Coefficients perturbation methods for higher-order differential equations

International Journal of Computer MathematicsVol. 86, No. 8, August 2009, 1453–1472

Coefficients perturbation methods for higher-orderdifferential equations

M.K. El Daou* and N.S. Al Mutawa

Applied Sciences Department, College of Technological Studies, Shuwaikh, Kuwait

(Received 16 January 2007; revised version received 07 September 2007; accepted 12 December 2007 )

In this paper, we develop a general approach for estimating and bounding the error committed when higher-order ordinary differential equations (ODEs) are approximated by means of the coefficients perturbationmethods. This class of methods was specially devised for the solution of Schrödinger equation by Ixaru in1984. The basic principle of perturbation methods is to find the exact solution of an approximation problemobtained from the original one by perturbing the coefficients of the ODE, as well as any supplementarycondition associated to it. Recently, the first author obtained practical formulae for calculating tight errorbounds for the perturbation methods when this technique is applied to second-order ODEs. This paperextends those results to the case of differential equations of arbitrary order, subjected to some specifiedinitial or boundary conditions. The results of this paper apply to any perturbation-based numerical techniquesuch as the segmented Tau method, piecewise collocation, Constant and Linear perturbation. We will focuson the Tau method and present numerical examples that illustrate the accuracy of our results.

Keywords: perturbation; Green’s function; series expansions; orthogonal polynomials

2000 AMS Subject Classification: 65L05; 65L70

1. Introduction

The coefficients perturbation methods form a class of efficient numerical techniques for approx-imating ordinary differential equations (ODES) associated with initial or boundary conditions.This class of methods was specially devised for the solution of Schrödinger equation (see Ixaru[6,7] for a comprehensive review). The basic principle of this technique is to solve analyticallyapproximation problems obtained from the original one by perturbing the coefficients of the differ-ential equation as well as the supplementary conditions associated to it. In recent years, due to thedevelopment of highly sophisticated computer algebra packages such as Mathematica, Maple

and Matlab, (see Gander and Hrebicek [5] and Wolfram [12]), there is again much interest in thistopic, for those packages enjoy the capability of integrating numeric and symbolic computations,and of processing complex formulae involving arithmetic and functional operations with a littleconcern regarding the stability against roundoff errors of recurrence relations.

*Corresponding author. Email: [email protected]

ISSN 0020-7160 print/ISSN 1029-0265 online© 2009 Taylor & FrancisDOI: 10.1080/00207160701874797http://www.informaworld.com

1454 M.K. El Daou and N.S. Al Mutawa

One advantage of the coefficients perturbation method is that it allows in general relativelylarge subintervals at which it can achieve accurate approximations. Another advantage is that thequality of approximation can be appreciated in advance in terms of the perturbations introducedin the coefficients, and therefore it can be regulated so that the errors are kept under a certainlevel. A computational device to measure this error was given by the first author [3] for the case ofsecond-order initial value problems. This paper is concerned with developing a general approachfor estimating and bounding the error committed when differential equations of higher order areapproximated by means of the coefficients perturbation methods.

To start with, let us consider the following ODE of order ν ∈ N:

(Dy)(x) := y(ν)(x) + pν−1(x)y(ν−1)(x) + · · · + p0(x)y(x) = f (x), x ∈ [a, b], (1)

where {pk(x); k = 0, 1, . . . , ν − 1} belong to Cν[a, b], the space of ν times continuously differ-entiable functions in interval [a, b], and y(k) stands for the kth derivative of y with respect to x.Suppose that there are ν supplementary conditions associated with (1) and defined as

Bk[y] = αk ∈ R; k = 0, 1, 2, . . . , ν − 1, (2)

where {Bk; k = 0, 1, . . . , ν − 1} are ν linearly independent linear functionals actingon the elements of Cν[a, b]. In the coefficients perturbation methods we replace{f (x), pk(x), αk; k = 0, 1, . . . , ν − 1} in Equations (1) and (2) by some prescribed approxima-tions {f (x), pk(x), αk; k = 0, 1, . . . , ν − 1} chosen so that the exact solution y of the resultingapproximation problem:

y(ν)(x) + pν−1(x)y(ν−1)(x) + · · · . + p0(x)y(x) = f (x), x ∈ [a, b], (3)

Bk[y] = αk, k = 0, 1, 2, . . . , ν − 1 (4)

can be obtained analytically in a closed form.Subtracting Equations (1) and (2) from (3) and (4) we find that the error function e(x) :=

y(x) − y(x) satisfies the differential equation:

e(ν)(x) + pν−1(x)e(ν−1)(x) + · · · . + p0(x)e(x) = F(x), x ∈ [a, b], (5)

Bk[e] = εk, k = 0, 1, 2, . . . , ν − 1

where εk := αk − αk , k = 0, 1, 2, . . . , ν − 1 and

F(x) := δf (x) − δpν−1y(ν−1) − δpν−2y

(ν−2) − · · · . − δp0y,

with δf := f − f and δpk := pk − pk , k = 0, 1, . . . , ν − 1.Our primary focus is on establishing an enclosure theorem (Theorem 3.2) from which follows

error estimates and bounds that allow to measure, with a great degree of accuracy, the deviationof the (unknown) exact solution y of problem (1) and (2) from the computable reference y that isdefined implicitly by Equations (3) and (4). These error bounds and estimates have a considerablepractical value because they do not involve uncomputable quantities, and their degree of accuracydoes not depend solely on the selected stepsize, but also on the order of the truncated seriesexpansions. The investigation carried out here will make an extensive use of the Green’s functionG(x, t) that is associated with the differential operator D given in (1). The Green’s function isrelated to the inverse of the differential operator and will play a substantial role in constructing aformal series representation for the error function.

A large class of highly efficient techniques for approximating initial or boundary-value prob-lems can be interpreted as coefficients perturbation methods. For instance, if {f (x), pk(x);

International Journal of Computer Mathematics 1455

k = 0, 1, 2, · · · , ν − 1} are chosen to be constant (respectively linear) then one obtains theConstant Perturbation or CP-method (resp. Linear Perturbation or LP-method) introduced ini-tially by Ixaru in [6] and later used to approximate Schrödinger equation and Sturm–liouvilleproblems (see Ledoux et al. [8, 9]). The CP-method (respectively LP) yields approximate solu-tions expressed in terms of exponential or trigonometric functions (respectively Airy functions).When {f , pk(x); k = 0, 1, 2, · · · , ν − 1} are polynomials of appropriate orders, then we obtainthe recursive Tau method (see Ortiz [10]) or the spectral Tau method (see Canuto et al. [4]). Theformer produces approximations in terms of a special polynomials basis associated with D calledcanonical polynomials basis, and the latter exhibits the desired approximation as a truncatedorthogonal polynomial series.

This paper is organized as follows. The next section presents an algorithm for constructing theTaylor series expansions of the Green’s function G(x, t). Section 3 is concerned with obtainingbounds and estimates for the local error; that is the error committed after one step only. Globalerror estimates will be investigated in Section 4. Section 5 is devoted to apply our results to theTau method as a special form of coefficients perturbation methods. Numerical tests illustratingthe accuracy and sharpness of our results will be given in Section 6.

2. Taylor series expansions of the Green’s function

From the theory of ODEs (see Birkhoff and Rota [1]), the Green’s function G(x, t) associatedwith the linear operator D satisfies the following initial value problem:

ν∑j=0

pν−j (x)G(ν−j)(x, t) = 0, t ≤ x ∈ [a, b], (pν(x) = 1) (6)

G(k)(t, t) = 0, k = 0, 1, . . . , ν − 2,

G(ν−1)(t, t) = 1,

where G(k)(x, t) := ∂kG(x, t)/∂xk denotes the kth partial derivative of G(x, t) with respectto x. The purpose of this section is to present a procedure that permits to construct explicitlythe coefficients of Taylor series expansions of G(x, t). This is summarized in the followingproposition:

Proposition 2.1 If {pk(x); k ≥ 0} ⊂ C∞[a, b], the space of infinitely differentiable functions in[a, b], then the Taylor series expansions of G(x, t) is

G(x, t) =∞∑

k=0

gk(t)

k! (x − t)k (7)

where the functions {gk(t); k = 0, 1, 2, · · · } are generated recursively in terms of {p0, p1, . . . , pν}by the following algorithm:

gk = 0 for k = 0, 1, . . . , ν − 2,

gν−1 = 1,


gν = −pν−1,

gν+m = −m−1∑k=0

ν∑j=0

(mk

)gk+jp

(m−k)j −

ν−1∑j=0

gm+jpj , for m = 1, 2, 3, . . . ,

Proof Due to the initial conditions associated with the error equation (6), we have

g0 = g1 = g2 = · · · = gν−2 = 0 and gν−1 = 1.

To find the other gk’s we proceed as follows: a repeated differentiation of (7) with respect to x

gives

G′(x, t) =∞∑

k=1

kgk(t)

k! (x − t)k−1 =∞∑

k=0

gk+1(t)

k! (x − t)k

G′′(x, t) =

∞∑k=0

gk+2(t)

k! (x − t)k,

...

G(j)(x, t) =∞∑

k=0

gk+j (t)

k! (x − t)k, j = 1, 2, 3, · · · ,

Inserting these expansions into (6) we gain

φ(x, t) :=ν∑

j=0

pj (x)

[ ∞∑k=0

gk+j (t)

k! (x − t)k

]= 0 for all (x, t) ∈ [a, b] × [a, b]. (8)

In particular, for x = t , we have φ(t, t) = 0, and therefore (8) reduces to

ν∑j=0

pj (t)gj (t) = 0,

which leads to

pν−1(t)gν−1(t) + pν(t)gν(t) = pν−1(t) + gν(t) = 0,

giving

gν(t) = −pν−1(t).

For gν+1, gν+2, . . ., we differentiate φ(x, t) repeatedly, and then use the fact that φ(x, t), and itsderivatives, are identically zero:

dm

dxm[φ(x, t)] = φ(m)(x, t) =

ν∑j=0

∞∑k=0

gk+j (t)dm

dxm

[1

k!pj (x)(x − t)k]

,

and in particular, when x = t ,

φ(m)(t, t) =ν∑

j=0

∞∑k=0

gk+j (t)dm

dxm

[1

k!pj (x)(x − t)k]

x=t

. (9)


Noting that the higher-order derivative of product is

dm

dxm[f (x)g(x)] =

m∑i=0

(m

i

)f (m−i)(x)g(i)(x) (10)

we can write, for a generic p(x),

dm

dxm

[1

k!p(x)(x − t)k]

=m∑

i=0

(m

i

)p(m−i)(x)

(x − t)k−i

(k − i)! ,

which, at x = t , gives

dm

dxm

[1

k!p(x)(x − t)k]

x=t

=

⎧⎪⎨⎪⎩(

m

k

)p(m−k)(t) if k ≤ m,

0 otherwise.

Due to this, (9) is written as

φ(m)(t, t) =ν∑

j=0

∞∑k=0

(m

k

)gk+j (t)p

(m−k)j (t)

=ν∑

j=0

m∑k=0

(m

k

)gk+jp

(m−k)j

=m∑

k=0

ν∑j=0

(m

k

)gk+jp

(m−k)j

=m−1∑k=0

ν∑j=0

(m

k

)gk+jp

(m−k)j +

ν∑j=0

(m

m

)gm+jpj

=m−1∑k=0

ν∑j=0

(m

k

)gk+jp

(m−k)j +

ν−1∑j=0

gm+jpj + gm+νpν.

Solving φ(m)(t, t) = 0 for gν+m(t) we find that

gν+m(t) = −m−1∑k=0

ν∑j=0

(m

k

)gk+jp

(m−k)j −

ν−1∑j=0

gm+jpj , (m = 1, 2, . . .).

This completes the proof of the proposition. �

3. Bounds and estimates for the local error

The local error is the error of the computed solution after one step. This section is concerned withderiving bounds for the local error at any point x ∈ [a, b] in terms of the values of e(j)(a), the errorat the left end a. This is achieved by combining Proposition 2.1 with the following proposition inwhich the exact error is given in terms of Green’s function:


Proposition 3.1 If {pk(x); k ≥ 0} ⊂ C∞[a, b], then for all x ∈ [a, b], and for all i =0, 1, 2, . . . , ν − 1, we have

e(i)(x) =∫ x

a

G(i)(x, t)F (t)dt +ν−1∑j=0

[(x − a)j−i

(j − i)! −∫ x

a

G(i)(x, t)p∗j (t)dt

]e(j)(a), (11)

where

p∗j (x) =

j∑i=0

(x − a)j−ipi(x)

(j − i)! .

with the convention that 1/j ! = 0 whenever j is negative.

Proof Let

e∗(x) = e(x) −ν−1∑i=0

e(i)(a)

i! (x − a)i .

Rewriting (5) in terms of e∗ we get

e∗(ν)(x) +ν−1∑j=0

pj (x)e∗(j)(x) = F ∗(x),

where

F ∗(x) := F(x) −ν−1∑j=0

ν−1∑i=j

(x − a)i−jpj (x)

(i − j)! e(i)(a).

Further, the first ν derivatives of e∗(x) are

e∗(k)(x) = e(k)(x) −ν−1∑i=k

e(i)(a)

(i − k)! (x − a)i−k, (k = 1, 2, . . . , ν − 1),

e∗(ν)(x) = e(ν)(x).

For all k = 0, 1, 2, . . . , ν − 1, we have

e∗(k)(a) = 0.

Thus, e∗(x) satisfies the initial value problem

e∗(ν)(x) +ν−1∑j=0

pj (x)e∗(j)(x) = F ∗(x),

e∗(k)(a) = 0, k = 0, 1, 2, . . . , ν − 1.

of which the exact solution is

e∗(x) =∫ x

x0

G(x, t)F ∗(t)dt.

Noting that∑m

j=0

∑mi=j Aij = ∑m

j=0

∑j

i=0 Aji , F ∗(x) can be written as

F ∗(x) = F(x) −ν−1∑j=0

p∗j (x)e∗(j)

(a) where p∗j (x) =

j∑i=0

(x − a)j−ipi(x)

(j − i)! ,


and therefore

e∗(x) =∫ x

a

G(x, t)

[F(t) −

ν−1∑j=0

p∗j (t)e

(j)(a)

]dt

=∫ x

a

G(x, t)F (t)dt −N−1∑j=0

e(j)(a)

∫ x

a

G(x, t)p∗j (t)dt,

from which (11) follows immediately. �

We now proceed to derive error bounds from (11). For any positive integers n and i, letG(i)

n (x, t) designate the nth partial sum of G(i)(x, t), and let R(i)n (x, t) stand for the nth residual

that corresponds to G(i)n (x, t); that is

G(i)n (x, t) :=

n∑k=0

gk+i (t)

k! (x − t)k and R(i)n (x, t) :=

∑k>n

gk+i (t)

k! (x − t)k.

Let

�(i)jn (x, a) := (x − a)j−i

(j − i)! −∫ x

a

G(i)n (x, t)p∗

j (t)dt.

We establish now the main result of this paper:

Theorem 3.2 Let ρ < min(1, 1/b − a) and let n ≥ 1. Then there exists a constant γ such that thelocal error committed by the coefficients perturbation methods satisfies the following enclosures:∣∣e(i)(x) − e(i)(x; n)

∣∣ ≤ w(i)(x; n); (i = 0, 1, . . . , ν − 1), for all x ∈ [a, b], (12)

where

e(i)(x; n) =∫ x

a

G(i)n (x, t)F (t)dt +

ν−1∑j=0

�(i)jn (x, a)e(j)(a), (13)

w(i)(x; n) = γ

∫ x

a

φ(i)n,ρ(x, t)

∣∣∣∣F(t) −ν−1∑j=0

p∗j (t)e

(j)(a)

∣∣∣∣dt, (14)

φn,ρ(x, t) = [ρ(x − t)]n+1

1 − ρ(x − t).

We say then that e(x; n) := (e(x, n), e′(x, n), . . . , eν−1(x, n)) is the nth estimate (vector) ofe(x) := (e(x), e′(x), . . ., eν−1(x)) with deviation w(x; n) := (w(x, n), w′(x, n), . . ., wν−1(x, n)).

Proof Subtracting (11) from (13) and taking absolute values, we find that:

|e(i)(x) − e(i)(x; n)| ≤∣∣∣∣∣∣∫ x

a

R(i)n (x, t)F (t)dt −

ν−1∑j=0

[∫ x

a

R(i)n (x, t)p∗

j (t)

]e(j)(a)

∣∣∣∣∣∣≤∫ x

a

∣∣R(i)n (x, t)

∣∣ ∣∣∣∣∣∣F(t) −ν−1∑j=0

p∗j (t)e

(j)(a)

∣∣∣∣∣∣ dt. (15)


The proof of Theorem 3.2 follows immediately once we show that for all i ≥ 0 and for all(x, t) ∈ [a, b] × [a, b] the following inequality holds:∣∣R(i)

n (x, t)∣∣ ≤ γφ(i)

n,ρ(x, t). (16)

For this, we need to apply Cauchy’s inequalities (see Davis [2, Chap. 1]): in view of the assump-tion that ρ < min(1, 1/b − a) and n ≥ 1, the application of Cauchy’s inequalities to G(i)(x, t) asa real-analytic function of x, implies that there exists a positive real number γ such that∥∥∥gk

k!∥∥∥ ≤ γρk, for all k > n

where {gk(t); k ≥ 0} are introduced in Proposition 2.1 and ‖u‖ := sup{|u(t)| : t ∈ a, b]}. Then,owing to the fact that ρ(x − t) ≤ ρ(b − a) < 1, we have

|Rn(x, t)| ≤∑k>n

∥∥∥gk

k!∥∥∥ (x − t)k ≤

∑k>n

γρk(x − t)k ≤ γ∑k>n

[ρ(x − t)]k

≤ γ [ρ(x − t)]n+1∞∑

k=0

[ρ(x − t)]k ≤ γ

( [ρ(x − t)]n+1

1 − ρ(x − t)

)≡ γφn,ρ(x, t).

This shows that (16) holds for i = 0. For arbitrary i,

R(i)n (x, t) =

∑k>n

gk(t)

k!di

dxi(x − t)k,

and therefore using the same arguments we arrive at:

∣∣R(i)n (x, t)

∣∣ ≤∑k>n

∥∥∥gk

k!∥∥∥ di

dxi(x − t)k ≤

∑k>n

γρk di

dxi(x − t)k ≤

∑k>n

γdi

dxi[ρk(x − t)k]

≤ γdi

dxi

[∑k>n

ρk(x − t)k

]≤ γ

di

dxi[φn,ρ(x, t)] = γφ(i)

n,ρ(x, t),

where we have used the well-known fact that the term-by-term differentiation of the infinite powerseries is possible because x − t ≤ b − a < 1. Thus (16) holds for all i, and consequently the proofof Theorem 3.2 follows if we use (16) in (15). �

It is clearly seen that the computation of e(x; n) and w(x; n) by means of algorithm (12)–(14)requires the exact initial error e(a). The next result assures that algorithm (12)–(14) remains validif e(a) is replaced by bounds instead of exact values.

Theorem 3.3 Suppose that the assumptions of Theorem 3.2 hold true. Suppose further that thereexist 2ν real numbers {�±

i ; i = 0, 1, . . . , ν − 1} such that

�−i ≤ e(i)(a) ≤ �+

i , i = 0, 1, 2, . . . , ν − 1. (17)


Then, for n ≥ 1 and for all x ∈ [a, b], we have

E(i)n [x; �si ] − W (i)

n [x; �] ≤ e(i)(x) ≤ E(i)n [x; �si ] + W (i)

n [x; �], (18)

where

E(i)n [x; �si ] =

∫ x

a


ν−1∑j=0

�(i)jn (x, a)�

sij

j , (19)

E(i)n [x; �si ] =

∫ x

a


ν−1∑j=0

�(i)jn (x, a)�

sij

j , (20)

W (i)n [x; �] = γ

∫ x

a

φ(i)n,ρ(x, t)

⎧⎨⎩|F(t)| +ν−1∑j=0

|p∗j (t)| max{|�+

j |, |�−j |}⎫⎬⎭ dt, (21)

and

sij ≡ si

j (x) := sign[�(i)jn (x, a)] and si

j = −sij .

Proof Let us assume without loss of generality that each pair (�−j , �+

j ) is either positive or

negative. Fix x ∈ [a, b] and let Ti = {j ; 0 ≤ j ≤ ν − 1 such that ; �(i)jn (x, a) > 0}. Multiplying

(17) by �(i)jn (x, a) we get:

�(i)jn (x, a)�−

j ≤ �(i)jn (x, a)e(j)(a) ≤ �

(i)jn (x, a)�+

j ifj ∈ Ti

�(i)jn (x, a)�+

j ≤ �(i)jn (x, a)e(j)(a) ≤ �

(i)jn (x, a)�−

j ifj /∈ Ti.

More compactly,

�(i)jn (x, a)�

sij

j ≤ �(i)jn (x, a)e(j)(a) ≤ �

(i)jn (x, a)�

sij

j , j = 0, 1, 2, . . . , ν − 1.

Summing up j = 0, 1, . . . , ν − 1 we obtain

ν−1∑j=0

�(i)jn (x, a)�

sij

j ≤ν−1∑j=0

�(i)jn (x, a)e(j)(a) ≤

ν−1∑j=0

�(i)jn (x, a)�

sij

j , j = 0, 1, 2, . . . , ν − 1,

which implies that

E(i)n [x; �si ] ≤ E(i)

n (x) ≤ E(i)n [x; �si ]. (22)

Further, since φn,ρ(x, a) ≥ 0 (by definition), we have

w(i)(x; n) ≤ γ

∫ x

a

φ(i)n,ρ(x, t)

⎧⎨⎩|F(t)| +ν−1∑j=0

|p∗j (t)| |e(j)(a)|

⎫⎬⎭ dt

≤ γ

∫ x

a

φ(i)n,ρ(x, t)

⎧⎨⎩|F(t)| +ν−1∑j=0

|p∗j (t)| max{|�+

j |, |�−j |}⎫⎬⎭ dt ≡ W (i)

n [x; �].

This, combined with (13), gives∣∣e(i)(x) − e(i)(x; n)∣∣ ≤ w(i)(x; n) ≤ W (i)

n [x; �] for all x ∈ [a, b], i = 0, 1, 2, . . . , ν − 1,


which, in turn, leads to

e(i)(x; n) − W (i)n [x; �] ≤ e(i)(x) ≤ e(i)(x; n) + W (i)

n [x; �],

and consequently, by (22)

E(i)n [x; �si ] − W (i)

n [x; �] ≤ e(i)(x; n) − W (i)n [x; �] ≤ e(i)(x) ≤ e(i)(x; n)

+ W (i)n [x; �] ≤ E(i)

n [x; �si ] − W (i)n [x; �].

This completes the proof of Theorem 3.3. �

4. Bounds and estimates for the global error

Let us reconsider (1) and (2) and let us assume that the linear functionals in (2) define two-pointboundary conditions of the following type

L y(a)� + R y(b)� = α� (23)

where L and R are two ν × ν matrices, and where the following notation is used:

(i) For any set {v0, v1, . . . , vν−1} of ν reals, v := (v0, v1, . . . , vν−1) and v� := Transpose of v.

(ii) For any function f ∈ Cν[a, b], f(x) := (f (x), f ′(x), . . . , f (ν−1)(x)).

Let us consider a partition π := {x0 = a < x1 < x2 < · · · < xm = b} of [a, b] where m ≥ 1.We wish to approximate y(x) by means of a segmented approximation y = ∑m

k=1 χkyk , where χk

is the characteristic function of subinterval [xk−1, xk] and where yk belongs to Cν[xk−1, xk] andimplicitly defined by the differential equation

y(ν)k + pν−1,ky

(ν−1)k + · · · + p0,kyk = fk, x ∈ [xk−1, xk] (24)

yk(xk−1) = yk−1(xk−1)

Ly0(x0)� + R ym(xm)� = α� (25)

where for all k = 1, 2, . . . , m and i = 0, 1, · · · , ν − 1, pi,k(x) (respectively fk(x)) is anapproximation of pi(x) (respectively f (x)) in [xk−1, xk], and α ≈ α.

In this section we analyse the global error; that is the error of the computed solution afterseveral steps. This error after k steps, denoted by ek(x) := y(x) − yk(x), satisfies the followingdifferential problems

e(ν)k + pν−1,ke

(ν−1)k + · · · + p0,kek = Fk, x ∈ [xk−1, xk] (26)

ek(xk−1) = ek−1(xk−1)

L e0(x0)� + R em(xm)� = ε� := α� − α� (27)

where

Fk(x) := δfk(x) − δpν−1,ky(ν−1)k − δpν−2,ky

(ν−2)k − · · · − δp0,kyk.


Then, for each k = 1, 2, . . . , m, Formulae (13) and (14) can be written as

e(i)k (x) =

∫ x

xk−1

G(i)n (x, t)Fk(t)dt +

ν−1∑j=0

�(i)jn (x, xk−1)e

(j)

k−1(xk−1), x ∈ [xk−1, xk] (28)

w(i)k (x) = γ

∫ x

xk−1

φ(i)n,ρ(x, t)

∣∣∣∣∣∣F(t) −ν−1∑j=0

p∗j (t)e

(j)

k−1(xk−1)

∣∣∣∣∣∣ dt, (29)

where we suppress indices n and set ek(x) ≡ ek(x; n) and wk(x) ≡ wk(x; n).To compute the nth error estimates in the first interval [x0, x1], apply (12)–(14), provided

that e(a), the error vector at x0 = a, is known. To estimate the error throughout the kth interval[xk−1, xk], k = 2, 3, . . . , m, we apply successively (28) and (29). But, it is clear that if the startingvalue e(a) is not available, one cannot proceed implementing Equation (28). Next we explain thatEquation (28) itself can be used to generate such starting values.

From the boundary conditions (27) it follows that

L e1(x0)� + R em(xm)� ≈ L e1(x0)

� + R em(xm)� = ε�. (30)

giving a system of ν linear algebraic equations with 2ν unknowns. In Corollary 4.1 below weobtain the other ν algebraic equations. To formulate this corollary, we need to use a matricialnotation: for i = 0, 1, . . . , ν − 1 and k = 1, 2, . . . , m, let

f(i)k :=

∫ xk

xk−1

G(i)n (xk, t)Fk(t)dt ψ

(i)jk := �

(i)jn (xk, xk−1).

fk = [fk f ′

k . . . f(ν−1)k

]�

Mk :=

⎡⎢⎢⎢⎣ψ0k ψ1k · · · ψν−1,k

ψ ′0k ψ ′

1k · · · ψ ′ν−1,k

... · · · . . . · · ·ψ

(ν−1)0k ψ

(ν−1)1k · · · ψ

(ν−1)ν−1,k

⎤⎥⎥⎥⎦Corollary 4.1 (i) The global error estimates at points {xk; k = 1, 2, . . . , m} are given by the

formula:

ek(xk)� = c�

k + Dk e1(x0)�, k = 1, 2, . . . , m (31)

where

c�k = f�

k +k−2∑j=0

[j∏

i=0

Mk−i

]f�k−j−1 and Dk =

k−1∏i=0

Mk−i

(ii) The estimates of the initial errors are given by the 2ν × 2ν algebraic system[Dm −Iν

L R

] [e1(x0)

�em(xm)�

]=[

c�m

ε�

]where Iν is the identity matrix.

Proof With the new notation, (28) can be written as

e(i)k (xk) = f

(i)k +

ν−1∑j=0

�(i)jk e

(j)

k−1(xk−1), i = 0, 1, 2, . . . , ν − 1, k = 1, 2, . . . , m.


In matricial notation the latter becomes

ek(xk) = fk + Mk ek−1(xk−1), k = 1, 2, . . . , m.

Iterating k times we obtain

ek(xk)� = f�k + Mk ek−1(xk−1)

�

= f�k + Mk[f�k−1 + Mk−1ek−2(xk−2)�]

= · · ·

= f�k +k−2∑j=0

[j∏

i=0

Mk−i

]f�k−j−1 +

[k−1∏i=0

Mk−i

]e1(x0)

�

= c�k + Dk e1(x0)

�.

and this gives part (i). To obtain part (ii) combine (30) with (31) and set k = m. �

5. Application to the Tau method

Let us demonstrate the results of the previous sections on a special form of Coefficients Pertur-bation Methods called the Tau method (Ortiz [10]). The main features of the TM are recallednow.

Reconsider problem (1) and (2) and let us assume that the coefficients of the differential operatorD, {pk(x); k = 0, 1, 2, . . . , ν − 1}, are polynomials written as:

pk(x) =k∑

j=0

pkjxj , pkk = 0, k := deg[pk].

The basic idea of TM is to perturb the right-hand side of Equation (1) by adding a perturbationH(x) so that the resulting initial value problem,

DuN(x) = u(ν)N + pν−1(x)u

(ν−1)N + · · · + p0(x)uN = f (x) + H(x), x ∈ [a, b] (32)

Bk[uN ] = αk, k = 0, 1, 2, . . . , ν − 1, (33)

has an exact polynomial solution uN(x), called Tau approximation. In this method, uN is expressedin terms of a special polynomial basis, uniquely associated with the given differential operatorD, called canonical polynomials basis.

5.1 Canonical polynomials

For all n ∈ N, we call Qn a canonical function of D if D[Qn] = xn. From Ortiz [10,11], the Qn’sare formally generated by the recursion:

Qn+h = 1

n(n − 1) · · · (n − k0 + 1)pk0,k0

⎡⎢⎣xn −ν∑

k=0k =k0

k∑j=0

pk,jn!(n − k)!Qn+j−k

⎤⎥⎦ , n = 0, 1, 2, . . .

(34)where h = k0 − k0, with k0 being the integer that maximizes the set {k − k; k = 0, 1, 2, . . . , ν}.That is max{k − k; k = 0, 1, 2, . . . , ν} = k0 − k0 = h. With formula (34) one can generate all the


Qn’s except the k0 canonical functions {Qi; i = 0, 1, 2, . . . , k0 − 1}. This observation impliesthat for all n ≥ 0 we can split each Qn into two parts and write it as:

Qn = Qn +k0−1∑i=0

rniQi,

where for all n ≥ 0

Qn+h = 1

n(n − 1) · · · (n − k0 + 1)pk0,k0

⎡⎢⎣xn −ν∑

k=0k =k0

k∑j=0

pk,jn!(n − k)!Qn+j−k

⎤⎥⎦ , (35)

rn+h,i = −1

n(n − 1) · · · (n − k0 + 1)pk0,k0

⎡⎢⎣ ν∑k=0k =k0

k∑j=0

pk,jn!(n − k)! rn+j−k,i

⎤⎥⎦ , (36)

and Qi = 0, rij = δj

i (Kronecker’s symbol) for i, j ∈ {0, 1, . . . , k0 − 1}. Clearly, the first part Qn

is a polynomial and it is called nth canonical polynomial of D.

5.2 Representation of Tau approximations

There are basically two steps in constructing any TM approximation. First, a perturbation H(x)

should be chosen. In practice, H(x) is a linear combination of a prescribed polynomial basis{Vk; k ∈ N} where

Vk(x) :=k∑

i=0

c(k)i xi, ck

k = 0; k ∈ N.

Since D has k0 undefined canonical polynomials, and uN is subject to ν supplementary condi-tions (33), H(x) must involve k0 + ν free parameters {τk; k = 0, 1, . . . , k0 + ν − 1} and may beexpressed as

H(x) ≡ HN(x) =k0+ν−1∑

k=0

τkVN+k, N ∈ N.

The values of {τk; k = 0, 1, . . . , k0 + ν − 1} are determined as explained below. With thischoice of H , and due to the linearity of D, uN is written as:

uN =d∑

k=0

fkQk +k0+ν−1∑

k=0

τk

⎡⎣N+k∑j=0

c(N+k)j Qj

⎤⎦=

d∑k=0

fkQk +k0+ν−1∑

k=0

τk

⎡⎣N+k∑j=0

c(N+k)j Qj

⎤⎦ (37)

+k0−1∑i=0

⎧⎨⎩d∑

k=0

fkrki +k0+ν−1∑

k=0

τk

⎡⎣N+k∑j=0

c(N+k)j rji

⎤⎦⎫⎬⎭ Qi . (38)


In order to obtain a pure polynomial solution uN , we force the coefficients of {Qi; i =0, 1, . . . , k0 − 1} in (38) to be zero. This leads to k0 algebraic equations:

k0+ν−1∑k=0

τk

⎡⎣N+k∑j=0

c(N+k)j rji

⎤⎦ = −d∑

k=0

fkrki, 0 ≤ i ≤ k0 − 1.

Further, conditions (33) imposed on (37) give ν algebraic equations:

k0+ν−1∑k=0

τk

⎡⎣N+k∑j=0

c(N+k)j Bi[Qj ]

⎤⎦ = αi −d∑

k=0

fkBi[Qk], i = 0, 1, 2, . . . , ν − 1.

Therefore, the free parameters {τ0, τ1, . . . , τk0+ν−1} satisfy the algebraic linear systems:

Pτ� = ρ� (39)

where

P = [Pik]0≤i,k≤k0+ν−1, τ = [τ0 τ1 · · · τk0+ν−1] and ρ = [ρ0 ρ1 · · · ρk0+ν−1],with

rj,k0+i = Bi[Qj ], i = 0, 1, . . . , ν − 1

ρi = αi−k0−

d∑k=0

fkrk,i , i = 0, 1, . . . , k0 + ν − 1

Pik :=N+k∑j=0

c(N+k)j rji; i, j = 0, 1, 2, . . . , k0 + ν − 1.

Thus the TM approximation uN is a polynomial of degree ≤ N expressed as:

uN(x) =d∑

k=0

fkQk +k0+ν−1∑

k=0

τk

⎡⎣N+k∑j=0

c(N+k)j Qj

⎤⎦ , x ∈ [a, b].

This approach of TM produces approximations on the whole interval [a, b]. If the TM is appliedto each subinterval [xk−1, xk], k = 1, 2, . . . , m, we obtain the segmented TM. This can be doneby solving (24) with fk(x) = f (x) + HN,k(x) where each HN,k(x) is of the form

HN,k(x) =k0+ν−1∑

i=0

τi,kVN+i,k, N ∈ N, x ∈ [xk−1, xk].

Then for each k = 1, 2, . . . , m, we compute a set of free parameters{τi,k; i = 0, 1, 2, . . . , k0 − 1}.

5.3 Error estimates for the Tau method

Based on the previous analysis, the TM can be interpreted as a coefficients perturbation methodwhere the perturbation occurs only in the right-hand side of the differential Equation (1), whereas


the differential operator coefficients {pk(x), k = 0, 1, . . . , ν}, as well as the supplementaryconditions (2), remain unperturbed. So we deduce from Theorem 3.2 (13):

Corollary 5.1 (i) The estimated local error of TM is given by

e(i)(x; n) =k0+ν−1∑

j=0

τj

[∫ x

a

G(i)n (x, t)VN+j (t)dt

], x ∈ [a, b].

(ii) The estimated global error of segmented TM is given by

e(i)k (x; n) =

k0+ν−1∑j=0

τj

[∫ x

xk−1

G(i)n (x, t)VN+j,k(t)dt

]

+ν−1∑j=0

�(i)jn (x, xk−1)e

(i)k−1(xk−1; n), x ∈ [xk−1, xk].

6. Numerical tests

We illustrate, through three examples, how to compute error estimates for the function and itsderivatives using the formulae established above.

Example 1 The accuracy of our estimates were tested on the following fourth-order initial valueproblem:

y iv(x) + (1 + 4x − 4x2)y′′ + (8 − 16x)y

′ − 8y = 0; x ∈ [0, 1] (40)

y(0) = y′(0) = 1; y

′′(0) = −1, y

′′′(0) = −5, (41)

the exact solution of which is y = exp(x − x2). We wish to approximate y(x) by the TM. In thisexample, the canonical polynomials {Qk; k ≥ 0} are generated by the recursive formula:

Qk = 1

−4k2 + 20k − 8

[xk − k(k − 1)(k − 2)(k − 3)Qk−4 − k(k − 1)Qk−2 − (4k2 + 4k)Qk−2

].

Clearly, all the Qk’s are known and therefore the Tau problem associated with Equation (40) and(41) takes the form

uivN + (1 + 4x − 4x2)u

′′N + (8 − 16x)u

′N − 8uN =

3∑j=0

τjT∗N+j (x); x ∈ [0, 1]

uN(0) = u′N(0) = 1; u

′′N(0) = −1, u

′′′N(0) = −5,

where T ∗k (x) stands for the shifted Chebyshev polynomial of degree k and {τ0, τ1, τ2, τ3} are free

parameters adjusted in such a way that uN(x) is an exact polynomial.


We solved this problem using Mathematica and computed the Tau approximation u6(x), (thatis N = 6), along with the Tau parameters {τ0, τ1, τ2, τ3}. We obtained:

u6(x) = 1 + x − x2

2− 5 x3

6+ 2441180769686772776 x4

57381999990135669435+ 94471434329677199056 x5

286909999950678347175

+ 8575707120575636432 x6

95636666650226115725− 49043840202022798048 x7

286909999950678347175

+ 4086982527675116032 x8

95636666650226115725− 94146237602816 x9

286909999950678347175

τ0 = 493313887821450187

22952799996054267774= 0.02149, τ1 = 34913244898857

306037333280723570320

= 1.14082E − 7

τ2 = 8979875921598732

19127333330045223145= −4.69479E − 4, τ3 = 1011336536749

918111999842170710960

= 1.101539395E − 9

The error estimates along with the corresponding maximum deviations {e(i)(x; 20), w(i)(x; 20),

i = 0, 1, 2, 3} are computed by means of (13) and (14) and Corollary 5.1 with δf =−∑3

j=0 τjT∗

6+j (x) and ρ = 0.7. These estimates compared to the exact errors at some selectedpoints of [0, 1] are displayed in Table 1. One can appreciate the accuracy of estimates and thesharpness of deviations.

Table 1. Approximation of IVP (40) and (41) by the TM: exact and estimated values of the functionsand derivatives errors and their deviations are obtained using Equations (13) and (14).

i e(i)(0.3) e(i)(0.4) e(i)(0.5)

e(i)(0.3, 20) e(i)(0.4, 20) e(i)(0.5, 20)

w(i)(0.3, 20) w(i)(0.4, 20) w(i)(0.5, 20)

0 2.54488515367019418E-6 6.418695272408658E-6 1.08532603188851E-52.54488515367019416E-6 6.418695272408652E-6 1.08532603188847E-5

1.305E-24 8.863E-22 1.388E-191 3.12493631504767337E-5 4.380563457646525E-5 4.35805453205367E-5

3.12493631504767338E-5 4.380563457646530E-5 4.35805453205385E-51.372E-22 7.084E-20 8.968E-18

2 1.910734291420456383E-4 5.01709131357965E-5 −2.5619480647254E-51.910734291420456380E-4 5.01709131357964E-5 −2.5619480647259E-5

1.379E-20 5.4195E-18 5.547E-163 −9.03484476657230565E-4 −1.4827153622330555E-3 2.161396693687E-4

−9.03484476657230563E-4 −1.4827153622330546E-3 2.161396693688E-41.325E-18 3.961E-16 3.279E-14

i e(i)(0.6) e(i)(0.8) e(i)(1)

e(i)(0.6, 20) e(i)(0.8, 20) e(i)(1, 20)

w(i)(0.6, 20) w(i)(0.8, 20) w(i)(1, 20)

0 1.52069540575555E-5 2.856204778E-5 5.2956938E-51.52069540575525E-5 2.856204779E-5 5.2956940E-5

8.62010-18 5.821E-15 9.725E-131 4.552202352187E-5 9.60401984E-5 1.451074842E-4

4.552202352180E-5 9.60401982E-5 1.451074540E-44.689E-16 2.431E-13 3.384E-11

2 9.36065554901E-5 3.141635406E-4 2.503519E-49.36065554903E-5 3.141635411E-4 2.503520E-4

2.444E-14 9.754E-12 1.140E-93 1.921278679082E-3 −6.40773089E-4 6.733406E-4

1.921278679085E-3 −6.40773088E-4 6.733407E-41.219E-12 3.759E-10 3.723E-8


Example 2 Our second example is concerned with the boundary-value problem

y iv(x) + 9x2

8πy

′′′ = 0, x ∈ [0, π ] (42)

y(0) = y′(0) = 0, y

′′′(0) = −9/8, y

′(π) = 0. (43)

which arises in a linearized treatment of the boundary-layer equations for laminar viscous flow.The third-order derivative of y is known exactly: y ′′′(x) = −9/8 exp(−3x3/8π). To solve thisproblem we applied the segmented TM on four equal subintervals (m = 4) using perturbations ofthe form

(τ0k + τ1kx + τ2kx2)P6,k(x),

where Pj,k stands for Legendre polynomial of degree j shifted to subinterval [xk−1, xk], wherexk = kπ/4, k = 0, 1, 2, 3, 4, and {τ0k, τ1k, τ2k} are free parameters adjusted in such a way thatthe Tau approximant is an exact polynomial on each [xk−1, xk].

Having found the segmented Tau approximants, we wish to compute the error estimates through(28) with n = 10: in this example, e(0) = e

′(0) = e

′′′(0) = 0 but e

′′(0) is not known. So it is

necessary to estimate e′′(0) in advance. For this, we applied Corollary 4.1 and found that e

′′(0) ≈

−1.140560E-11. Using this value in (28) we calculated the estimated and exact errors for thethird derivative at the partition points and at some selected points. The numerical results are listedin Table 2.

Example 3 Finally, let us consider a BVP with nonpolynomial coefficients:

y iv(x) + 6

(1 + x)3y

′ = 0, x ∈ [0, 1], (44)

y(0) = 0, y′(0) = 1, y

′′(0) = −1, y(1) = ln 2, (45)

whose the exact solution is y = ln(1 + x). Let {0 = x0 < x1 < · · · < xm = 1} be a uniform par-tition of [0, 1], and let us approximate p(x) := 6/(1 + x)3 on each subinterval [xk−1, xk] by

Table 2. Approximation of BVP (42) and (43) by the segmented TM: [0, π ] is partitioned into four equal subintervals.Exact and estimated errors in y

′′′are obtained using (28). Estimated errors of y are also listed.

Estimation error in y′′′

Approximate y(x)

xi Exact error in y′′′

Estimation error in y(x)

π

105.264093174431E-8 0.37618380846

π

2−4.17E-12 0.829780381067

5.264093174439E-8 −1.94E-10 −4.14E-12 −1.989E-11π

55.339984556E-8 0.64194755212

3π

5−2.88868333E-7 0.7312071849

5.339984531E-8 −1.168E-10 −2.88868315E-7 1.071E-9π

41.093E-12 0.7342072644486

3π

4−7.86E-11 0.49545413709

1.089E-12 −7.878E-12 −7.82E-11 −3.712E-113π

10−9.3363140E-8 0.80029604277

4π

5−1.923922E-7 0.4024493724

−9.3363143E-8 1.632E-10 −1.923920E-7 1.99E-102π

5−9.8246939E-8 0.85863972554

9π

10−3.116653E-7 0.2051512893

−9.8246942E-8 3.527E-10 −3.116654E-7 1.223E-9π 6.733E-10 0

6.746E-10 0


the chord joining the end points; that is pk(x) = (p(xk) − p(xk−1))(x − xk−1)/(xk − xk−1) +p(xk−1). Then we consider the following sequence of segmented Tau problems:

uivN,1 + p1(x)u

′N,1 = HN,1(x) x ∈ [x0, x1]

uN,1(0) = 0, u′N,1(0) = 1; u

′′N,1(0) = −1

uivN,k + pk(x)u

′N,k = HN,k(x) x ∈ [xk−1, xk]

uN,k(xk−1) = uN,k−1(xk−1), k = 2, 3, . . . , m − 1

uivN,m + pm(x)u

′N,m = HN,m(x) x ∈ [xm−1, xm]

uN,m(xm−1) = uN,m−1(xm−1), uN,m(1) = ln 2

using perturbations

HN,k(x) := (τ0k + τ1kx + τ2kx2 + τ3kx

3)PN,k(x), k = 1, 2, . . . , m

For m = 5, (number of steps), N = 6 (order of Tau perturbation) we obtained the followingsegmented Tau approximations:

u6,1 = x − 0.5x2 + 0.337537307221907 x3 − 0.249999989477165 x4 + 0.155323631060760 x5

− 0.0519774500857725 x6 + 0.0223141807358504 x7 − 0.00998459470962153 x8

+ 0.00305413873879217 x9

τ0,1 = 9.45797966406387497108259E − 8

u6,2 = 0.0000146331031990459 + 0.999629694515629 x − 0.496227816302932 x2

+ 0.318076182847629 x3 − 0.198151780912712 x4 + 0.0928169009595729 x5

− 0.0300906106441770 x6 + 0.0112929525406314 x7 − 0.00365753549675781 x8

+ 0.000666328168298515 x9

τ0,2 = 2.27175040892825534095330E − 8

u6,3 = 0.000205435230514429 + 0.997190065563841 x − 0.483596668673571 x2

+ 0.284543959005339 x3 − 0.150716597464354 x4 + 0.0589640940548238 x5

− 0.0178021274165942 x6 + 0.00565033911385786 x7 − 0.00140162478752145 x8

+ 0.000181868981915180 x9

τ0,3 = 7.20972763305847888025073E − 9

u6,4 = 0.000872317976875229 + 0.991452474498003 x − 0.463496485552888 x2

+ 0.248014001783112 x3 − 0.114295092543167 x4 + 0.0389694641210883 x5

− 0.0107531666396617 x6 + 0.00290768603411532 x7 − 0.000581525145586628 x8

+ 0.0000586275189184990 x9

τ0,4 = 2.78051372793862068962417E − 9


Table 3. Approximation of BVP (44) and (45) by the segmented TM: [0, 1]is partitioned into five equal subinterval. Exact and estimated errors in thefunction and its derivatives are obtained using (28).

e(xi , 8) e′(xi , 8) e

′′(xi , 8) e

′′′(xi , 8)

xi e(xi ) e′(xi ) e

′′(xi ) e

′′′(xi )

0 0 0 0 −2.555E-20 0 0 −2.522E-2

1

10−3.92E-6 −1.12E-4 −1.95E-3 −1.01E-2

−3.86E-6 −1.10E-4 −1.93E-3 −1.00E-21

5−2.58E-5 −3.29E-4 −2.146E-3 2.93E-3

−2.54E-5 −3.23E-4 −2.113E-3 2.78E-31

2−1.90E-4 −6.1E-4 −4.26E-4 1.579E-2

−1.88E-4 −6.0E-4 −4.24E-4 1.562E-23

5−2.44E-4 −4.26E-4 2.69E-3 1.777E-2

−2.41E-4 −4.24E-4 2.64E-3 1.771E-27

10−2.70E-4 −6.67E-5 4.51E-3 1.87E-2

−2.67E-4 −7.08E-5 4.45E-3 1.88E-24

5−2.51E-4 4.81E-4 6.437E-3 1.949E-2

−2.48E-4 4.70E-4 6.392E-3 1.974E-29

10−1.68E-4 1.22E-3 8.396E-3 1.98E-2

−1.66E-4 1.21E-3 8.383E-3 2.02E-21 4.628E-9 2.164E-3 1.040E-2 2.00E-2

0 2.149E-3 1.043E-2 2.06E-2

u6,5 = 0.00229016020992610 + 0.982227101781927 x − 0.438929823118906 x2

+ 0.213712720395026 x3 − 0.0873473761070088 x4 + 0.0265318114780622 x5

− 0.00665516498220164 x6 + 0.00155680800748385 x7 − 0.000260551224845670 x8

+ 0.0000214941204825831 x9

τ0,5 = 1.23041263900426751184282E − 9

Through Formula (28), we obtained error estimates at some selected points, with n = 8 andδfk = −Hk(x) − (6/(1 + x)3 − pk(x)). Table 3 exhibits the obtained numerical results.

7. Conclusion

We have developed formulae for computing estimates and bounds for the error committed bynumerical methods based on coefficients perturbations. Those bounds are (i) accurate and sharpeven when we choose a considerably large steplength, (ii) easy to compute using symbolic pack-ages such as Mathemtica and Maple and (iii) apply to discrete or finite difference schemes thatcan be simulated in terms of coefficients perturbation methods.

Acknowledgements

This work is supported by the Public Authority for Applied Education and Training in Kuwait, Project No. TS-06-07.


References

[1] G. Birkhoff and G. Rota, Ordinary Differential Equations, J. Wiley, New York, 1989.[2] J.P. Davis, Interpolation and Approximation, Dover Publications, New York, 1975.[3] M.K. El-Daou, Computable error bounds for coefficients perturbation methods, Computing 69 (2002), pp. 305–317.[4] C. Canuto, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, Berlin,

Heidelberg, New York, 2006.[5] W. Gander and J. Hrebicek, Solving Problems in Scientific Computing Using Maple and Matlab, Springer, Berlin,

New York, 2004.[6] L. Ixaru, Numerical Methods for Differential Equations and Applications, D. Reidel Publishing Co, Dordrecht, 1984.[7] ———, Coefficients perturbation methods for Schrödinger equation, J. Comput. Appl. Math. 125 (2000),

pp. 347–357.[8] V. Ledoux, M. Van Daele, and G. Vanden Berghe, CP methods of higher order for Sturm-Liouville and Schrödinger

equations, Comput. Phys. Commun. 162 (2004), pp. 151–165.[9] V. Ledoux et al., Solution of the Schrödinger equation by a high order perturbation method based on a linear

reference potential, Comput. Phys. Commun. 175 (2006), pp. 424–439.[10] E.L. Ortiz, The Tau Method, SIAM J. Numer. Anal. 6 (1969), pp. 480–492.[11] ———, Canonical polynomials in the Lanczos Tau Method, in Studies in Numerical Analysis, B.K.P. Scaife, eds,

Academic Press, New York, 1974, pp. 73–93.[12] S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, Addison Wesley, Redwood City,

California, 2000.

Documents

Coefficients perturbation methods for higher-order differential equations