CPU elapsed time.Clearly,our diagonalized spectral method(DSM)costs much less CPU time than that of the classical Hermite spectral method(CSM).

Table 1:Condition numbers of the classical Hermite Galerkin spectral methods.

Table 2:A comparison of the CPU time.

Figure 5:Errors of scheme(4.3)with solutions of exponential decay.

Figure 6:Errors of scheme(4.3)with solutions of algebraic decay.

Figure 7:Errors of scheme(4.16)with solutions of exponential decay.

Figure 8:Errors of scheme(4.16)with solutions of algebraic decay.

5 Concluding remarks

In this paper,we proposed the Laguerre dual-Petrov-Galerkin and Hermite Galerkin spectral methods for odd-order differential equations in unbounded domains.We also constructed some Sobolev bi-orthogonal basis functions,which lead to the diagonalization of discrete systems.Numerical results demonstrate the effectiveness of the suggested approaches.Particularly,the suggested methods can be extended to some variable coefficient differential equations with the coefficients being certain polynomials.