Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol101.Issue1-2.1999:

Journal of Computational and Applied Mathematics 101 (1999) 189–202

Continuous numerical solutions of coupled mixed partial
dierential systems using Fer’s factorization
Sergio Blanes a; ∗ , Lucas Jodar b
a

Departament de Fsica Teorica, Universitat de Valencia, Dr. Moliner 50, 46100-Burjassot, Valencia, Spain
Departamento de Matematica Aplicada, Universidad Politecnica, Camino de Vera, 22 012-Valencia, Spain

b

Received 4 May 1998; received in revised form 18 September 1998

Abstract
In this paper continuous numerical solutions expressed in terms of matrix exponentials are constructed to approximate
time-dependent systems of the type ut − A(t)ux x − B(t)u = 0; 0 ¡ x ¡ p; t ¿ 0, u(0; t) = u(p; t) = 0; u(x; 0) = f (x); 06
x6p. After truncation of an exact series solution, the numerical solution is constructed using Fer’s factorization. Given
 ¿ 0 and t0 ; t1 ; with 0 ¡ t0 ¡ t1 and D(t0 ; t1 ) = {(x; t); 06x6p; t0 6t6t1 } the error of the approximated solution with
c 1999 Elsevier
respect to the exact series solution is less than  uniformly in D(t0 ; t1 ). An algorithm is also included.

Science B.V. All rights reserved.
AMS classication: 65M15, 34A50, 35C10, 35A50
Keywords: Mixed time-dependent partial dierential systems; Accurate solution; A priori error bounds; Fer’s
factorization; Algorithm

1. Introduction
Systems of partial dierential equations are frequent in many dierent problems such as the study
of heat conduction and diusion problems [21], or in the analysis of pollutant migration through soil
modelling coupled thermoelastoplastic hydraulic response clays [22]. In the evaluation of coupled
microwave heating processes the constant coecient model often leads to misleading results due to
the complexity of the eld distribution within the over and the variation in dielectric properties of
material with temperature, moisture content, density and other properties [23, 15, Ch. 3].
In this paper we consider mixed problems for time-dependent systems of the type
ut (x; t) − A(t)ux x (x; t) − B(t)u(x; t) = 0;
∗

0 ¡ x ¡ p; t ¿ 0;

Corresponding author. Tel:. 34-6-3983148; fax: 34-6-3642345; e-mail: blanes@evalvx.uv.es.

c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 1 9 - 2

(1)

190

S. Blanes, L. Jodar / Journal of Computational and Applied Mathematics 101 (1999) 189–202

u(0; t) = u(p; t) = 0;
u(x; 0) = f (x);

t ¿ 0;

(2)

06x6p;

(3)

where the unknown u(x; t) and the right-hand side f (x) are vectors in Cr , and A(t); B(t) are continuous Cr×r valued functions such that
there exists a positive number % such that for all t¿0
and every eigenvalue z of (A(t) + AH (t))=2; z¿% ¿ 0

(4)

where AH (t) denotes the Hermitian conjugate of the matrix A(t). Problem (1)–(3) has been treated
in [25] for the case where A(t) = A and B(t) = B are constant matrices, and in [19] for the case
where B(t) = 0 and A(t) is an analytic matrix function.
The aim of this paper is not the comparison with respect to other discrete methods but the
construction of exact and continuous numerical solutions of problem (1)–(3) in terms of matrix
exponentials, with a prexed accuracy in a bounded subdomain. In spite of the expensive cost of the
numerical computation of matrix exponentials the proposed method has several possible important
advantages:
(i) It permits the determination of a priori error bounds for the constructed analytic-numerical
solutions, in terms of the available information of the problem.
(ii) The matrix coecient A(t) does not need to be an analytic function, but only continuous, see
[19].
(iii) With respect to discrete methods, the constructed approximation is dened simultaneously for

all the points (x; t) of the prexed subdomain, and not only at a discrete mesh of points.
The organization of the paper is as follows. Section 2 deals with a revisited version of the error
analysis developed in [4], adapted to the problem
V ′ (t) = [B(t) − 2 A(t)]V(t);

V(a0 ) = V0 ;

a0 6t6a1 ;

(5)

using Fer’s factorization to approximate the solution of (5) by matrix exponentials. In particular,
using the concept of logarithmic norm, error bounds given in [4] are improved. In Section 3 an
exact series solution of Problem (1)–(3), is constructed under hypothesis (4), using a separation
of variables technique. Given an admissible error  ¿ 0 and t1 ¿ t0 ¿ 0 we propose a truncation
strategy so that the error of the truncated series be less than  in
D(t0 ; t1 ) = {(x; t);

06x6p;

0 ¡ t0 6t6t1 } :

(6)

Section 4 deals with the construction of Fer’s approximations to each of the exact solutions of
vector problems of the type
"

Tn′ (t) = B(t) −



n
p

2

#

A(t) Tn (t);

Tn (0) = cn ; 16n6n0 ;

(7)

where
cn =

2
p

Z

0

p

f (x) sin



nx
d x;
p


(8)

S. Blanes, L. Jodar / Journal of Computational and Applied Mathematics 101 (1999) 189–202

191

is the sine Fourier series coecient of f (x) and n0 is the truncation index. Given  ¿ 0 we determine
the index m of Fer’s approximations Tn[m] (t), so that the error of the numerical approximation of
Problem (1)–(3), after replacing the exact solution Tn (t) of Problem (7)–(8) by its Fer’s approximation Tn[m] (t), be smaller than  uniformly in D(t0 ; t1 ), when the whole interval of integration is
split in subintervals. Section 5 provides the algorithm and an illustrative example.
Throughout this paper the set of all the eigenvalues of a matrix D in Cr×r is denoted by (D) and
the spectral radius of D, denoted by (D) is the maximum of the set {|z|; z ∈ (D)}. We denote
by kDk the 2-norm of D, [13, p. 56; 16, p. 295]:
kDk = sup

y6=0

kDyk2
= max{|!|1=2 ; ! ∈ (DH D)};
kyk2

where for a vector y ∈ Cr ; kyk2 = (yH y)1=2 is the usual euclidean norm of y. In accordance with
[12, p. 110; 14, p. 59], the logarithmic norm (D) is dened by
(D) =

kI + hDk − 1
;
h→0; h ¿ 0
h
lim

and satises
|(D)|6kDk;

(9)

(D) = (D) for ¿0;
(

(D) = max !; ! ∈ 

(10)

D + DH

2

!)

:

(11)

2. Fer approximation to V ′ = [B(t) − 2 A(t)]V
The interest in recovering qualitative features of the exact solution of time dependent matrix

dierential equations, and in particular the fundamental solution of a time-dependent linear system,
has claimed the attention of many authors who have developed methods based on Lie groups.
Although Lie groups are not mentioned in [17], the well-known method of the iterated commutators is
a rediscovering by Iserles of Fer’s approach, and is now being intensively investigated as a powerful
tool to treat both linear and nonlinear dierential equations on Lie groups and other manifolds
[3, 27]. Application of Fer’s approximation as a symplectic integrator may be found in [6], and may
be found as a tool for solving certain initial value problems for linear partial dierential equations
in [7]. Apart from [11] other recent relevant works related to Fer’s method are [4, 8, 20].
In this section, starting from recent results of [4], we introduce some improvements in the error
analysis of Fer’s method addressed to nd a connection between the order of approximation and a
prexed accuracy.
Fer’s algorithm approximates the solution V(t) of the matrix initial problem
V ′ (t) = S(t)V(t)

V (0) = I;

(12)

by a product of matrix exponentials. Convergence of the approximations appear already in Fer’s
original work [11]. Our starting point is Section 3 of [4] where, using a slightly dierent argument,

192

S. Blanes, L. Jodar / Journal of Computational and Applied Mathematics 101 (1999) 189–202

the convergence region is enlarged and upper error bounds are improved. The aim of this section is
to improve some of the results given in [4] using the concept of the logarithmic norm of a matrix
and introducing a dierent iterative mapping. We recall that Fer’s expansion is generated by the
following recursive scheme:
V = eF1 eF2 · · · eFm Vm ;

Vm′ = Sm (t)Vm ;

Vm (0) = I; m = 1; 2; 3 : : : ;

(13)

where Fm (t); Sm (t) are given by
Fm+1 (t) =

Z

t

Sm (s) d s;

S0 (t) = S(t); m = 0; 1; 2 : : :

0

Sm+1 =

Z

1

dx

Z

x

d u e−(1−u)Fm+1 [Sm ; Fm+1 ] e(1−u)Fm+1

(14)

0

0

where [P; Q] = PQ − QP.
When after m steps we impose Vm (t) = I we are left with an approximation V [m] (t) to the exact
solution V(t). Let us consider that the matrix S(t) is bounded and k S(t) k is a piecewise continuous
function such that k S(t) k 6k(t) ≡ k (0) (t). Fer’s algorithm, Eqs. (13)–(14), provides thenR a recursive
t
relation among corresponding bounds k (m) (t) for kSm (t)k. Let us denote K (m) (t; 0) ≡ 0 k (m) (s)ds.
Using the mapping
M (x) =

Z

0

x

1 − e2s (1 − 2s)
d s;
2s

presented in [4], we can take as bounds (for kFm+1 (t)k), K (m+1) (t; 0) = M (K (m) (t; 0)) and the convergence is assured if K (m) (t; 0) ¡  with  = 0:8604065. But, considering that for 0 ¡ x ¡ 
M (x) ¡ x2 = ¡ x;

(15)

we can use the mapping
K (m+1) (t; 0) = G K (m) (t; 0) ;



G(x) = x2 =:

(16)

Then limm→∞ K (m) (t; 0) = 0 if 0 is a stable xed point for the iteration and K (0) is within its basin
of attraction. It is clear that x = 0 is a stable xed point of x = G(x) and x =  is the next, unstable,
xed point [10]. Thus we have still assured the convergence of Fer’s expansion for values of time
t such that
Z

0

t

kS(s)k d s6K (0) (t; 0) ¡ 0:8604065:

(17)

Note that if K (m) (t; 0) ¡  then K (m+1) (t; 0) ¡ K (m) (t; 0). Expression (16) is simpler than using the
mapping M (x). Further it has the same convergence domain and provides the iteration
m

K (m+1) = (K (0) =)2 :
The next result provides a priori error bounds of the theoretical solution of the matrix problem
V ′ (t) = [B(t) − 2 A(t)]V(t);

V(a0 ) = V0 ; a0 6t6a1 ;  ¿ 0:

(18)

S. Blanes, L. Jodar / Journal of Computational and Applied Mathematics 101 (1999) 189–202

193

Theorem 1. Under hypothesis (4); let (a0 ; a1 ); (a0 ; a1 ) be positive numbers dened by
(

(a0 ; a1 ) = min z ∈ 
(

A(t) + AH (t)

2

B(t) + BH (t)

(a0 ; a1 )¿ max 

2

!

!

;

)

;

(19)

a0 6t6a1 ;
)

a0 6t6a1 :

(20)

a0 6t6a1 :

(21)

Then the solution V(t) of (18) satises
kV(t)k6kV0 ke(t−a0 )[(a0 ; a1 )− (a0 ; a1 )] ;
2

Proof. By [12, p. 114] the solution V(t) of (18) satises
kV(t)k6kV0 k exp

t

Z

a0



 B(s) − 2 A(s) d s ;



By (11) one gets
(

 B(s) − 2 A(s) = max z ∈ 



B(t) + BH (t)

2

(22)

a0 6t6a1 :

−

H
2 A(t) + A (t)

2

!)

(23)

;

and by [2, p. 246] it follows that


B(t) + BH (t)

Gi (s) =

2

−

2 A(t)

+ AH (t)
2




! ∈ C; ! + 2 i


(

!

⊂

r
[

Gi (s);

16i6r;

(24)

i=1

!
A(t) + AH (t) 
 6

2

B(t) + BH (t)

2

!)

(25)

;

where i A(t) + AH (t)=2 is the ith eigenvalue of the matrix (A(t) + AH (t))=2. By (22)–(25) and
(19)–(20), it follows that



2




 B(s) −  A(s) 6 

B(t) + BH (t)

2

!

2

(

−  min z ∈ 

6 (a0 ; a1 ) − 2 (a0 ; a1 );

A(t) + AH (t)

2

!)

a0 6s6a1 :

(26)

Thus the result is established.
Now from [4], see also [3], if V(t) is the solution of
V ′ (t) = [B(t) − 2 A(t)]V(t);

V(a0 ) = V0 ; a0 6t6a1 = a0 + h;

and V [m] (t) is the Fer approximation of order m of V(t), for a0 6t6a1 one gets


V(t) − V [m] (t)
6 kV0 k K (m) (a1 ; a0 ) eK (0) (a1 ; a0 )+2K (m) (a1 ; a0 ) ;

(27)

194

S. Blanes, L. Jodar / Journal of Computational and Applied Mathematics 101 (1999) 189–202

but in the problem we are considering it is preferable to take a rened bound. From Fer’s algorithm
we have V = V [m] Vm , so V − V [m] = VDm with Dm = I − Vm−1 . If we consider that
Vm−1

then


′

= −Vm−1 Sm ;

Dm (t) = −

Z

(28)

t

Vm−1 (s)Sm (s) ds:

a0

(29)

Taking norms and considering kVm−1 (s)k 6 eK
kV(t) − V [m] (t)k 6 kVkeK

(m)

(a1 ; a0 )

(m)

(a1 ; a0 )

we have kDm (t)k 6 eK

K (m) (a1 ; a0 );

(m)

(a1 ; a0 )

K (m) (a1 ; a0 ), then
(30)

where
K (0) (s; a0 ) ¿

Z

s

a0

kB(t) − 2 A(t)k dt:

(31)

By (30) and Theorem 1 for a0 6 t 6 a1 one gets
2

kV(t) − V [m] (t)k 6 kV0 ke(t−a0 )[(a0 ; a1 )− (a0 ; a1 )] eK

(m)

(a1 ; a0 )

K (m) (a1 ; a0 ):

(32)

Remark 2. Once we approximate the solution of (12) by the Fer approximation V [m] (t) = eF1 (t) · · ·
eFm (t) , it is necessary to compute the matrix exponentials eFi (t) , where matrices Fi (t) are related by
(14). Numerous algorithms for computing matrix exponentials have been proposed, but most of them
are of dubious numerical quality, as is pointed in [24]. In accordance with [24], scaling and squaring
with Pade approximants and a careful implementation of Parlett’s Schur decomposition method, see
[13, Ch. 11], were found the less dubious of the nineteen methods scrutinized. A promising method
for computing matrix exponentials has been recently proposed in [9].

3. Exact and approximated theoretical solutions
The eigenfunction method suggests to seek a series solution of Problem (1)–(3) of the form
nx
u(x; t) =
Tn (t) sin
;
p
n¿1


X



(33)

where Tn (t) is the Cr -valued solution of the initial value Problem (7)–(8). The solution of the vector
problem can be written in the form
Tn (t) = Un (t)cn ;

(34)

where Un is the solution of the matrix initial value problem
"

Un′ (t) = B(t) −



n
p

2

#

A(t) Un (t);

Un (0) = I:

(35)

S. Blanes, L. Jodar / Journal of Computational and Applied Mathematics 101 (1999) 189–202

195

By the Riemann–Lebesgue lemma, there exists a constant M such that
kcn k 6 M;

n ¿ 1;

(36)

and by Theorem 1 and (34), solution of (7)–(8) satises
2

kTn (t)k 6 M et[(0; t1 )−(n=p) (0; t1 )]

(37)

t1 (0; t1 ) −t0 (n=p)2 (0; t1 )

e

6 Me

;

0 ¡ t0 6 t 6 t1 ;

kTn′ (t)k 6 (kB(t)k + (n=p)2 kA(t)k)kTn (t)k
6 b(t1 ) +



n
p

2

!

(38)
(39)

2

a(t1 ) M et1 (0; t1 ) e−t0 (n=p) (0; t1 ) ;

where
a(t1 ) = max{kA(t)k; 0 6 t 6 t1 };

b(t1 ) = max{kB(t)k; 0 6 t 6 t1 }:

(40)

Hence the series appearing taking termwise partial dierentiation in (33), once with respect to t,


X
nx
′
;
Tn (t) sin
p
n¿1
and twice with respect to x,
X n
n¿1

−

nx
Tn (t) cos
;
p
p


X  n 2
n¿1

p



nx
Tn (t) sin
;
p




are uniformly convergent in D(t0 ; t1 ) = {(x; t); 0 6 x 6 p; 0 ¡ t0 6 t 6 t1 }. By the derivation theorem of functional series [1, p. 402], one gets that u(x; t) is termwise partially dierentiable with
respect to the variable t and twice termwise partially dierentiable with respect to the variable x.
By (7)–(8) it follows that
ut (x; t) = B(t)u(x; t) + A(t)uxx (x; t);

0 6 x 6 p; t ¿ 0:

If each component fj of f = (f1 ; : : : ; fr )T satises one of the conditions:
(i) fj is locally of bounded variation at every point x in [0; p]:
(ii) fj admits one-side derivatives (fj′ )R (x) and (fj′ )L (x) at
every point x in [0; p];
and f is continuous in [0; p] with f (0) = f (p) = 0, then by [5, p. 57] it follows that


X
nx
f (x) =
cn sin
= u(x; 0); 0 6 x 6 p:
p
n¿1
Thus the following result has been established:

(41)

(42)

196

S. Blanes, L. Jodar / Journal of Computational and Applied Mathematics 101 (1999) 189–202

Theorem 3. Let A(t); B(t) continuous Cr×r -valued functions such that condition (4) is satised.
Let f (x) be continuous in [0; p] with f (0) = f (p)) = 0 and let each component fj of f satisfy one
of the conditions of (41). Then u(x; t) dened by (33) when Tn (t) satises (7)–(8), is a solution
of Problem (1)–(3).
Given  ¿ 0 we are interested, under hypotheses of Theorem 3, in the determination of an index
n0 so that



X

nx


T (t) sin
¡ ;


n¿n n
p
2
0

(x; t) ∈ D(t0 ; t1 ):

(43)

From (38), considering that
X

2

e−t0 (n=p) (0; t1 ) 6

Z

∞

n0

n ¿ n0

√

where erfc(t) = (2= )
that

R∞
t

2
n0 p
p
e−t0 (x=p) (0; t1 ) dx = √
t0 (0; t1 )
erfc
p
2 t0 (0; t1 )





2

e−x dx is the complementary error function, and from (43) it follows

n0 p
M et1 (0; t1 ) p
erfc
t0 (0; t1 ) :
kTn (t)k 6 √
p
2 t0 (0; t1 )
n ¿ n0




X

Hence, taking the rst positive integer n 0 such that
√


n p
 t0 (0; t1 )
t0 (0; t1 ) ¡
erfc
;
p
M et1 (0; t1 ) p

(44)

(45)

inequality (43) holds and




n0

X
nx



Tn (t) sin
¡ ;
u(x; t) −


p
2
n=1

(x; t) ∈ D(t0 ; t1 ):

(46)

Thus the following result has been established:
Corollary 4. Under hypothesis of Theorem 3, let t1 ¿ t0 ¿ 0;  ¿ 0 and let D(t0 ; t1 ) be dened by
(6). If u(x; t) is the exact series solution of Problem (1)–(3) given by Theorem 3, and n 0 is the
rst positive integer satisfying (45), then
n0
X

nx
u(x; t; n 0 ) =
Tn (t) sin
p
n=1




(47)

is an approximation satisfying (46).
4. Continuous numerical solution
Section 2 was concerned with the study of the local error, in the convergence domain, using
Fer’s approximation for the solution of Problem (7). If we are interested in the approximation of

S. Blanes, L. Jodar / Journal of Computational and Applied Mathematics 101 (1999) 189–202

197

Problem (7) using Fer’s algorithm in an interval [t0 ; t1 ] where Fer’s method is not convergent, then
we may split the interval in subintervals with guaranteed convergence. In this section we address
the following question: Given  ¿ 0; 0 ¡ t0 ¡ t1 and n 0 given by Corollary 4, how to determine
the order m of the Fer’s approximation Tn[m] (t) of the exact solution Tn (t) of Problem (7)–(8) such
that the error with respect to the exact solution be smaller than =(2n 0 ) when the whole interval of
integration is split in N subintervals and the algorithm is used in each interval, i.e. to look for m
such that

kTn (t) − Tn[m] (t)k ¡
; 0 ¡ t0 6 t 6 t1 ; 1 6 n 6 n 0 :
(48)
2n 0
Let h ¿ 0 and consider the partition 0 = h0 ¡ h1 ¡ · · · ¡ hN = t1 ; where hj = jh; 0 6 j 6 N and
Nh = t1 . If hj 6 t 6 hj+1 , then we can write
Tn (t) = Un (t; 0)cn ;
"

(49)

d
Un (t; hj ) = B(t) −
dt

Un (hj ; hj ) = I,



n
p

2

#

A(t) Un (t; hj );

(50)

hj 6 t 6 hj+1 ;

Un (t; 0) = Un (t; hi )Un (hi ; hi−1 ) · · · Un (h; 0);

(51)

hi 6 t 6 hi+1 :

Let us introduce the notation
Un (t; hj ) = Un; j (t);

0 6 j 6 N − 1;

[m]

Un[m] (t; hj ) = Un; j (t);

(52)

hj 6 t 6 hj+1 ;

where for simplicity we will write Un; j (hj+1 ) = Un; j (h); Un;[m]j (hj+1 ) = Un;[m]j (h). Thus for hi 6 t 6 hi+1 ;
0 6 i 6 N − 1 we can write
[m]

[m]

[m]

Un (t; 0) − Un[m] (t; 0) = Un; i (t)Un; i−1 (h) · · · Un; 0 (h) − Un; i (t)Un; i−1 (h) · · · Un; 0 (h)

= (Un; i (t) − Un;[m]i (t))Un; i−1 (h) · · · Un; 0 (h)

+Un;[m]i (t)(Un; i−1 (h) − Un;[m]i−1 (h))Un; i−2 (h) · · · Un; 0 (h)
+···

+Un;[m]i (t) · · · Un;[m]i−j+1 (h)(Un; i−j (h) − Un;[m]i−j (h))Un; i−j−1 (h) · · · Un; 0 (h)
+···

+Un;[m]i (t) · · · Un;[m]2 (h)(Un; 0 (h) − Un;[m]0 (h)):
Hence
kUn (t; 0) − Un[m] (t; 0)k 6

i
X
j=0

kUn;[m]i (t)k · · · kUn;[m]i−j+1 (h)kkUn; i−j (h) − Un;[m]i−j (h)k

× kUn; i−j−1 (h)k · · · kUn; 0 (h)k;

hi 6 t 6 hi+1 :

(53)

198

S. Blanes, L. Jodar / Journal of Computational and Applied Mathematics 101 (1999) 189–202

By (30), for hj 6 t 6 hj+1 it follows that
(m)

kUn;[m]j (t)k 6 kUn; j (t)keKn; j (t; hj ) ;

(54)

kUn; i−j (t) − Un;[m]i−j (t)k 6 kUn; i−j (t)kKn;(m)
i−j (t; hi−j )e

Kn;(m)
(t; hi−j )
i−j

:

(55)

By (54)–(55) one gets
kUn;[m]i (t)k · · · kUn;[m]i−j+1 (h)kkUn; i−j (h) − Un;[m]i−j (h)k

6 kUn; i (t)k · · · kUn; i−j+1 (h)kkUn; i−j (h)kKn;(m)
i−j (hi−j+1 ; hi−j )

(m)
(m)
× exp(Kn;(m)
i (t; hi ) + Kn; i−1 (hi ; hi−1 ) + · · · + Kn; i−j (hi−j+1 ; hi−j )):

(56)

Note that by Theorem 1, for 0 ¡ t0 6 t 6 t1 one gets
2

kUn; i (t)k · · · kUn; 0 (h)k 6 et1 (0; t1 )−t0 (n=p) (0; t1 ) :

(57)

By (53)–(56) and (57) it follows
2

kUn (t; 0) − Un[m] (t; 0)k 6 et1 (0; t1 )−t0 (n=p) (0; t1 )
×

i
X
j=0

(m)
(m)
(m)
Kn;(m)
i−j (hi−j+1 ; hi−j ) exp(Kn; i (t; hi ) + Kn; i−1 (hi ; hi−1 ) + · · · + Kn; i−j (hi−j+1 ; hi−j ))

hi 6 t 6 hi+1 ; N0 =

t0
6 i 6 N − 1:
h

(58)

Let 0 ¡  ¡ 1, and let n 0 be given by Corollary 4, then take h ¿ 0 and select the integer N such
that
N ¿ t1

a(t1 )(n 0 =p)2 + b(t1 )
;


t1
:
N

(59)

1 6 n 6 n0;

(60)

h=

Then taking
n =

a(t1 )(n=p)2 + b(t1 )
;
a(t1 )(n 0 =p)2 + b(t1 )

where n 6  and n 0 = , one can consider

"
#
 2
 2
Z t


n
n


(0)
A(s)
ds ¡ Kn; j (t; hj ) = h a(t1 )
+ b(t1 ) ¡ n ;
B(s) −

p
p
jh

jh 6 t 6 (j + 1)h; 0 6 j 6 N − 1; 1 6 n 6 n 0 :

(61)

By (60) and (61) it follows that
Kn;(0)j (t; hj ) 6 Kn;(0)j (hj+1 ; hj ) ¡ n ;
Kn;(m)
j (t; hj ) = 

Kn;(0)j (t; hj )


!2m

;

(62)
(63)

S. Blanes, L. Jodar / Journal of Computational and Applied Mathematics 101 (1999) 189–202

Kn;(0)j (hj+1 ; hj )
(m)
Kn;(m)
(t;
h
)
6
K
(h
;
h
)
=

j
j+1
j
j
n; j

m

n; m () = 2n ;

!2m

6 n; m ();

199

(64)

lim Kn;(m)
j (t; hj ) = 0:

(65)

m→∞

Under hypothesis (59), by (58), (65) it follows that
2

kUn (t; 0) − Un[m] (t; 0)k 6 et1 (0; t1 )−t0 (n=p) (0; t1 ) n; m () en; m ()

i
X

e jn; m() ;

j=0

e(i+1)n; m () − 1
en; m () − 1
ih 6 t 6 (i + 1)h; N0 6 i 6 N − 1:
2

= et1 (0; t1 )−t0 (n=p) (0; t1 ) n; m () en; m ()

(66)

Note that if  ¿ 0; x ¿ 0, by the mean value theorem ex − 1 = xes for some s ∈ ]0; x[. Hence,
using that e x − 1 ¿ x, one gets
ex − 1
6  es 6  ex ;
ex − 1

x ¿ 0;  ¿ 0:

(67)

By (66), (67) one gets
2

kUn (t; 0) − Un[m] (t; 0)k6et1 (0; t1 )−t0 (n=p) (0; t1 ) (i + 1)n; m () e(i+2)n; m () ;
and by (34) it follows that
2

kTn (t) − Tn[m] (t)k6kcn kNet1 (0; t1 )−t0 (n=p) (0; t1 ) n; m ()e(N +1)n; m () ;
0 ¡ t0 6t6t1 ;

16n6n0 :

(68)

Let n be the unique root of equation
2

xe

(N +1)x

et0 (n=p) (0; t1 )
;
=
2kc n kN et1 (0; t1 ) n0

16n6n0

(69)

and let mn be the rst positive integer m satisfying
m

2 ¿

ln( n )
ln(n )

(70)

;

then
n
2 ln(n ) ¡ ln
;

m





m

n; m () = n2 ¡ n ;

(71)

and
kTn (t) − Tn[m] (t)k6


;
2n0

06t6t1 ;

16n6n0 :

(72)

200

S. Blanes, L. Jodar / Journal of Computational and Applied Mathematics 101 (1999) 189–202

5. The algorithm and an example
We begin this section summarizing the algorithm proposed in Sections 3 and 4 for the construction
of a continuous numerical solution of Problem (1)–(3), such that under hypothesis (4) satises



n0

X
nx


[m]
Tn (t) sin
6;
u(x; t) −

p
n=1

(x; t) ∈ D(t0 ; t1 ):

(73)

STEP 1. Truncated theoretical approximation.
Given 0 ¡ t0 ¡ t1 ;  ¿ 0:
• Compute M given by (36) and (0; t1 ); (0; t1 ) given by (19) and (20).
• Take n0 as the rst positive integer n satisfying (45).
STEP 2. Construction of continuous numerical solution.
Given n0 ; let 0 ¡  ¡ 1 xed and  = 0:8604065:
• Compute a(t1 ); b(t1 ) given by (40).
• Select h ¿ 0 and an integer N such that Nh = t1 ; and satisfying (59).
• Compute the root n of Eq. (69) for 16n6n0 :
• Given n take the rst positive integer mn satisfying (70) for 16n6n0 :
• Compute Tn[m] (t)
for 16n6n0 :
Pn0
• u(x; t; n0 ; m) = n = 1 Tn[m] (t) sin(nx=p) is an approximate solution satisfying (73) uniformly
for (x; t) ∈ D(t0 ; t1 ):
Example. Let us consider Problem (1)–(3) in C2×2 where
A(t) =



1 t
e
4

1
1 −t
e
4

1



;

B(t) =

1
− 41 et

− 41 et
1

!

and f (x) any function in C2 satisfying the hypotheses of Theorem 3, such that ||cn || ¡ M with M = 1:
We will consider p = 1; 0 ¡ t0 6t6t1 with t0 = 15 and t1 = 1 and the error bound  = 10−3 . Then we
can choose
(0; 1) = 1 − 81 (e + e−1 );

(0; 1) = 1 + 18 (e + e−1 ):
The next step is to look for the minimum n0 satisfying (45). That happens for n0 = 3 where
√


t0 (0; 1)
n0  p
−6
−4
t0 (0; 1) = 2:9885 × 10 ¡ 1:5539 × 10 = 
:
erfc
p
M et1 (0;1)
Following the second step we choose  = 111 , and taking
a(t1 ) = b(t1 ) = 1 + 41 e
the number of steps to consider is
t1

b(t1 ) +



n0 
p



2

a(t1 )

= 1928: 8 ¡ 2000 = N

S. Blanes, L. Jodar / Journal of Computational and Applied Mathematics 101 (1999) 189–202

201

or equivalently h = 1=2000: The values of n are
1 = 1:100 × 10−2 ;

2 = 4:097 × 10−2 ;

3 =  = 111 :

Now we compute the root n of Eq. (69) for 16n63; giving us
1 = 7:009 × 10−8 ;

2 = 2:649 × 10−6 ;

3 = 4:577 × 10−4 :

Finally we take the rst positive integer m n satisfying (70) for 16n63: We obtain
2m1 ¿ 3:619;

2m2 ¿ 3:972;

2m3 ¿ 3:144

and so m1 = m2 = m3 = 2: To sum up, using the Fer’s factorization to second order for the three rst
terms of the series, the error of the approximate solution will be smaller than  = 10−3 : For  = 10−4
(with n0 = 3) and taking now  = 1=22:8 we can choose N = 4000. Following the same calculations
we nd also m1 = m2 = m3 = 2:
Remark 5. Note that previous algorithm involves a free parameter  with 0 ¡  ¡ 1 and its choice
is signicant. In accordance with (59) if  → 0 then h must tend to 0 also. This means that we need
to apply Fer’s approximation more times, in more subintervals in which the original interval [t0 ; t1 ]
is split. If  → 1, then we apply Fer’s approximation a minor number of times but, by (70), the
integer m proposed by the algorithm as the order of the Fer’s approximation to construct Tn[m] (t);
increases as well as  → 1. Taking into account the complexity of the nested integrals (13)–(14),
in general, we suggest to take  ¡ 12 . Of course the optimal value of  is also depending on the
interval [t0 ; t1 ] and the minimum of the positive eigenvalues of the real part of A(t), see Condition
(4).
Acknowledgements
S.B. has been supported by DGICYT under contract no. PB92-0820 and by EEC network no.
ERBCHRXCT940456. The work of L.J. is supported by DGICYT under contract no. PB96-1321C02-02 and Generalitat Valenciana under contract no. GV-C-CN-10057=96.
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